2. filter_functions package
2.1. Submodules
2.2. filter_functions.analytic module
This file provides functions for the analytical solutions to some of the dynamical decoupling sequences. Note that the filter functions given here differ by a factor of 1/omega**2 from those defined in this package due to different conventions. See for example [Cyw08]. Depending on the definition of the noise Hamiltonian one might also get different results. The functions here agree for
2.2.1. Functions
FID()
Free Induction Decay / Ramsey pulse
SE()
Spin Echo
PDD()
Periodic Dynamical Decoupling
CPMG()
Carr-Purcell-Meiboom-Gill Sequence
CDD()
Concatenated Dynamical Decoupling
UDD()
Uhrig Dynamical Decoupling
References
- Cyw08
Cywiński, Ł., Lutchyn, R. M., Nave, C. P., & Das Sarma, S. (2008). How to enhance dephasing time in superconducting qubits. Physical Review B - Condensed Matter and Materials Physics, 77(17), 1–11. https://doi.org/10.1103/PhysRevB.77.174509
2.3. filter_functions.basis module
This module defines the Basis class, a subclass of NumPy’s ndarray, to represent operator bases.
2.3.1. Classes
Basis
The operator basis as an array of shape (d**2, d, d) with d the dimension of the Hilbert space
2.3.2. Functions
normalize()
Function to normalize a
Basis
instanceexpand()
Function to expand an array of operators in a given basis
ggm_expand()
Fast function to expand an array of operators in a Generalized Gell-Mann basis
- class Basis(basis_array: Sequence, traceless: Optional[bool] = None, btype: Optional[str] = None, labels: Optional[Sequence[str]] = None)[source]
Bases:
ndarray
Class for operator bases. There are several ways to instantiate a Basis object:
by just calling this constructor with a (possibly incomplete) array of basis matrices. No checks regarding orthonormality or hermiticity are performed.
by calling one of the classes alternative constructors (classmethods):
pauli()
: Pauli operator basisggm()
: Generalized Gell-Mann basisfrom_partial()
Generate an complete basis from partial elements
These bases guarantee the following properties:
hermitian
orthonormal
[traceless] (can be controlled by a flag)
Since Basis is a subclass of NumPy’s
ndarray
, it inherits all of its attributes, e.g.shape
. The following attributes behave slightly differently to a ndarray, howeverA == B
isTrue
if all elements evaluate almost equal, i.e. equivalent tonp.allclose(A, B)
.basis.T
transposes the last two axes ofbasis
. For a full basis, this corresponds to transposing each element individually. For a basis element, it corresponds to normal transposition.
- Parameters
- basis_array: array_like, shape (n, d, d)
An array or list of square matrices that are elements of an operator basis spanning \(\mathbb{C}^{d\times d}\). n should be smaller than or equal to d**2.
- traceless: bool, optional (default: auto)
Controls whether a traceless basis is forced. Here, traceless means that the first element of the basis is the identity and the remaining elements are matrices of trace zero. If an element of
basis_array
is neither traceless nor the identity andtraceless == True
, an exception will be raised. Defaults toTrue
if basis_array is traceless andFalse
if not.- btype: str, optional (default: ``’custom’``)
A string describing the basis type. For example, a basis created by the factory method
pauli()
has btype ‘pauli’.- labels: sequence of str, optional
A list of labels for the individual basis elements. Defaults to ‘C_0’, ‘C_1’, …
- Attributes
- Other than the attributes inherited from ``ndarray``, a ``Basis``
- instance has the following attributes:
- btype: str
Basis type.
- labels: sequence of str
The labels for the basis elements.
- d: int
Dimension of the space spanned by the basis.
- H: Basis
Hermitian conjugate.
- isherm: bool
If the basis is hermitian.
- isorthonorm: bool
If the basis is orthonormal.
- istraceless: bool
If the basis is traceless except for an identity element
- iscomplete: bool
If the basis is complete, ie spans the full space.
- sparse: COO, shape (n, d, d)
Representation in the COO format supplied by the
sparse
package.- four_element_traces: COO, shape (n, n, n, n)
Traces over all possible combinations of four elements of self. This is required for the calculation of the error transfer matrix and thus cached in the Basis instance.
- Most of the attributes above are properties which are lazily
- evaluated and cached.
Methods
Other than the methods inherited from ``ndarray``, a ``Basis``
instance has the following methods:
normalize(b)
Normalizes the basis (used internally when creating a basis from elements)
tidyup(eps_scale=None)
Cleans up floating point errors in-place to make zeros actual zeros.
eps_scale
is an optional argument multiplied to the data type’seps
to get the absolute tolerance.Constructor.
- property four_element_traces: COO
Return all traces of the form \(\mathrm{tr}(C_i C_j C_k C_l)\) as a sparse COO array for \(i,j,k,l > 0\) (i.e. excluding the identity).
- classmethod from_partial(partial_basis_array: Sequence, traceless: Optional[bool] = None, btype: Optional[str] = None, labels: Optional[Sequence[str]] = None) Basis [source]
Generate complete and orthonormal basis from a partial set.
The basis is completed using singular value decomposition to determine the null space of the expansion coefficients of the partial basis with respect to another complete basis.
- Parameters
- partial_basis_array: array_like
A sequence of basis elements.
- traceless: bool, optional
If a traceless basis should be generated (i.e. the first element is the identity and all the others have trace zero).
- btype: str, optional
A custom identifier.
- labels: Sequence[str], optional
A list of custom labels for each element. If len(labels) == len(partial_basis_array), the newly created elements get labels ‘C_i’.
- Returns
- basis: Basis, shape (d**2, d, d)
The orthonormal basis.
- Raises
- ValueError
If the given elements are not orthonormal.
- ValueError
If the given elements are not traceless but traceless==True.
- ValueError
If not len(partial_basis_array) or d**2 labels were given.
- classmethod ggm(d: int) Basis [source]
Returns a generalized Gell-Mann basis in \(d\) dimensions [Bert08] where the elements \(\Lambda_i\) are normalized with respect to the Hilbert-Schmidt inner product,
\[\begin{split}\langle\Lambda_i,\Lambda_j\rangle &= \mathrm{Tr}\,\Lambda_i^\dagger\Lambda_j \\ &= \delta_{ij}.\end{split}\]- Parameters
- d: int
The dimensionality of the space spanned by the basis
- Returns
- basis: Basis
The Basis object representing the GGM.
References
- Bert08
Bertlmann, R. A., & Krammer, P. (2008). Bloch vectors for qudits. Journal of Physics A: Mathematical and Theoretical, 41(23). https://doi.org/10.1088/1751-8113/41/23/235303
- classmethod pauli(n: int) Basis [source]
Returns a Pauli basis for \(n\) qubits, i.e. the basis spans the space \(\mathbb{C}^{d\times d}\) with \(d = 2^n\):
\[\mathcal{P} = \{I, X, Y, Z\}^{\otimes n}.\]The elements \(\sigma_i\) are normalized with respect to the Hilbert-Schmidt inner product,
\[\begin{split}\langle\sigma_i,\sigma_j\rangle &= \mathrm{Tr}\,\sigma_i^\dagger\sigma_j \\ &= \delta_{ij}.\end{split}\]- Parameters
- n: int
The number of qubits.
- Returns
- basis: Basis
The Basis object representing the Pauli basis.
- property sparse: COO
Return the basis as a sparse COO array
- expand(M: Union[ndarray, Basis], basis: Union[ndarray, Basis], normalized: bool = True, hermitian: bool = False, tidyup: bool = False) ndarray [source]
Expand the array M in the basis given by basis.
- Parameters
- M: array_like
The square matrix (d, d) or array of square matrices (…, d, d) to be expanded in basis
- basis: array_like
The basis of shape (m, d, d) in which to expand.
- normalized: bool {True}
Wether the basis is normalized.
- hermitian: bool (default: False)
If M is hermitian along its last two axes, the result will be real.
- tidyup: bool {False}
Whether to set values below the floating point eps to zero.
- Returns
- coefficients: ndarray
The coefficient array with shape (…, m) or (m,) if M was 2-d
Notes
For an orthogonal matrix basis \(\mathcal{C} = \big\{C_k\in \mathbb{C}^{d\times d}: \langle C_k,C_l\rangle_\mathrm{HS} \propto \delta_{kl}\big\}_{k=0}^{d^2-1}\) with the Hilbert-Schmidt inner product as implemented by
dot_HS()
and \(M\in\mathbb{C}^{d\times d}\), the expansion of \(M\) in terms of \(\mathcal{C}\) is given by\[\begin{split}M &= \sum_j c_j C_j, \\ c_j &= \frac{\mathrm{tr}\big(M C_j\big)} {\mathrm{tr}\big(C_j^\dagger C_j\big)}.\end{split}\]
- ggm_expand(M: Union[ndarray, Basis], traceless: bool = False, hermitian: bool = False) ndarray [source]
Expand the matrix M in a Generalized Gell-Mann basis [Bert08]. This function makes use of the explicit construction prescription of the basis and thus makes do without computing the expansion coefficients as the overlap between the matrix and each basis element.
- Parameters
- M: array_like
The square matrix (d, d) or array of square matrices (…, d, d) to be expanded in a GGM basis.
- traceless: bool (default: False)
Include the basis element proportional to the identity in the expansion. If it is known beforehand that M is traceless, the corresponding coefficient is zero and thus doesn’t need to be computed.
- hermitian: bool (default: False)
If M is hermitian along its last two axes, the result will be real.
- Returns
- coefficients: ndarray
The real coefficient array with shape (d**2,) or (…, d**2)
References
- Bert08
Bertlmann, R. A., & Krammer, P. (2008). Bloch vectors for qudits. Journal of Physics A: Mathematical and Theoretical, 41(23). https://doi.org/10.1088/1751-8113/41/23/235303
- normalize(b: Basis) Basis [source]
Return a copy of the basis b normalized with respect to the Frobenius norm [Gol85]:
\(||A||_F = \left[\sum_{i,j} |a_{i,j}|^2\right]^{1/2}\)
or equivalently, with respect to the Hilbert-Schmidt inner product as implemented by
dot_HS()
.References
- Gol85
G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
2.4. filter_functions.numeric module
This module defines the functions to calculate everything related to filter functions.
2.4.1. Functions
calculate_control_matrix_from_atomic()
Calculate the control matrix from those of atomic pulse sequences
calculate_control_matrix_from_scratch()
Calculate the control matrix from scratch
calculate_control_matrix_periodic()
Calculate the control matrix for a periodic Hamiltonian
calculate_noise_operators_from_atomic()
Calculate the interaction picture noise operators from atomic segments. Same calculation as
calculate_control_matrix_from_atomic()
except in Hilbert space.calculate_noise_operators_from_scratch()
Calculate the interaction picture noise operators from scratch. Same calculation as
calculate_control_matrix_from_scratch()
except in Hilbert space.calculate_cumulant_function()
Calculate the cumulant function for a given
PulseSequence
object.calculate_decay_amplitudes()
Calculate the decay amplitudes, corresponding to first order terms of the Magnus expansion
calculate_frequency_shifts()
Calculate the frequency shifts, corresponding to second order terms of the Magnus expansion
calculate_filter_function()
Calculate the filter function from the control matrix
calculate_second_order_filter_function()
Calculate the second order filter function used to compute the frequency shifts.
calculate_pulse_correlation_filter_function()
Calculate the pulse correlation filter function from the control matrix
diagonalize()
Diagonalize a Hamiltonian
error_transfer_matrix()
Calculate the error transfer matrix of a pulse up to a unitary rotation
infidelity()
Function to compute the infidelity of a pulse defined by a
PulseSequence
instance for a given noise spectral density and frequencies
- calculate_control_matrix_from_atomic(phases: ndarray, control_matrix_atomic: ndarray, propagators_liouville: ndarray, show_progressbar: bool = False, which: str = 'total') ndarray [source]
Calculate the control matrix from the control matrices of atomic segments.
- Parameters
- phases: array_like, shape (n_dt, n_omega)
The phase factors for \(g\in\{0, 1, \dots, G-1\}\).
- control_matrix_atomic: array_like, shape (n_dt, n_nops, d**2, n_omega)
The pulse control matrices for \(g\in\{1, 2, \dots, G\}\).
- propagators_liouville: array_like, shape (n_dt, n_nops, d**2, d**2)
The transfer matrices of the cumulative propagators for \(g\in\{0, 1, \dots, G-1\}\).
- show_progressbar: bool, optional
Show a progress bar for the calculation.
- which: str, (‘total’, ‘correlations’)
Compute the total control matrix (the sum of all time steps) or the correlation control matrix (first axis holds each pulses’ contribution)
- Returns
- control_matrix: ndarray, shape ([n_pls,] n_nops, d**2, n_omega)
The control matrix \(\tilde{\mathcal{B}}(\omega)\).
See also
calculate_control_matrix_from_scratch
Control matrix from scratch.
liouville_representation
Liouville representation for a given basis.
Notes
The control matrix is calculated by evaluating the sum
\[\tilde{\mathcal{B}}(\omega) = \sum_{g=1}^G e^{i\omega t_{g-1}} \tilde{\mathcal{B}}^{(g)}(\omega)\mathcal{Q}^{(g-1)}.\]
- calculate_control_matrix_from_scratch(eigvals: ndarray, eigvecs: ndarray, propagators: ndarray, omega: Sequence[float], basis: Basis, n_opers: Sequence[Union[ndarray, Qobj]], n_coeffs: Sequence[Sequence[float]], dt: Sequence[float], t: Optional[Sequence[float]] = None, show_progressbar: bool = False, cache_intermediates: bool = False, out: Optional[ndarray] = None) Union[ndarray, Tuple[ndarray, Dict[str, ndarray]]] [source]
Calculate the control matrix from scratch, i.e. without knowledge of the control matrices of more atomic pulse sequences.
- Parameters
- eigvals: array_like, shape (n_dt, d)
Eigenvalue vectors for each time pulse segment g with the first axis counting the pulse segment, i.e.
eigvals == array([D_0, D_1, ...])
.- eigvecs: array_like, shape (n_dt, d, d)
Eigenvector matrices for each time pulse segment g with the first axis counting the pulse segment, i.e.
eigvecs == array([V_0, V_1, ...])
.- propagators: array_like, shape (n_dt+1, d, d)
The propagators \(Q_g = P_g P_{g-1}\cdots P_0\) as a (d, d) array with d the dimension of the Hilbert space.
- omega: array_like, shape (n_omega,)
Frequencies at which the pulse control matrix is to be evaluated.
- basis: Basis, shape (d**2, d, d)
The basis elements in which the pulse control matrix will be expanded.
- n_opers: array_like, shape (n_nops, d, d)
Noise operators \(B_\alpha\).
- n_coeffs: array_like, shape (n_nops, n_dt)
The sensitivities of the system to the noise operators given by n_opers at the given time step.
- dt: array_like, shape (n_dt)
Sequence duration, i.e. for the \(g\)-th pulse \(t_g - t_{g-1}\).
- t: array_like, shape (n_dt+1), optional
The absolute times of the different segments. Can also be computed from dt.
- show_progressbar: bool, optional
Show a progress bar for the calculation.
- cache_intermediates: bool, optional
Keep and return intermediate terms of the calculation that can be reused in other computations (second order or gradients). Otherwise the sum is performed in-place. The default is False.
- out: ndarray, optional
A location into which the result is stored. See
numpy.ufunc()
.
- Returns
- control_matrix: ndarray, shape (n_nops, d**2, n_omega)
The control matrix \(\tilde{\mathcal{B}}(\omega)\)
- intermediates: dict[str, ndarray]
Intermediate results of the calculation. Only if cache_intermediates is True.
See also
calculate_control_matrix_from_atomic
Control matrix from concatenation.
calculate_control_matrix_periodic
Control matrix for periodic system.
Notes
The control matrix is calculated according to
\[\tilde{\mathcal{B}}_{\alpha k}(\omega) = \sum_{g=1}^G e^{i\omega t_{g-1}} s_\alpha^{(g)}\mathrm{tr}\left( [\bar{B}_\alpha^{(g)}\circ I(\omega)] \bar{C}_k^{(g)} \right)\]where
\[\begin{split}I^{(g)}_{nm}(\omega) &= \int_0^{t_l - t_{g-1}}\mathrm{d}t\, e^{i(\omega+\omega_n-\omega_m)t} \\ &= \frac{e^{i(\omega+\omega_n-\omega_m) (t_l - t_{g-1})} - 1} {i(\omega+\omega_n-\omega_m)}, \\ \bar{B}_\alpha^{(g)} &= V^{(g)\dagger} B_\alpha V^{(g)}, \\ \bar{C}_k^{(g)} &= V^{(g)\dagger} Q_{g-1} C_k Q_{g-1}^\dagger V^{(g)},\end{split}\]and \(V^{(g)}\) is the matrix of eigenvectors that diagonalizes \(\tilde{\mathcal{H}}_n^{(g)}\), \(B_\alpha\) the \(\alpha\)-th noise operator \(s_\alpha^{(g)}\) the noise sensitivity during interval \(g\), and \(C_k\) the \(k\)-th basis element.
- calculate_control_matrix_periodic(phases: ndarray, control_matrix: ndarray, total_propagator_liouville: ndarray, repeats: int, check_invertible: bool = True) ndarray [source]
Calculate the control matrix of a periodic pulse given the phase factors, control matrix and transfer matrix of the total propagator, total_propagator_liouville, of the atomic pulse.
- Parameters
- phases: ndarray, shape (n_omega,)
The phase factors \(e^{i\omega T}\) of the atomic pulse.
- control_matrix: ndarray, shape (n_nops, d**2, n_omega)
The control matrix \(\tilde{\mathcal{B}}^{(1)}(\omega)\) of the atomic pulse.
- total_propagator_liouville: ndarray, shape (d**2, d**2)
The transfer matrix \(\mathcal{Q}^{(1)}\) of the atomic pulse.
- repeats: int
The number of repetitions.
- check_invertible: bool, optional
Check for all frequencies if the inverse \(\mathbb{I} - e^{i\omega T} \mathcal{Q}^{(1)}\) exists by calculating the determinant. If it does not exist, the sum is evaluated explicitly for those points. The default is True.
- Returns
- control_matrix: ndarray, shape (n_nops, d**2, n_omega)
The control matrix \(\tilde{\mathcal{B}}(\omega)\) of the repeated pulse.
Notes
The control matrix is computed as
\[\begin{split}\tilde{\mathcal{B}}(\omega) &= \tilde{\mathcal{B}}^{(1)}(\omega) \sum_{g=0}^{G-1} \left(e^{i\omega T}\right)^g \\ &= \tilde{\mathcal{B}}^{(1)}(\omega)\bigl( \mathbb{I} - e^{i\omega T} \mathcal{Q}^{(1)}\bigr)^{-1}\bigl( \mathbb{I} - \bigl(e^{i\omega T} \mathcal{Q}^{(1)}\bigr)^G\bigr).\end{split}\]with \(G\) the number of repetitions.
- calculate_cumulant_function(pulse: PulseSequence, spectrum: Optional[ndarray] = None, omega: Optional[Sequence[float]] = None, n_oper_identifiers: Optional[Sequence[str]] = None, which: str = 'total', second_order: bool = False, decay_amplitudes: Optional[ndarray] = None, frequency_shifts: Optional[ndarray] = None, show_progressbar: bool = False, memory_parsimonious: bool = False, cache_intermediates: Optional[bool] = None) ndarray [source]
Calculate the cumulant function \(\mathcal{K}(\tau)\).
The error transfer matrix is obtained from the cumulant function by exponentiation, \(\langle\tilde{\mathcal{U}}\rangle = \exp\mathcal{K}(\tau)\).
- Parameters
- pulse: PulseSequence
The
PulseSequence
instance for which to compute the cumulant function.- spectrum: array_like, shape ([[n_nops,] n_nops,] n_omega), optional
The noise power spectral density in units of inverse frequencies as an array of shape (n_omega,), (n_nops, n_omega), or (n_nops, n_nops, n_omega). In the first case, the same spectrum is taken for all noise operators, in the second, it is assumed that there are no correlations between different noise sources and thus there is one spectrum for each noise operator. In the third and most general case, there may be a spectrum for each pair of noise operators corresponding to the correlations between them. n_nops is the number of noise operators considered and should be equal to
len(n_oper_identifiers)
.- omega: array_like, shape (n_omega,), optional
The frequencies at which to evaluate the filter functions.
- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which to evaluate the cumulant function. The default is all.
- which: str, optional
Which decay amplitudes should be calculated, may be either ‘total’ (default) or ‘correlations’. See
infidelity()
and Notes. Note that the latter is not available for the second order terms.- second_order: bool, optional
Also take into account the frequency shifts \(\Delta\) that correspond to second order Magnus expansion and constitute unitary terms. Default
False
.- decay_amplitudes, array_like, shape ([[n_pls, n_pls,] n_nops,] n_nops, d**2, d**2), optional
A precomputed cumulant function. If given, spectrum, omega are not required.
- frequency_shifts, array_like, shape ([[n_pls, n_pls,] n_nops,] n_nops, d**2, d**2), optional
A precomputed frequency shift. If given, spectrum, omega are not required for second order terms.
- show_progressbar: bool, optional
Show a progress bar for the calculation of the control matrix.
- memory_parsimonious: bool, optional
Trade memory footprint for performance. See
calculate_decay_amplitudes()
. The default isFalse
.- cache_intermediates: bool, optional
Keep and return intermediate terms of the calculation of the control matrix that can be reused in other computations (second order or gradients). Otherwise the sum is performed in-place. Default is True if second_order=True, else False.
- Returns
- cumulant_function: ndarray, shape ([[n_pls, n_pls,] n_nops,] n_nops, d**2, d**2)
The cumulant function. The individual noise operator contributions chosen by
n_oper_identifiers
are on the third to last axis / axes, depending on whether the noise is cross-correlated or not. Ifwhich == 'correlations'
, the first two axes correspond to the contributions of the pulses in the sequence.
See also
calculate_decay_amplitudes
Calculate the \(\Gamma_{\alpha\beta,kl}\)
error_transfer_matrix
Calculate the error transfer matrix \(\exp\mathcal{K}\).
infidelity
Calculate only infidelity of a pulse.
pulse_sequence.concatenate
Concatenate
PulseSequence
objects.calculate_pulse_correlation_filter_function
Notes
The cumulant function is given by
\[\begin{split}K_{\alpha\beta,ij}(\tau) = -\frac{1}{2} \sum_{kl}\biggl( &\Delta_{\alpha\beta,kl}\left( T_{klji} - T_{lkji} - T_{klij} + T_{lkij} \right) \\ + &\Gamma_{\alpha\beta,kl}\left( T_{klji} - T_{kjli} - T_{kilj} + T_{kijl} \right) \biggr)\end{split}\]Here, \(T_{ijkl} = \mathrm{tr}(C_i C_j C_k C_l)\) is a trivial function of the basis elements \(C_i\), and \(\Gamma_{\alpha\beta,kl}\) and \(\Delta_{\alpha\beta,kl}\) are the decay amplitudes and frequency shifts which correspond to first and second order in the Magnus expansion, respectively. Since the latter induce coherent errors, we can approximately neglect them if we assume that the pulse has been experimentally calibrated.
For a single qubit and represented in the Pauli basis, the above reduces to
\[\begin{split}K_{\alpha\beta,ij}(\tau) = \begin{cases} - \sum_{k\neq i}\Gamma_{\alpha\beta,kk} &\quad\mathrm{if}\: i = j, \\ - \Delta_{\alpha\beta,ij} + \Delta_{\alpha\beta,ji} + \Gamma_{\alpha\beta,ij} &\quad\mathrm{if}\: i\neq j, \end{cases}\end{split}\]for \(i\in\{1,2,3\}\) and \(K_{0j} = K_{i0} = 0\).
Lastly, the pulse correlation cumulant function resolves correlations in the cumulant function of a sequence of pulses \(g = 1,\dotsc,G\) such that the following holds:
\[K_{\alpha\beta,ij}(\tau) = \sum_{g,g'=1}^G K_{\alpha\beta,ij}^{(gg')}(\tau).\]
- calculate_decay_amplitudes(pulse: PulseSequence, spectrum: ndarray, omega: Sequence[float], n_oper_identifiers: Optional[Sequence[str]] = None, which: str = 'total', show_progressbar: bool = False, cache_intermediates: bool = False, memory_parsimonious: bool = False) ndarray [source]
Get the decay amplitudes \(\Gamma_{\alpha\beta, kl}\) for noise sources \(\alpha,\beta\) and basis elements \(k,l\).
- Parameters
- pulse: PulseSequence
The
PulseSequence
instance for which to compute the decay amplitudes.- spectrum: array_like, shape ([[n_nops,] n_nops,] n_omega)
The noise power spectral density. If 1-d, the same spectrum is used for all noise operators. If 2-d, one (self-) spectrum for each noise operator is expected. If 3-d, should be a matrix of cross-spectra such that
spectrum[i, j] == spectrum[j, i].conj()
.- omega: array_like,
The frequencies at which to calculate the filter functions.
- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which to calculate the decay amplitudes. The default is all.
- which: str, optional
Which decay amplitudes should be calculated, may be either ‘total’ (default) or ‘correlations’. See
infidelity()
and Notes.- show_progressbar: bool, optional
Show a progress bar for the calculation.
- cache_intermediates: bool, optional
Keep and return intermediate terms of the calculation that are useful in other places (if control matrix not already cached).
- memory_parsimonious: bool, optional
For large dimensions, the integrand
\[\tilde{\mathcal{B}}^\ast_{\alpha k}(\omega) S_{\alpha\beta}(\omega)\tilde{\mathcal{B}}_{\beta l}(\omega)\]can consume quite a large amount of memory if set up for all \(\alpha,\beta,k,l\) at once. If
True
, it is only set up and integrated for a single \(k\) at a time and looped over. This is slower but requires much less memory. The default isFalse
.
- Returns
- decay_amplitudes: ndarray, shape ([[n_pls, n_pls,] n_nops,] n_nops, d**2, d**2)
The decay amplitudes.
- Raises
- ValueError
If spectrum has incompatible shape.
See also
infidelity
Compute the infidelity directly.
pulse_sequence.concatenate
Concatenate
PulseSequence
objects.calculate_frequency_shifts
Second order (unitary) terms.
calculate_pulse_correlation_filter_function
Notes
The total decay amplitudes are given by
\[\Gamma_{\alpha\beta, kl} = \int\frac{\mathrm{d}\omega}{2\pi} \tilde{\mathcal{B}}^\ast_{\alpha k}(\omega) S_{\alpha\beta}(\omega)\tilde{\mathcal{B}}_{\beta l}(\omega).\]If pulse correlations are taken into account, they are given by
\[\Gamma_{\alpha\beta, kl}^{(gg')} = \int \frac{\mathrm{d}\omega}{2\pi} S_{\alpha\beta}(\omega) F_{\alpha\beta, kl}^{(gg')}(\omega).\]
- calculate_filter_function(control_matrix: ndarray, which: str = 'fidelity') ndarray [source]
Compute the filter function from the control matrix.
- Parameters
- control_matrix: array_like, shape (n_nops, d**2, n_omega)
The control matrix.
- whichstr, optional
Which filter function to return. Either ‘fidelity’ (default) or ‘generalized’ (see Notes).
- Returns
- filter_function: ndarray, shape (n_nops, n_nops, [d**2, d**2,] n_omega)
The filter functions for each noise operator correlation. The diagonal corresponds to the filter functions for uncorrelated noise sources.
See also
calculate_control_matrix_from_scratch
Control matrix from scratch.
calculate_control_matrix_from_atomic
Control matrix from concatenation.
calculate_pulse_correlation_filter_function
Pulse correlations.
Notes
The generalized filter function is given by
\[F_{\alpha\beta,kl}(\omega) = \tilde{\mathcal{B}}_{\alpha k}^\ast(\omega) \tilde{\mathcal{B}}_{\beta l}(\omega),\]where \(\alpha,\beta\) are indices counting the noise operators \(B_\alpha\) and \(k,l\) indices counting the basis elements \(C_k\).
The fidelity filter function is obtained by tracing over the basis indices:
\[F_{\alpha\beta}(\omega) = \sum_{k} F_{\alpha\beta,kk}(\omega).\]
- calculate_frequency_shifts(pulse: PulseSequence, spectrum: ndarray, omega: Sequence[float], n_oper_identifiers: Optional[Sequence[str]] = None, show_progressbar: bool = False) ndarray [source]
Get the frequency shifts \(\Delta_{\alpha\beta, kl}\) for noise sources \(\alpha,\beta\) and basis elements \(k,l\).
- Parameters
- pulse: PulseSequence
The
PulseSequence
instance for which to compute the frequency shifts.- spectrum: array_like, shape ([[n_nops,] n_nops,] n_omega)
The two-sided noise power spectral density. If 1-d, the same spectrum is used for all noise operators. If 2-d, one (self-) spectrum for each noise operator is expected. If 3-d, should be a matrix of cross-spectra such that
spectrum[i, j] == spectrum[j, i].conj()
.- omega: array_like,
The frequencies. Note that the frequencies are assumed to be symmetric about zero.
- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which to calculate the frequency shifts. The default is all.
- show_progressbar: bool, optional
Show a progress bar for the calculation.
- Returns
- Delta: ndarray, shape ([n_nops,] n_nops, d**2, d**2)
The frequency shifts.
- Raises
- ValueError
If spectrum has incompatible shape.
See also
calculate_second_order_filter_function
Corresponding filter function.
calculate_decay_amplitudes
First order (dissipative) terms.
infidelity
Compute the infidelity directly.
pulse_sequence.concatenate
Concatenate
PulseSequence
objects.calculate_pulse_correlation_filter_function
Notes
The total frequency shifts are given by
\[\Delta_{\alpha\beta, kl} = \int_{-\infty}^\infty \frac{\mathrm{d}{\omega}}{2\pi} S_{\alpha\beta}(\omega) F_{\alpha\beta,kl}^{(2)}(\omega)\]with \(F_{\alpha\beta,kl}^{(2)}(\omega)\) the second order filter function.
- calculate_noise_operators_from_atomic(phases: ndarray, noise_operators_atomic: ndarray, propagators: ndarray, show_progressbar: bool = False) ndarray [source]
Calculate the interaction picutre noise operators from atomic segments.
- Parameters
- phases: array_like, shape (n_dt, n_omega)
The phase factors for \(g\in\{0, 1, \dots, G-1\}\).
- noise_operators_atomic: array_like, shape (n_dt, n_nops, d, d, n_omega)
The noise operators in the interaction picture of the g-th pulse, i.e. for \(g\in\{1, 2, \dots, G\}\).
- propagators: array_like, shape (n_dt, d, d)
The cumulative propagators of the pulses \(g\in\{0, 1, \dots, G-1\}\).
- show_progressbar: bool, optional
Show a progress bar for the calculation.
- Returns
- noise_operators: ndarray, shape (n_omega, n_nops, d, d)
The interaction picture noise operators \(\tilde{B}_\alpha(\omega)\).
See also
calculate_noise_operators_from_scratch
Compute the operators from scratch.
calculate_control_matrix_from_atomic
Same calculation in Liouville space.
Notes
The noise operators are calculated by evaluating the sum
\[\tilde{B}_\alpha(\omega) = \sum_{g=1}^G e^{i\omega t_{g-1}} Q_{g-1}^\dagger\tilde{B}_\alpha^{(g)}(\omega) Q_{g-1}.\]The control matrix then corresponds to the coefficients of expansion in an operator basis \(\{C_k\}_k\):
\[\tilde{\mathcal{B}}_{k\alpha}(\omega) = \mathrm{tr}(\tilde{B}_\alpha(\omega) C_k).\]Due to differences in implementation (for performance reasons), the axes of the result are transposed compared to the control matrix:
>>> ctrlmat = calculate_control_matrix_from_atomic(...) >>> ctrlmat.shape (n_nops, d**2, n_omega) >>> noiseops = calculate_noise_operators_from_atomic(...) >>> noiseops.shape (n_omega, n_nops, d, d) >>> ctrlmat_from_noiseops = ff.basis.expand(noiseops, basis) >>> np.allclose(ctrlmat, ctrlmat_from_noiseops.transpose(1, 2, 0)) True
- calculate_noise_operators_from_scratch(eigvals: ndarray, eigvecs: ndarray, propagators: ndarray, omega: Sequence[float], n_opers: Sequence[Union[ndarray, Qobj]], n_coeffs: Sequence[Sequence[float]], dt: Sequence[float], t: Optional[Sequence[float]] = None, show_progressbar: bool = False, cache_intermediates: bool = False) Union[ndarray, Tuple[ndarray, Dict[str, ndarray]]] [source]
Calculate the noise operators in interaction picture from scratch.
- Parameters
- eigvals: array_like, shape (n_dt, d)
Eigenvalue vectors for each time pulse segment g with the first axis counting the pulse segment, i.e.
eigvals == array([D_0, D_1, ...])
.- eigvecs: array_like, shape (n_dt, d, d)
Eigenvector matrices for each time pulse segment g with the first axis counting the pulse segment, i.e.
eigvecs == array([V_0, V_1, ...])
.- propagators: array_like, shape (n_dt+1, d, d)
The propagators \(Q_g = P_g P_{g-1}\cdots P_0\) as a (d, d) array with d the dimension of the Hilbert space.
- omega: array_like, shape (n_omega,)
Frequencies at which the pulse control matrix is to be evaluated.
- n_opers: array_like, shape (n_nops, d, d)
Noise operators \(B_\alpha\).
- n_coeffs: array_like, shape (n_nops, n_dt)
The sensitivities of the system to the noise operators given by n_opers at the given time step.
- dt: array_like, shape (n_dt)
Sequence duration, i.e. for the \(g\)-th pulse \(t_g - t_{g-1}\).
- t: array_like, shape (n_dt+1), optional
The absolute times of the different segments. Can also be computed from dt.
- show_progressbar: bool, optional
Show a progress bar for the calculation.
- cache_intermediates: bool, optional
Keep and return intermediate terms of the calculation that can be reused in other computations (second order or gradients). Otherwise the sum is performed in-place. The default is False.
- Returns
- noise_operators: ndarray, shape (n_omega, n_nops, d, d)
The interaction picture noise operators \(\tilde{B}_\alpha(\omega)\).
- intermediates: dict[str, ndarray]
Intermediate results of the calculation. Only if cache_intermediates is True.
See also
calculate_noise_operators_from_atomic
Compute the operators from atomic segments.
calculate_control_matrix_from_scratch
Same calculation in Liouville space.
Notes
The interaction picture noise operators are calculated according to
\[\tilde{B}_\alpha(\omega) = \sum_{g=1}^G e^{i\omega t_{g-1}} s_\alpha^{(g)} P^{(g)\dagger}\left[ \bar{B}^{(g)}_\alpha \circ I^{(g)}(\omega) \right] P^{(g)}\]where
\[\begin{split}I^{(g)}_{nm}(\omega) &= \int_0^{t_g - t_{g-1}}\mathrm{d}t\, e^{i(\omega+\omega_n-\omega_m)t} \\ &= \frac{e^{i(\omega+\omega_n-\omega_m) (t_g - t_{g-1})} - 1} {i(\omega+\omega_n-\omega_m)}, \\ \bar{B}_\alpha^{(g)} &= V^{(g)\dagger} B_\alpha V^{(g)}, \\ P^{(g)} &= V^{(g)\dagger} Q_{g-1},\end{split}\]and \(V^{(g)}\) is the matrix of eigenvectors that diagonalizes \(\tilde{\mathcal{H}}_n^{(g)}\), \(B_\alpha\) the \(\alpha\)-th noise operator, and \(s_\alpha^{(g)}\) the noise sensitivity during interval \(g\).
The control matrix then corresponds to the coefficients of expansion in an operator basis \(\{C_k\}_k\):
\[\tilde{\mathcal{B}}_{k\alpha}(\omega) = \mathrm{tr}(\tilde{B}_\alpha(\omega) C_k).\]Due to differences in implementation (for performance reasons), the axes of the result are transposed compared to the control matrix:
>>> ctrlmat = calculate_control_matrix_from_scratch(...) >>> ctrlmat.shape (n_nops, d**2, n_omega) >>> noiseops = calculate_noise_operators_from_scratch(...) >>> noiseops.shape (n_omega, n_nops, d, d) >>> ctrlmat_from_noiseops = ff.basis.expand(noiseops, basis) >>> np.allclose(ctrlmat, ctrlmat_from_noiseops.transpose(1, 2, 0)) True
- calculate_pulse_correlation_filter_function(control_matrix: ndarray, which: str = 'fidelity') ndarray [source]
Compute pulse correlation filter function from control matrix.
- Parameters
- control_matrix: array_like, shape (n_pulses, n_nops, d**2, n_omega)
The control matrix.
- whichstr, optional
Which filter function to return. Either ‘fidelity’ (default) or ‘generalized’ (see Notes).
- Returns
- filter_function_pc: ndarray, shape (n_pls, n_pls, n_nops, n_nops, [d**2, d**2,] n_omega)
The pulse correlation filter functions for each pulse and noise operator correlations. The first two axes hold the pulse correlations, the second two the noise correlations.
- whichstr, optional
Which filter function to return. Either ‘fidelity’ (default) or ‘generalized’ (see Notes).
See also
calculate_control_matrix_from_scratch
Control matrix from scratch.
calculate_control_matrix_from_atomic
Control matrix from concatenation.
calculate_filter_function
Regular filter function.
Notes
The generalized pulse correlation filter function is given by
\[F_{\alpha\beta,kl}^{(gg')}(\omega) = \bigl[ \mathcal{Q}^{(g'-1)\dagger} \tilde{\mathcal{B}}^{(g')\dagger}(\omega) \bigr]_{k\alpha} \bigl[ \tilde{\mathcal{B}}^{(g)}(\omega)\mathcal{Q}^{(g-1)} \bigr]_{\beta l} e^{i\omega(t_{g-1} - t_{g'-1})},\]with \(\tilde{\mathcal{B}}^{(g)}\) the control matrix of the \(g\)-th pulse. The fidelity pulse correlation function is obtained by tracing out the basis indices,
\[F_{\alpha\beta}^{(gg')}(\omega) = \sum_{k} F_{\alpha\beta,kk}^{(gg')}(\omega)\]
- calculate_second_order_filter_function(eigvals: ndarray, eigvecs: ndarray, propagators: ndarray, omega: Sequence[float], basis: Basis, n_opers: Sequence[Union[ndarray, Qobj]], n_coeffs: Sequence[Sequence[float]], dt: Sequence[float], intermediates: Optional[Dict[str, ndarray]] = None, show_progressbar: bool = False) ndarray [source]
Calculate the second order filter function for frequency shifts.
- Parameters
- eigvals: array_like, shape (n_dt, d)
Eigenvalue vectors for each time pulse segment l with the first axis counting the pulse segment, i.e.
eigvals == array([D_0, D_1, ...])
.- eigvecs: array_like, shape (n_dt, d, d)
Eigenvector matrices for each time pulse segment l with the first axis counting the pulse segment, i.e.
eigvecs == array([V_0, V_1, ...])
.- propagators: array_like, shape (n_dt+1, d, d)
The propagators \(Q_l = P_l P_{l-1}\cdots P_0\) as a (d, d) array with d the dimension of the Hilbert space.
- omega: array_like, shape (n_omega,)
Frequencies at which the pulse control matrix is to be evaluated.
- basis: Basis, shape (d**2, d, d)
The basis elements in which the pulse control matrix will be expanded.
- n_opers: array_like, shape (n_nops, d, d)
Noise operators \(B_\alpha\).
- n_coeffs: array_like, shape (n_nops, n_dt)
The sensitivities of the system to the noise operators given by n_opers at the given time step.
- dt: array_like, shape (n_dt)
Sequence duration, i.e. for the \(l\)-th pulse \(t_l - t_{l-1}\).
- intermediates: Dict[str, ndarray], optional
Intermediate terms of the calculation of the control matrix that can be reused here. If None (default), they are computed from scratch.
- show_progressbar: bool, optional
Show a progress bar for the calculation.
- Returns
- second_order_filter_function: ndarray, shape (n_nops, n_nops, d**2, d**2, n_omega)
The second order filter function.
See also
calculate_frequency_shifts
Integrate over filter function times spectrum.
calculate_decay_amplitudes
First order (dissipative) terms.
infidelity
Compute the infidelity directly.
pulse_sequence.concatenate
Concatenate
PulseSequence
objects.calculate_pulse_correlation_filter_function
Notes
The second order filter function is given by
\[F_{\alpha\beta, kl}^{(2)} = \sum_{g=1}^G\left[ \mathcal{G}_{\alpha k}^{(g)\ast}(\omega) \sum_{g'=1}^{g-1}\mathcal{G}_{\beta l}^{(g')}(\omega) + s_\alpha^{(g)}\bar{B}_{\alpha,ij}^{(g)}\bar{C}_{k,ji}^{(g)} I_{ijmn}^{(g)}(\omega)\bar{C}_{l,nm}^{(g) \bar{B}_{\beta,mn}^{(g)}s_\beta^{(g)}} \right]\]with
\[\begin{split}\mathcal{G}^{(g)}(\omega) &= e^{i\omega t_{g-1}}\mathcal{B}^{(g)}(\omega) \mathcal{Q}^{(g-1)}, \\ I_{ijmn}^{(g)}(\omega) &= \int_{t_{g-1}}^{t_g}\mathrm{d}{t} e^{i\Omega_{ij}^{(g)}(t - t_{g-1}) - i\omega t} \int_{t_{g-1}}^{t}\mathrm{d}{t'} e^{i\Omega_{mn}^{(g)}(t' - t_{g-1}) + i\omega t'}.\end{split}\]
- diagonalize(hamiltonian: ndarray, dt: Sequence[float]) Tuple[ndarray] [source]
Diagonalize a Hamiltonian.
Diagonalize the Hamiltonian which is piecewise constant during the times given by dt and return eigenvalues, eigenvectors, and the cumulative propagators \(Q_l\). Note that we calculate in units where \(\hbar\equiv 1\) so that
\[U(t, t_0) = \mathcal{T}\exp\left( -i\int_{t_0}^t\mathrm{d}t'\mathcal{H}(t') \right).\]- Parameters
- hamiltonian: array_like, shape (n_dt, d, d)
Hamiltonian of shape (n_dt, d, d) with d the dimensionality of the system
- dt: array_like
The time differences
- Returns
- eigvals: ndarray
Array of eigenvalues of shape (n_dt, d)
- eigvecs: ndarray
Array of eigenvectors of shape (n_dt, d, d)
- propagators: ndarray
Array of cumulative propagators of shape (n_dt+1, d, d)
- error_transfer_matrix(pulse: Optional[PulseSequence] = None, spectrum: Optional[ndarray] = None, omega: Optional[Sequence[float]] = None, n_oper_identifiers: Optional[Sequence[str]] = None, second_order: bool = False, cumulant_function: Optional[ndarray] = None, show_progressbar: bool = False, memory_parsimonious: bool = False, cache_intermediates: bool = False) ndarray [source]
Compute the error transfer matrix up to unitary rotations.
- Parameters
- pulse: PulseSequence
The
PulseSequence
instance for which to compute the error transfer matrix.- spectrum: array_like, shape ([[n_nops,] n_nops,] n_omega)
The two-sided noise power spectral density in units of inverse frequencies as an array of shape (n_omega,), (n_nops, n_omega), or (n_nops, n_nops, n_omega). In the first case, the same spectrum is taken for all noise operators, in the second, it is assumed that there are no correlations between different noise sources and thus there is one spectrum for each noise operator. In the third and most general case, there may be a spectrum for each pair of noise operators corresponding to the correlations between them. n_nops is the number of noise operators considered and should be equal to
len(n_oper_identifiers)
.- omega: array_like, shape (n_omega,)
The frequencies at which to calculate the filter functions.
- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which to evaluate the error transfer matrix. The default is all. Note that, since in general contributions from different noise operators won’t commute, not selecting all noise operators results in neglecting terms of order \(\xi^4\).
- second_order: bool, optional
Also take into account the frequency shifts \(\Delta\) that correspond to second order Magnus expansion and constitute unitary terms. Default
False
.- cumulant_function: ndarray, shape ([[n_pls, n_pls,] n_nops,] n_nops, d**2, d**2)
A precomputed cumulant function. If given, pulse, spectrum, omega are not required.
- show_progressbar: bool, optional
Show a progress bar for the calculation of the control matrix.
- memory_parsimonious: bool, optional
Trade memory footprint for performance. See
calculate_decay_amplitudes()
. The default isFalse
.- cache_intermediates: bool, optional
Keep and return intermediate terms of the calculation of the control matrix (if it is not already cached) that can be reused for second order or gradients. Can consume large amount of memory, but speed up the calculation.
- Returns
- error_transfer_matrix: ndarray, shape (d**2, d**2)
The error transfer matrix. The individual noise operator contributions are summed up before exponentiating as they might not commute.
See also
calculate_cumulant_function
Calculate the cumulant function \(\mathcal{K}\)
calculate_decay_amplitudes
Calculate the \(\Gamma_{\alpha\beta,kl}\)
infidelity
Calculate only infidelity of a pulse.
Notes
The error transfer matrix is given by
\[\tilde{\mathcal{U}} = \exp\mathcal{K}(\tau)\]with \(\mathcal{K}(\tau)\) the cumulant function (see
calculate_cumulant_function()
). For Gaussian noise this expression is exact when taking into account the decay amplitudes \(\Gamma\) and frequency shifts \(\Delta\). As the latter effects coherent errors it can be neglected if we assume that the experimenter has calibrated their pulse.For non-Gaussian noise the expression above is perturbative and includes noise up to order \(\xi^2\) and hence \(\tilde{\mathcal{U}} = \mathbb{1} + \mathcal{K}(\tau) + \mathcal{O}(\xi^4)\) (although it is evaluated as a matrix exponential in any case).
Given the above expression of the error transfer matrix, the entanglement fidelity is given by
\[\mathcal{F}_\mathrm{e} = \frac{1}{d^2}\mathrm{tr}\,\tilde{\mathcal{U}}.\]
- infidelity(pulse: PulseSequence, spectrum: Union[Sequence[float], Callable], omega: Union[Sequence[float], Dict[str, Union[int, str]]], n_oper_identifiers: Optional[Sequence[str]] = None, which: str = 'total', show_progressbar: bool = False, cache_intermediates: bool = False, return_smallness: bool = False, test_convergence: bool = False) Union[ndarray, Any] [source]
Calculate the leading order entanglement infidelity.
This function calculates the infidelity approximately from the leading peturbation (see Notes). To compute it exactly for Gaussian noise and vanishing coherent errors (second order Magnus terms), use
error_transfer_matrix()
to obtain it from the full process matrix.- Parameters
- pulse: PulseSequence
The
PulseSequence
instance for which to calculate the infidelity for.- spectrum: array_like, shape ([[n_nops,] n_nops,] omega) or callable
The two-sided noise power spectral density in units of inverse frequencies as an array of shape (n_omega,), (n_nops, n_omega), or (n_nops, n_nops, n_omega). In the first case, the same spectrum is taken for all noise operators, in the second, it is assumed that there are no correlations between different noise sources and thus there is one spectrum for each noise operator. In the third and most general case, there may be a spectrum for each pair of noise operators corresponding to the correlations between them. n_nops is the number of noise operators considered and should be equal to
len(n_oper_identifiers)
.If test_convergence is
True
, a function handle to compute the power spectral density from a sequence of frequencies is expected.- omega: array_like or dict
The frequencies at which the integration is to be carried out. If test_convergence is
True
, a dict with possible keys (‘omega_IR’, ‘omega_UV’, ‘spacing’, ‘n_min’, ‘n_max’, ‘n_points’), where all entries are integers except forspacing
which should be a string, either ‘linear’ or ‘log’. ‘n_points’ controls how many steps are taken.- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which to calculate the infidelity contribution. If given, the infidelities for each noise operator will be returned. Otherwise, all noise operators will be taken into account.
- which: str, optional
Which infidelities should be calculated, may be either ‘total’ (default) or ‘correlations’. In the former case, one value is returned for each noise operator, corresponding to the total infidelity of the pulse (or pulse sequence). In the latter, an array of infidelities is returned where element (i,j) corresponds to the infidelity contribution of the correlations between pulses i and j (see Notes). Note that this option is only available if the pulse correlation filter functions have been computed during concatenation (see
calculate_pulse_correlation_filter_function()
andconcatenate()
).- show_progressbar: bool, optional
Show a progressbar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep and return intermediate terms of the calculation of the control matrix (if it is not already cached) that can be reused for second order or gradients. The default is False.
- return_smallness: bool, optional
Return the smallness parameter \(\xi\) for the given spectrum.
- test_convergence: bool, optional
Test the convergence of the integral with respect to the number of frequency samples. Returns the number of frequency samples and the corresponding fidelities. See spectrum and omega for more information.
- Returns
- infid: ndarray, shape ([[n_pls, n_pls,], n_nops,] n_nops)
Array with the infidelity contributions for each spectrum spectrum on the last axis or axes, depending on the shape of spectrum and which. If
which
iscorrelations
, the first two axes are the individual pulse contributions. If spectrum is 2-d (3-d), the last axis (two axes) are the individual spectral contributions. Only if test_convergence isFalse
.- n_samples: array_like
Array with number of frequency samples used for convergence test. Only if test_convergence is
True
.- convergence_infids: array_like
Array with infidelities calculated in convergence test. Only if test_convergence is
True
.
See also
calculate_decay_amplitudes
Calculate the full matrix of first order terms.
error_transfer_matrix
Calculate the full process matrix.
plotting.plot_infidelity_convergence
Convenience function to plot results.
pulse_sequence.concatenate
Concatenate
PulseSequence
objects.calculate_pulse_correlation_filter_function.
Notes
The infidelity is given by
\[\begin{split}\mathcal{I}_{\alpha\beta} &= 1 - \frac{1}{d^2}\mathrm{tr}\:\tilde{\mathcal{U}} \\ &= \frac{1}{d}\int_{-\infty}^{\infty} \frac{\mathrm{d}\omega}{2\pi}S_{\alpha\beta}(\omega) F_{\alpha\beta}(\omega) + \mathcal{O}\big(\xi^4\big) \\ &= \sum_{g,g'=1}^G \mathcal{I}_{\alpha\beta}^{(gg')}\end{split}\]with \(S_{\alpha\beta}(\omega)\) the two-sided noise spectral density and \(F_{\alpha\beta}(\omega)\) the first-order filter function for noise sources \(\alpha,\beta\). The noise spectrum may include correlated noise sources, that is, its entry at \((\alpha,\beta)\) corresponds to the correlations between sources \(\alpha\) and \(\beta\). \(\mathcal{I}_{\alpha\beta}^{(gg')}\) are the correlation infidelities that can be computed by setting
which='correlations'
.To convert to the average gate infidelity, use the following relation given by Horodecki et al. [Hor99] and Nielsen [Nie02]:
\[\mathcal{I}_\mathrm{avg} = \frac{d}{d+1}\mathcal{I}.\]The smallness parameter is given by
\[\xi^2 = \sum_\alpha\left[ \lvert\lvert B_\alpha\rvert\rvert^2 \int_{-\infty}^\infty\frac{\mathrm{d}\omega}{2\pi} S_\alpha(\omega)\left(\sum_gs_\alpha^{(g)}\Delta t_g \right)^2 \right].\]Note that in practice, the integral is only evaluated on the interval \(\omega\in[\omega_\mathrm{min},\omega_\mathrm{max}]\).
References
- Hor99
Horodecki, M., Horodecki, P., & Horodecki, R. (1999). General teleportation channel, singlet fraction, and quasidistillation. Physical Review A - Atomic, Molecular, and Optical Physics, 60(3), 1888–1898. https://doi.org/10.1103/PhysRevA.60.1888
- Nie02
Nielsen, M. A. (2002). A simple formula for the average gate fidelity of a quantum dynamical operation. Physics Letters, Section A: General, Atomic and Solid State Physics, 303(4), 249–252. https://doi.org/10.1016/S0375-9601(02)01272-0
2.5. filter_functions.plotting module
This module provides various plotting functions.
2.5.1. Functions
plot_bloch_vector_evolution()
Plot the evolution of the Bloch vector on a QuTiP-generated Bloch sphere
plot_filter_function()
Plot the filter function of a given
PulseSequence
plot_infidelity_convergence()
Helper function called by
infidelity()
to plot the convergence of the infidelityplot_pulse_correlation_filter_function()
Plot the pulse correlation filter function of a given
PulseSequence
plot_pulse_train()
Plot the pulse train of a given
PulseSequence
plot_cumulant_function()
Plot the cumulant function of a
PulseSequence
for a given spectrum as an image.
- plot_bloch_vector_evolution(pulse: PulseSequence, psi0: Optional[Union[ndarray, Qobj]] = None, b: Optional[Bloch] = None, n_samples: Optional[int] = None, cmap: Union[Colormap, str] = 'winter', add_cbar: bool = False, show: bool = True, return_Bloch: bool = False, cbar_kwargs: Optional[dict] = None, **bloch_kwargs) Union[None, Bloch] [source]
Plot the evolution of the Bloch vector under the given pulse sequence.
- Parameters
- pulse: PulseSequence
The PulseSequence instance whose control Hamiltonian determines the time evolution of the Bloch vector.
- psi0: Qobj or array_like, optional
The initial state before the pulse is applied. Defaults to \(|0\rangle\).
- b: qutip.Bloch, optional
If given, the QuTiP Bloch instance on which to plot the time evolution.
- n_samples: int, optional
The number of time points to be sampled.
- cmap: matplotlib colormap, optional
The colormap for the trajectory. Requires
matplotlib >= 3.3.0
.- add_cbar: bool, optional
Add a colorbar encoding the time evolution to the figure. Default is false.
- show: bool, optional
Whether to show the sphere (by calling
b.make_sphere()
).- return_Bloch: bool, optional
Whether to return the
qutip.bloch.Bloch
instance- cbar_kwargs: dict, optional
A dictionary with keyword arguments to be fed into the colorbar constructor (if
add_cbar == True
).- **bloch_kwargs: optional
Keyword arguments to be fed into the qutip.Bloch constructor (if b not given).
- Returns
- b: qutip.Bloch
The qutip.Bloch instance
- Raises
- ValueError
If the pulse is for more than one qubit
See also
qutip.bloch.Bloch
Qutip’s Bloch sphere implementation.
- plot_cumulant_function(pulse: Optional[PulseSequence] = None, spectrum: Optional[ndarray] = None, omega: Optional[Sequence[float]] = None, cumulant_function: Optional[ndarray] = None, n_oper_identifiers: Optional[Sequence[int]] = None, second_order: bool = False, colorscale: str = 'linear', linthresh: Optional[float] = None, basis_labels: Optional[Sequence[str]] = None, basis_labelsize: Optional[int] = None, cbar_label: str = 'Cumulant Function', cbar_labelsize: Optional[int] = None, fig: Optional[Figure] = None, grid: Optional[ImageGrid] = None, cmap: Optional[Union[Colormap, str]] = None, grid_kw: Optional[dict] = None, cbar_kw: Optional[dict] = None, imshow_kw: Optional[dict] = None, **figure_kw) Tuple[Figure, ImageGrid] [source]
Plot the cumulant function for a given noise spectrum as an image.
The cumulant function generates the error transfer matrix \(\tilde{\mathcal{U}}\) exactly for Gaussian noise and to second order for non-Gaussian noise.
The function may be called with either a
PulseSequence
, a spectrum, and a list of frequencies in which case the cumulant function is calculated for those parameters, or with a precomputed cumulant function.As of now, only auto-correlated spectra are implemented.
- Parameters
- pulse: PulseSequence
The pulse sequence.
- spectrum: ndarray
The two-sided noise spectrum.
- omega: array_like
The frequencies for which to evaluate the error transfer matrix.
- cumulant_function: ndarray, shape (n_nops, d**2, d**2)
A precomputed cumulant function. If given, pulse, spectrum, omega are not required.
- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which the cumulant function should be plotted. All identifiers can be accessed via
pulse.n_oper_identifiers
. Defaults to all.- second_order: bool, optional
Also take into account the frequency shifts \(\Delta\) that correspond to second order Magnus expansion and constitute unitary terms. Default
False
.- colorscale: str, optional
The scale of the color code (‘linear’ or ‘log’ (default))
- linthresh: float, optional
The threshold below which the colorscale will be linear (only for ‘log’) colorscale
- basis_labels: array_like (str), optional
Custom labels for the elements of the cumulant function (the basis elements). Grabbed from the basis by default.
- basis_labelsize: int, optional
The size in points for the basis labels.
- cbar_label: str, optional
The label for the colorbar. Default: ‘Cumulant Function’.
- cbar_labelsize: int, optional
The size in points for the colorbar label.
- fig: matplotlib figure, optional
A matplotlib figure instance to plot in
- grid: matplotlib ImageGrid, optional
An ImageGrid instance to use for plotting.
- cmap: matplotlib colormap, optional
The colormap for the matrix plot.
- grid_kw: dict, optional
Dictionary with keyword arguments passed to the ImageGrid constructor.
- cbar_kw: dict, optional
Dictionary with keyword arguments passed to the colorbar constructor.
- imshow_kw: dict, optional
Dictionary with keyword arguments passed to imshow.
- figure_kw: optional
Keyword argument dictionaries that are fed into the
matplotlib.pyplot.figure()
function if no fig instance is specified.
- Returns
- fig: matplotlib figure
The matplotlib figure instance used for plotting.
- grid: matplotlib ImageGrid
The ImageGrid instance used for plotting.
- plot_filter_function(pulse: PulseSequence, omega: Optional[Sequence[float]] = None, n_oper_identifiers: Optional[Sequence[int]] = None, fig: Optional[Figure] = None, axes: Optional[Axes] = None, xscale: str = 'log', yscale: str = 'linear', omega_in_units_of_tau: bool = True, cycler: Optional[Cycler] = None, plot_kw: dict = {}, subplot_kw: Optional[dict] = None, gridspec_kw: Optional[dict] = None, **figure_kw) Tuple[Figure, Axes, Legend] [source]
Plot the fidelity filter function(s) of the given PulseSequence for positive frequencies. As of now only the diagonal elements of \(F_{\alpha\beta}\) are implemented, i.e. the filter functions corresponding to uncorrelated noise sources.
- Parameters
- pulse: PulseSequence
The pulse sequence whose filter function to plot.
- omega: array_like, optional
The frequencies at which to evaluate the filter function. If not given, the pulse sequence’s omega attribute is used (if set) or sensible values are chosen automatically (if
None
)- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which the filter function should be plotted. All identifiers can be accessed via
pulse.n_oper_identifiers
. Defaults to all.- fig: matplotlib figure, optional
A matplotlib figure instance to plot in
- axes: matplotlib axes, optional
A matplotlib axes instance to use for plotting.
- xscale: str, optional
x-axis scaling. One of (‘linear’, ‘log’).
- yscale: str, optional
y-axis scaling. One of (‘linear’, ‘log’).
- omega_in_units_of_tau: bool, optional
Plot \(\omega\tau\) or just \(\omega\) on x-axis.
- cycler: cycler.Cycler, optional
A Cycler instance used to set the style cycle if multiple lines are to be drawn
- plot_kw: dict, optional
Dictionary with keyword arguments passed to the plot function
- subplot_kw: dict, optional
Dictionary with keyword arguments passed to the subplots constructor
- gridspec_kw: dict, optional
Dictionary with keyword arguments passed to the gridspec constructor
- figure_kw: optional
Keyword argument dictionaries that are fed into the
matplotlib.pyplot.subplots()
function if no fig instance is specified.
- Returns
- fig: matplotlib figure
The matplotlib figure instance used for plotting.
- axes: matplotlib axes
The matplotlib axes instance used for plotting.
- legend: matplotlib legend
The matplotlib legend instance in the plot.
- Raises
- ValueError
If an invalid number of n_oper_labels were given
- plot_infidelity_convergence(n_samples: Sequence[int], infids: Sequence[float], axes: Optional[Axes] = None) Tuple[Figure, Axes] [source]
Plot the convergence of the infidelity integral. The function arguments are those returned by
infidelity()
with the test_convergence flag set toTrue
.- Parameters
- n_samples: array_like
Array with the number of samples at which the integral was evaluated
- infids: array_like, shape (n_samples, [n_oper_inds, optional])
Array with the calculated infidelities for each noise operator on the second axis or the second axis already traced out.
- axes: sequence of two matplotlib axes, optional
Two axes that the result is plotted in.
- Returns
- fig: matplotlib figure
The matplotlib figure instance used for plotting.
- axes: matplotlib axes
The matplotlib axes instances used for plotting.
- plot_pulse_correlation_filter_function(pulse: PulseSequence, n_oper_identifiers: Optional[Sequence[int]] = None, fig: Optional[Figure] = None, xscale: str = 'log', yscale: str = 'linear', omega_in_units_of_tau: bool = True, cycler: Optional[Cycler] = None, plot_kw: dict = {}, subplot_kw: Optional[dict] = None, gridspec_kw: Optional[dict] = None, **figure_kw) Tuple[Figure, Axes, Legend] [source]
Plot the fidelity pulse correlation filter functions of the given PulseSequence if they were computed during concatenation for positive frequencies.
Returns a figure with n by n subplots where n is the number of pulses that were concatenated. As of now only the diagonal elements of \(F_{\alpha\beta}\) are implemented, i.e. the filter functions corresponding to uncorrelated noise sources.
- Parameters
- pulse: PulseSequence
The pulse sequence whose filter function to plot.
- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which the filter function should be plotted. All identifiers can be accessed via
pulse.n_oper_identifiers
. Defaults to all.- fig: matplotlib figure, optional
A matplotlib figure instance to plot in
- xscale: str, optional
x-axis scaling. One of (‘linear’, ‘log’).
- yscale: str, optional
y-axis scaling. One of (‘linear’, ‘log’).
- omega_in_units_of_tau: bool, optional
Plot \(\omega\tau\) or just \(\omega\) on x-axis.
- cycler: cycler.Cycler, optional
A Cycler instance used to set the style cycle if multiple lines are to be drawn in one subplot. Used for all subplots.
- plot_kw: dict, optional
Dictionary with keyword arguments passed to the plot function
- subplot_kw: dict, optional
Dictionary with keyword arguments passed to the subplots constructor
- gridspec_kw: dict, optional
Dictionary with keyword arguments passed to the gridspec constructor
- figure_kw: optional
Keyword argument dictionaries that are fed into the
matplotlib.pyplot.subplots()
function if no fig instance if specified.
- Returns
- fig: matplotlib figure
The matplotlib figure instance used for plotting.
- axes: matplotlib axes
The matplotlib axes instances used for plotting.
- legend: matplotlib legend
The matplotlib legend instance in the plot.
- Raises
- CalculationError
If the pulse correlation filter function was not computed during concatenation.
- plot_pulse_train(pulse: PulseSequence, c_oper_identifiers: Optional[Sequence[int]] = None, fig: Optional[Figure] = None, axes: Optional[Axes] = None, cycler: Optional[Cycler] = None, plot_kw: Optional[dict] = {}, subplot_kw: Optional[dict] = None, gridspec_kw: Optional[dict] = None, **figure_kw) Tuple[Figure, Axes, Legend] [source]
Plot the pulsetrain of the
PulseSequence
pulse.- Parameters
- pulse: PulseSequence
The pulse sequence whose pulse train to plot.
- c_oper_identifiers: array_like, optional
The identifiers of the control operators for which the pulse train should be plotted. All identifiers can be accessed via
pulse.c_oper_identifiers
. Defaults to all.- fig: matplotlib figure, optional
A matplotlib figure instance to plot in
- axes: matplotlib axes, optional
A matplotlib axes instance to use for plotting.
- cycler: cycler.Cycler, optional
A Cycler instance used to set the style cycle if multiple lines are to be drawn
- plot_kw: dict, optional
Dictionary with keyword arguments passed to the plot function
- subplot_kw: dict, optional
Dictionary with keyword arguments passed to the subplots constructor
- gridspec_kw: dict, optional
Dictionary with keyword arguments passed to the gridspec constructor
- figure_kw: optional
Keyword argument dictionaries that are fed into the
matplotlib.pyplot.subplots()
function if no fig instance is specified.
- Returns
- fig: matplotlib figure
The matplotlib figure instance used for plotting.
- axes: matplotlib axes
The matplotlib axes instance used for plotting.
- legend: matplotlib legend
The matplotlib legend instance in the plot.
- Raises
- ValueError
If an invalid number of c_oper_labels were given
2.6. filter_functions.pulse_sequence module
This module defines the PulseSequence class, the package’s central object.
2.6.1. Classes
PulseSequence
The pulse sequence defined by a Hamiltonian
2.6.2. Functions
concatenate()
Function to concatenate different
PulseSequence
instances and efficiently compute their joint filter functionconcatenate_periodic()
Function to more efficiently concatenate many versions of the same
PulseSequence
instances and compute their joint filter functionextend()
Function to map several
PulseSequence
instances to different qubits, efficiently scaling up cached attributes.
- class PulseSequence(*args, **kwargs)[source]
Bases:
object
A class for pulse sequences and their filter functions.
The Hamiltonian is separated into a control and a noise part with
\[\begin{split}\mathcal{H}_c &= \sum_i a_i(t) A_i \\ \mathcal{H}_n &= \sum_j s_j(t) b_j(t) B_j\end{split}\]where \(A_i\) and \(B_j\) are hermitian operators and \(b_j(t)\) are classically fluctuating noise variables captured in a power spectral density and not needed at instantiation of a
PulseSequence
.- Parameters
- H_c: list of lists
A nested list of n_cops nested lists as taken by QuTiP functions (see for example
qutip.propagator.propagator()
) describing the control part of the Hamiltonian. The i-th entry of the list should be a list consisting of the i-th operator \(A_i\) making up the control Hamiltonian and a list or array \(a_i(t)\) describing the magnitude of that operator during the time intervals dt. Optionally, the list may also include operator identifiers. That is, H_c should look something like this:H = [[c_oper1, c_coeff1, c_oper_identifier1], [c_oper2, c_coeff2, c_oper_identifier2], ...]
The operators may be given either as NumPy arrays or QuTiP Qobjs and each coefficient array should have the same number of elements as dt, and should be given in units of \(\hbar\). If not every sublist (read: operator) was given a identifier, they are automatically filled up with ‘A_i’ where i is the position of the operator.
- H_n: list of lists
A nested list of n_nops nested lists as taken by QuTiP functions (see for example
qutip.propagator.propagator()
) describing the noise part of the Hamiltonian. The j-th entry of the list should be a list consisting of the j-th operator \(B_j\) making up the noise Hamiltonian and a list or array describing the sensitivity \(s_j(t)\) of the system to the noise operator during the time intervals dt. Optionally, the list may also include operator identifiers. That is, H_n should look something like this:H = [[n_oper1, n_coeff1, n_oper_identifier1], [n_oper2, n_coeff2, n_oper_identifier2], ...]
The operators may be given either as NumPy arrays or QuTiP Qobjs and each coefficient array should have the same number of elements as dt, and should be given in units of \(\hbar\). If not every sublist (read: operator) was given a identifier, they are automatically filled up with ‘B_i’ where i is the position of the operator.
- dt: array_like, shape (n_dt,)
The segment durations of the Hamiltonian (i.e. durations of constant control). Internally, the control operation is taken to start at \(t_0\equiv 0\), i.e. the edges of the constant control segments are at times
t = [0, *np.cumsum(dt)]
.- basis: Basis, shape (d**2, d, d), optional
The operator basis in which to calculate. If a Generalized Gell-Mann basis (see
ggm()
) is chosen, some calculations will be faster for large dimensions due to a simpler basis expansion. However, when extending the pulse sequence to larger qubit registers, cached filter functions cannot be retained since the GGM basis does not factor into tensor products. In this case a Pauli basis is preferable.
Notes
Due to the heavy use of NumPy’s
einsum()
function, results have a floating point error of ~1e-13.Examples
A rotation by \(\pi\) around the axis between x and y preceeded and followed by a period of free evolution with the system subject to dephasing noise.
>>> import qutip as qt; import numpy as np >>> H_c = [[qt.sigmax(), [0, np.pi, 0]], [qt.sigmay(), [0, np.pi, 0]]] >>> # Equivalent pulse: >>> # H_c = [[qt.sigmax() + qt.sigmay(), [0, np.pi, 0]]] >>> # The noise sensitivity is constant >>> H_n = [[qt.sigmaz()/np.sqrt(2), [1, 1, 1], 'Z']] >>> dt = [1, 1, 1] >>> # Free evolution between t=0 and t=1, rotation between t=1 and t=2, >>> # and free evolution again from t=2 to t=3. >>> pulse = PulseSequence(H_c, H_n, dt) >>> pulse.c_oper_identifiers ['A_0', 'A_1'] >>> pulse.n_oper_identifiers ['Z'] >>> omega = np.logspace(-1, 2, 500) >>> F = pulse.get_filter_function(omega) # shape (1, 500) >>> # Plot the resulting filter function: >>> from filter_functions import plotting >>> fig, ax, leg = plotting.plot_filter_function(pulse)
- Attributes
- c_opers: ndarray, shape (n_cops, d, d)
Control operators. Note that they are stored sorted by their corresponding identifiers.
- n_opers: ndarray, shape (n_nops, d, d)
Noise operators. Note that they are stored sorted by their corresponding identifiers.
- c_oper_identifers: sequence of str
Identifiers for the control operators of the system. Stored sorted.
- n_oper_identifers: sequence of str
Identifiers for the noise operators of the system. Stored sorted.
- c_coeffs: ndarray, shape (n_cops, n_dt)
Control parameters in units of \(\hbar\). Note that they are stored sorted by their corresponding identifiers.
- n_coeffs: ndarray, shape (n_nops, n_dt)
Noise sensitivities in units of \(\hbar\). Note that they are stored sorted by their corresponding identifiers.
- dt: ndarray, shape (n_dt,)
Time steps
- t: ndarray, shape (n_dt + 1,)
Absolute times taken to start at \(t_0\equiv 0\)
- tau: float
Total duration. Equal to t[-1].
- d: int
Dimension of the Hamiltonian
- basis: Basis, shape (d**2, d, d)
The operator basis used for calculation
- nbytes: int
An estimate of the memory consumed by the PulseSequence instance and its attributes
- If the Hamiltonian was diagonalized, the eigenvalues and -vectors as
- well as the cumulative propagators are cached:
- eigvals: ndarray, shape (n_dt, d)
Eigenvalues \(D^{(g)}\)
- eigvecs: ndarray, shape (n_dt, d, d)
Eigenvectors \(V^{(g)}\)
- propagators: ndarray, shape (n_dt+1, d, d)
Cumulative propagators \(Q_g\)
- total_propagator: ndarray, shape (d, d)
The total propagator \(Q\) of the pulse alone. That is, \(|\psi(\tau)\rangle = Q|\psi(0)\rangle\).
- total_propagator_liouville: array_like, shape (d**2, d**2)
The transfer matrix for the total propagator of the pulse. Given by
liouville_representation(pulse.total_propagator, pulse.basis)
.- Furthermore, when the filter function is calculated, the frequencies
- are cached as well as other relevant quantities.
Methods
get_filter_function_derivative
(omega[, ...])Calculate the pulse sequence's filter function derivative.
cleanup(method=’conservative’)
Delete cached attributes
is_cached(attr)
Checks if a given attribute of the
PulseSequence
is cacheddiagonalize()
Diagonalize the Hamiltonian of the pulse sequence, computing eigenvalues and -vectors as well as cumulative propagators
get_control_matrix(omega, show_progressbar=False)
Calculate the control matrix for frequencies omega
get_filter_function(omega, which=’fidelity’, show_progressbar=False)
Calculate the filter function for frequencies omega
get_pulse_correlation_filter_function(which=’fidelity’)
Get the pulse correlation filter function (only possible if computed during concatenation)
propagator_at_arb_t(t)
Calculate the propagator at arbitrary times
Initialize a PulseSequence instance.
- cache_control_matrix(omega: Sequence[float], control_matrix: Optional[ndarray] = None, show_progressbar: bool = False, cache_intermediates: bool = False) None [source]
Cache the control matrix and the frequencies it was calculated for.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies for which to cache the filter function.
- control_matrix: array_like, shape ([n_nops,] n_nops, d**2, n_omega), optional
The control matrix for the frequencies omega. If
None
, it is computed.- show_progressbar: bool
Show a progress bar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep intermediate terms of the calculation that are also required by other computations. Only applies if control_matrix is not supplied.
- cache_filter_function(omega: Sequence[float], control_matrix: Optional[ndarray] = None, filter_function: Optional[ndarray] = None, which: str = 'fidelity', order: int = 1, show_progressbar: bool = False, cache_intermediates: bool = False) None [source]
Cache the filter function. If control_matrix.ndim == 4, it is taken to be the ‘pulse correlation control matrix’ and summed along the first axis. In that case, also the pulse correlation filter function is calculated and cached. Total phase factors and transfer matrices of the the cumulative propagator are also cached so they can be reused during concatenation.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies for which to cache the filter function.
- control_matrix: array_like, shape ([n_nops,] n_nops, d**2, n_omega), optional
The control matrix for the frequencies omega. If
None
, it is computed and the filter function derived from it.- filter_function: array_like, shape (n_nops, n_nops, [d**2, d**2,] n_omega), optional
The filter function for the frequencies omega. If
None
, it is computed from R in the caseorder == 1
and from scratch else.- which: str, optional
Which filter function to cache. Either ‘fidelity’ (default) or ‘generalized’.
- order: int, optional
First or second order filter function.
- show_progressbar: bool, optional
Show a progress bar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep intermediate terms of the calculation that are also required by other computations.
See also
PulseSequence.get_filter_function
Getter method
- cache_total_phases(omega: Sequence[float], total_phases: Optional[ndarray] = None) None [source]
Cache the total phase factors for this pulse and omega.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies for which to cache the phase factors.
- total_phases: array_like, shape (n_omega,), optional
The total phase factors for the frequencies omega. If
None
, they are computed.
- cleanup(method: str = 'conservative') None [source]
Delete cached byproducts of the calculation of the filter function that are not necessarily needed anymore in order to free up memory.
- Parameters
- method: optional
If set to ‘conservative’ (the default), only the following attributes are deleted:
_eigvals
_eigvecs
_propagators
If set to ‘greedy’, all of the above as well as the following attributes are deleted:
_total_propagator
_total_propagator_liouville
_total_phases
_control_matrix
_control_matrix_pc
If set to ‘all’, all of the above as well as the following attributes are deleted:
omega
_filter_function
_filter_function_gen
_filter_function_pc
_filter_function_pc_gen
_filter_function_2
_intermediates[‘control_matrix_step’]
If set to ‘frequency dependent’ only attributes that are functions of frequency are initalized to
None
.Note that if this
PulseSequence
is concatenated with another one, some of the attributes might need to be calculated again, resulting in slower execution of the concatenation.
- get_control_matrix(omega: Sequence[float], show_progressbar: bool = False, cache_intermediates: bool = False) ndarray [source]
Get the control matrix for the frequencies omega. If it has been cached for the same frequencies, the cached version is returned, otherwise it is calculated from scratch.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies at which to evaluate the control matrix.
- show_progressbar: bool
Show a progress bar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep intermediate terms of the calculation that are also required by other computations.
- Returns
- control_matrix: ndarray, shape (n_nops, d**2, n_omega)
The control matrix for the noise operators.
- get_filter_function(omega: Sequence[float], which: str = 'fidelity', order: int = 1, show_progressbar: bool = False, cache_intermediates: bool = False) ndarray [source]
Get the first or second order filter function.
The filter function is cached so it doesn’t need to be calculated twice for the same frequencies.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies at which to evaluate the filter function.
- which: str, optional
Which filter function to return. Either ‘fidelity’ (default) or ‘generalized’ (see Notes). Only if
order == 1
.- order: int, optional
First or second order filter function.
- show_progressbar: bool, optional
Show a progress bar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep intermediate terms of the calculation that are also required by other computations.
- Returns
- filter_function: ndarray, shape (n_nops, n_nops, [d**2, d**2,] n_omega)
The filter function for each combination of noise operators as a function of omega.
Notes
The first-order generalized filter function is given by
\[F_{\alpha\beta,kl}(\omega) = \tilde{\mathcal{B}}_{\alpha k}^\ast(\omega) \tilde{\mathcal{B}}_{\beta l}(\omega),\]where \(\alpha,\beta\) are indices counting the noise operators \(B_\alpha\) and \(k,l\) indices counting the basis elements \(C_k\).
The fidelity filter function is obtained by tracing over the basis indices:
\[F_{\alpha\beta}(\omega) = \sum_{k} F_{\alpha\beta,kk}(\omega).\]
- get_filter_function_derivative(omega: Sequence[float], control_identifiers: Optional[Sequence[str]] = None, n_oper_identifiers: Optional[Sequence[str]] = None, n_coeffs_deriv: Optional[Sequence[Sequence[float]]] = None) ndarray [source]
Calculate the pulse sequence’s filter function derivative.
- Parameters
- omega: array_like, shape (n_omega,)
Frequencies at which the pulse control matrix is to be evaluated.
- control_identifiers: Sequence[str], shape (n_ctrl,)
Sequence of strings with the control identifiers to distinguish between control and drift Hamiltonian. The default is None, in which case the derivative is computed for all known non-noise operators.
- n_oper_identifiers: Sequence[str], shape (n_nops,)
Sequence of strings with the noise identifiers for which to compute the derivative contribution. The default is None, in which case it is computed for all known noise operators.
- n_coeffs_deriv: array_like, shape (n_nops, n_ctrl, n_dt)
The derivatives of the noise susceptibilities by the control amplitudes. The rows and columns should be in the same order as the corresponding identifiers above. Defaults to None, in which case the coefficients are assumed to be constant and hence their derivative vanishing.
Warning
Internally, control and noise terms of the Hamiltonian are stored alphanumerically sorted by their identifiers. If the noise and/or control identifiers above are not explicitly given, the rows and/or columns of this parameter need to be sorted in the same fashion.
- Returns
- filter_function_deriv: ndarray, shape (n_nops, n_t, n_ctrl, n_omega)
The regular filter functions’ derivatives for variation in each control contribution. Sorted in the same fashion as n_coeffs_deriv or, if not given, alphanumerically by the identifiers.
- get_pulse_correlation_control_matrix() ndarray [source]
Get the pulse correlation control matrix if it was cached.
- get_pulse_correlation_filter_function(which: str = 'fidelity') ndarray [source]
Get the pulse correlation filter function given by
\[F_{\alpha\beta}^{(gg')}(\omega) = e^{i\omega(t_{g-1} - t_{g'-1})} \tilde{\mathcal{B}}^{(g)}(\omega)\mathcal{Q}^{(g-1)} \mathcal{Q}^{(g'-1)\dagger} \tilde{\mathcal{B}}^{(g')\dagger}(\omega),\]where \(g,g'\) index the pulse in the sequence and \(\alpha,\beta\) index the noise operators, if it was computed during concatenation. Since the calculation requires the individual pulse’s control matrices and phase factors, which are not retained after concatenation, the pulse correlation filter function cannot be computed afterwards.
Note that the frequencies for which the filter function was calculated are not stored.
- Returns
- filter_function_pc: ndarray, shape (n_pls, n_pls, n_nops, n_nops, n_omega)
The pulse correlation filter function for each noise operator as a function of omega. The first two axes correspond to the pulses in the sequence, i.e. if the concatenated pulse sequence is \(C\circ B\circ A\), the first two axes are arranged like
\[\begin{split}F_{\alpha\beta}^{(gg')} &= \begin{pmatrix} F_{\alpha\beta}^{(AA)} & F_{\alpha\beta}^{(AB)} & F_{\alpha\beta}^{(AC)} \\ F_{\alpha\beta}^{(BA)} & F_{\alpha\beta}^{(BB)} & F_{\alpha\beta}^{(BC)} \\ F_{\alpha\beta}^{(CA)} & F_{\alpha\beta}^{(CB)} & F_{\alpha\beta}^{(CC)} \end{pmatrix}\end{split}\]for \(g,g'\in\{A, B, C\}\).
- get_total_phases(omega: Sequence[float]) ndarray [source]
Get the (cached) total phase factors for this pulse and omega.
- property nbytes: int
Return an estimate of the amount of memory consumed by this object (or, more precisely, the array attributes of this object).
- concatenate(pulses: Iterable[PulseSequence], calc_pulse_correlation_FF: bool = False, calc_filter_function: Optional[bool] = None, which: str = 'fidelity', omega: Optional[Sequence[float]] = None, show_progressbar: bool = False) PulseSequence [source]
Concatenate an arbitrary number of pulses. Note that pulses are concatenated left-to-right, that is,
\[\mathtt{concatenate((A, B))} \equiv B \circ A\]so that \(A\) is executed before \(B\) when applying the concatenated pulse.
- Parameters
- pulses: sequence of PulseSequences
The PulseSequence instances to be concatenated. If any of the instances have a cached filter function, the filter function for the composite pulse will also be calculated in order to make use of the speedup gained from concatenating the filter functions. If omega is given, calculation of the composite filter function is forced.
- calc_pulse_correlation_FF: bool, optional
Switch to control whether the pulse correlation filter function (see
PulseSequence.get_pulse_correlation_filter_function()
) is calculated. If omega is not given, the cached frequencies of all pulses need to be equal.- calc_filter_function: bool, optional
Switch to force the calculation of the filter function to be carried out or not. Overrides the automatic behavior of calculating it if at least one pulse has a cached control matrix. If
True
and no pulse has a cached control matrix, a list of frequencies must be supplied as omega.- which: str, optional
Which filter function to compute. Either ‘fidelity’ (default) or ‘generalized’ (see
PulseSequence.get_filter_function()
andPulseSequence.get_pulse_correlation_filter_function()
).- omega: array_like, optional
Frequencies at which to evaluate the (pulse correlation) filter functions. If
None
, an attempt is made to use cached frequencies.- show_progressbar: bool
Show a progress bar for the calculation of the control matrix.
- Returns
- pulse: PulseSequence
The concatenated pulse.
- concatenate_periodic(pulse: PulseSequence, repeats: int) PulseSequence [source]
Concatenate a pulse sequence pulse whose Hamiltonian is periodic repeats times. Although performing the same task, this function is much faster for concatenating many identical pulses with filter functions than
concatenate()
.Note that for large dimensions, the calculation of the control matrix using this function might be very memory intensive.
- Parameters
- pulse: PulseSequence
The
PulseSequence
instance to be repeated. If it has a cached filter function, the filter function for the new pulse will also be computed.- repeats: int
The number of repetitions
- Returns
- newpulse: PulseSequence
The concatenated
PulseSequence
See also
concatenate
Concatenate arbitrary PulseSequences.
Notes
The total control matrix is given by
\[\begin{split}\tilde{\mathcal{B}}(\omega) &= \tilde{\mathcal{B}}^{(1)}(\omega) \sum_{g=0}^{G-1} \left(e^{i\omega T}\right)^g \\ &= \tilde{\mathcal{B}}^{(1)}(\omega)\bigl( \mathbb{I} - e^{i\omega T} \mathcal{Q}^{(1)}\bigr)^{-1}\bigl( \mathbb{I} - \bigl(e^{i\omega T} \mathcal{Q}^{(1)}\bigr)^G\bigr).\end{split}\]with \(T\) the period of the control Hamiltonian and \(G\) the number of periods. The last equality is valid only if \(\mathbb{I} - e^{i\omega T}\mathcal{Q}^{(1)}\) is invertible.
- extend(pulse_to_qubit_mapping: Sequence[Sequence[Optional[Union[PulseSequence, Sequence[int], int, Mapping[str, str]]]]], N: Optional[int] = None, d_per_qubit: int = 2, additional_noise_Hamiltonian: Optional[Sequence[Sequence[Union[ndarray, Qobj, Sequence[float]]]]] = None, cache_diagonalization: Optional[bool] = None, cache_filter_function: Optional[bool] = None, omega: Optional[Sequence[float]] = None, show_progressbar: bool = False) PulseSequence [source]
Map one or more pulse sequences to different qubits.
- Parameters
- pulse_to_qubit_mapping: sequence of mapping tuples
A sequence of tuples with the first entry a
PulseSequence
instance and the second anint
or tuple ofint
s indicating the qubits that thePulseSequence
should be mapped to. A mapping of operator identifiers may optionally be given as a third element of each tuple. By default, the index of the qubit the operator is mapped to is appended to its identifier.Pulse sequences defined for multiple qubits may also be extended to non-neighboring qubits. Note that for multi-qubit pulses the order of the qubits is respected, i.e. mapping a pulse to (1, 0) is different from mapping it to (0, 1).
- N: int
The total number of qubits the new
PulseSequence
should be defined for. By default, this is inferred frompulse_to_qubit_mapping
.- d_per_qubit: int
The size of the Hilbert space a single qubit requires.
- additional_noise_Hamiltonian: list of lists
Additional noise operators and corresponding sensitivities for the new pulse sequence.
- cache_diagonalization: bool
Force diagonalizing the new pulse sequence. By default, diagonalization is cached if all pulses in
pulse_to_qubit_mapping
have been diagonalized since it is much cheaper to get the relevant quantities as tensor products from the mapped pulses instead of diagonalizing the new pulse.- cache_filter_function: bool
Force computing the filter functions for the new pulse sequence. Noise operators of individual pulses will be extended to the new Hilbert space. By default, this is done if all pulses in
pulse_to_qubit_mapping
have their filter functions cached.Note that extending the filter functions is only possible if they the mapped pulses are using a separable basis like the Pauli basis.
- omega: array_like
Frequencies for which to compute the filter functions if
cache_filter_function == True
. Defaults toNone
, in which case the cached frequencies of the individual pulses need to be the same.- show_progressbar: bool
Show a progress bar for the calculation of the control matrix.
- Returns
- newpulse: PulseSequence
The new pulse sequence on the larger qubit register. The noise operators (and possibly filter functions) are stored in the following order: first those of the multi-qubit pulses in the order they appeared in
pulse_to_qubit_mapping
, then those of the single-qubit pulses, and lastly any additional ones that may be given byadditional_noise_Hamiltonian
.
See also
remap
Map PulseSequence to a different qubit.
concatenate
Concatenate PulseSequences (in time).
concatenate_periodic
Periodically concatenate a PulseSequence.
Examples
>>> import filter_functions as ff >>> I, X, Y, Z = ff.util.paulis >>> X_pulse = ff.PulseSequence([[X, [np.pi/2], 'X']], ... [[X, [1], 'X'], [Z, [1], 'Z']], ... [1], basis=ff.Basis.pauli(1)) >>> XX_pulse = ff.extend([(X_pulse, 0), (X_pulse, 1)]) >>> XX_pulse.d 4 >>> XIX_pulse_1 = ff.extend([(X_pulse, 0), (X_pulse, 2)]) >>> XIX_pulse_1.d 8 >>> XXI_pulse = ff.extend([(X_pulse, 0), (X_pulse, 1)], N=3) >>> XXI_pulse.d 8
Filter functions are automatically cached if they are for mapped pulses:
>>> omega = ff.util.get_sample_frequencies(X_pulse) >>> X_pulse.cache_filter_function(omega) >>> XX_pulse = ff.extend([(X_pulse, 0), (X_pulse, 1)]) >>> XX_pulse.is_cached('filter_function') True
This behavior can also be overriden manually:
>>> XX_pulse = ff.extend([(X_pulse, 0), (X_pulse, 1)], ... cache_filter_function=False) >>> XX_pulse.is_cached('filter_function') False
Mapping pulses to non-neighboring qubits is also possible:
>>> Y_pulse = ff.PulseSequence([[Y, [np.pi/2], 'Y']], ... [[Y, [1], 'Y'], [Z, [1], 'Z']], ... [1], basis=ff.Basis.pauli(1)) >>> XXY_pulse = ff.extend([(XX_pulse, (0, 1)), (Y_pulse, 2)]) >>> XYX_pulse = ff.extend([(XX_pulse, (0, 2)), (Y_pulse, 1)])
Additionally, pulses can have the order of the qubits they are defined for permuted (see
remap()
):>>> Z_pulse = ff.PulseSequence([[Z, [np.pi/2], 'Z']], [[Z, [1], 'Z']], ... [1], basis=ff.Basis.pauli(1)) >>> XY_pulse = ff.extend([(X_pulse, 0), (Y_pulse, 1)]) >>> YZX_pulse = ff.extend([(XY_pulse, (2, 0)), (Z_pulse, 1)])
Control and noise operator identifiers can be mapped according to a specified mapping:
>>> YX_pulse = ff.extend([(X_pulse, 1, {'X': 'IX', 'Z': 'IZ'}), ... (Y_pulse, 0, {'Y': 'YI', 'Z': 'ZI'})]) >>> YX_pulse.c_oper_identifiers array(['IX', 'YI'], dtype='<U2') >>> YX_pulse.n_oper_identifiers array(['IX', 'IZ', 'YI', 'ZI'], dtype='<U2')
We can also add an additional noise Hamiltonian:
>>> H_n = [[ff.util.tensor(Z, Z, Z), [1], 'ZZZ']] >>> XYX_pulse = ff.extend([(XX_pulse, (0, 2)), (Y_pulse, 1)], ... additional_noise_Hamiltonian=H_n) >>> 'ZZZ' in XYX_pulse.n_oper_identifiers True
- remap(pulse: PulseSequence, order: Sequence[int], d_per_qubit: int = 2, oper_identifier_mapping: Optional[Mapping[str, str]] = None) PulseSequence [source]
Remap a PulseSequence by changing the order of qubits in the register. Cached attributes are automatically attempted to be retained.
Caution
This function simply permutes the order of the tensor product elements of control and noise operators. Thus, the resultant pulse will have its filter functions defined for different noise operators than the original one.
- Parameters
- pulse: PulseSequence
The pulse whose qubit order should be permuted.
- order: sequence of ints
A list of permutation indices. E.g., if pulse is defined for two qubits,
order == [1, 0]
will reverse the order of qubits.- d_per_qubit: int (default: 2)
The size of the Hilbert space a single qubit inhabitates.
- oper_identifier_mapping: dict_like
A mapping that maps operator identifiers from the old pulse to the remapped pulse. The default is the identity mapping.
- Returns
- remapped_pulse: PulseSequence
A new
PulseSequence
instance with the order of the qubits permuted according to order.
See also
extend
Map PulseSequences to composite Hilbert spaces.
util.tensor_transpose
Transpose the order of a tensor product.
Examples
>>> X, Y = util.paulis[1:3] >>> XY, YX = util.tensor(X, Y), util.tensor(Y, X) >>> pulse = PulseSequence([[XY, [np.pi/2], 'XY']], [[YX, [1], 'YX']], [1], ... Basis.pauli(2)) >>> mapping = {'XY': 'YX', 'YX': 'XY'} >>> remapped_pulse = remap(pulse, (1, 0), oper_identifier_mapping=mapping) >>> target_pulse = PulseSequence([[YX, [np.pi/2], 'YX']], ... [[XY, [1], 'XY']], [1], Basis.pauli(2)) >>> remapped_pulse == target_pulse True
Caching of attributes is automatically handled >>> remapped_pulse.is_cached(‘filter_function’) False >>> pulse.cache_filter_function(util.get_sample_frequencies(pulse)) >>> remapped_pulse = remap(pulse, (1, 0)) >>> remapped_pulse.is_cached(‘filter_function’) True
2.7. filter_functions.gradient module
This module implements functions to calculate filter function and infidelity derivatives. Currently only auto-correlated noise (i.e. no cross-correlations) is implemented.
- Throughout this documentation the following notation will be used:
n_dt denotes the number of time steps,
n_cops the number of all control operators,
n_ctrl the number of accessible control operators (if identifiers are provided, otherwise n_ctrl=n_cops),
n_nops the number of noise operators,
n_omega the number of frequency samples, and
d the dimension of the system.
2.7.1. Functions
calculate_derivative_of_control_matrix_from_scratch()
Calculate the derivative of the control matrix from scratch.
calculate_canonical_filter_function_derivative()
Compute the filter function derivative from the control matrix.
infidelity_derivative()
Calculate the infidelity derivative.
- calculate_derivative_of_control_matrix_from_scratch(omega: Sequence[float], propagators: ndarray, eigvals: ndarray, eigvecs: ndarray, basis: Basis, t: Sequence[float], dt: Sequence[float], n_opers: Sequence[Union[ndarray, Qobj]], n_coeffs: Sequence[Sequence[float]], c_opers: Sequence[Union[ndarray, Qobj]], n_coeffs_deriv: Optional[Sequence[Sequence[float]]] = None, intermediates: Optional[Dict[str, ndarray]] = None) ndarray [source]
Calculate the derivative of the control matrix from scratch.
- Parameters
- omega: array_like, shape (n_omega,)
Frequencies, at which the pulse control matrix is to be evaluated.
- propagators: array_like, shape (n_dt+1, d, d)
The propagators \(Q_g = P_g P_{g-1}\cdots P_0\) as a (d, d) array with d the dimension of the Hilbert space.
- eigvals: array_like, shape (n_dt, d)
Eigenvalue vectors for each time pulse segment g with the first axis counting the pulse segment, i.e.
HD == array([D_0, D_1, ...])
.- eigvecs: array_like, shape (n_dt, d, d)
Eigenvector matrices for each time pulse segment g with the first axis counting the pulse segment, i.e.
HV == array([V_0, V_1, ...])
.- basis: Basis, shape (d**2, d, d)
The basis elements, in which the pulse control matrix will be expanded.
- t: array_like, shape (n_dt+1), optional
The absolute times of the different segments.
- dt: array_like, shape (n_dt)
Sequence duration, i.e. for the \(g\)-th pulse \(t_g - t_{g-1}\).
- n_opers: array_like, shape (n_nops, d, d)
Noise operators \(B_\alpha\).
- n_coeffs: array_like, shape (n_nops, n_dt)
The sensitivities of the system to the noise operators given by n_opers at the given time step.
- c_opers: array_like, shape (n_ctrl, d, d)
Control operators \(H_k\) with respect to which the derivative is computed.
- n_coeffs_deriv: array_like, shape (n_nops, n_ctrl, n_dt)
The derivatives of the noise susceptibilities by the control amplitudes. Defaults to None.
- intermediates: Dict[str, ndarray], optional
Optional dictionary containing intermediate results of the calculation of the control matrix.
- Returns
- ctrlmat_deriv: ndarray, shape (n_ctrl, n_omega, n_dt, n_nops, d**2)
Partial derivatives of the total control matrix with respect to each control direction \(\frac{\partial\mathcal{B}_{\alpha k}(\omega)}{\partial u_h(t_{g'})}\).
See also
_liouville_derivative
_control_matrix_at_timestep_derivative
Notes
The derivative of the control matrix is calculated according to
\[\frac{\partial\mathcal{B}_{\alpha k}(\omega)}{\partial u_h(t_{g'})} = \sum_{g=1}^G \mathrm{e}^{\mathrm{i}\omega t_{g-1}}\cdot\left(\sum_j \left[\frac{\partial\mathcal{B}_{\alpha j}^{(g)}(\omega)} {\partial u_h(t_{g'})} \cdot \mathcal{Q}_{jk}^{(g-1)} + \mathcal{B}_{\alpha j}^{(g)}(\omega) \cdot\frac{\partial \mathcal{Q}_{jk}^{(g-1)}} {\partial u_h(t_{g'})} \right] \right)\]
- calculate_filter_function_derivative(ctrlmat: ndarray, ctrlmat_deriv: ndarray) ndarray [source]
Compute the filter function derivative from the control matrix.
- Parameters
- ctrlmat: array_like, shape (n_nops, d**2, n_omega)
The control matrix.
- ctrlmat_deriv: array_like, shape (n_nops, d**2, n_t, n_ctrl, n_omega)
The derivative of the control matrix.
- Returns
- filter_function_deriv: ndarray, shape (n_nops, n_dt, n_ctrl, n_omega)
The regular filter functions’ derivatives for variation in each control contribution \(\frac{\partial F_\alpha(\omega)}{\partial u_h(t_{g'})}\).
Notes
The filter function derivative is calculated according to
\[\frac{\partial F_\alpha(\omega)}{\partial u_h(t_{g'})} = 2 \mathrm{Re}\left(\sum_k \mathcal{B}_{\alpha k}^\ast(\omega) \frac{\partial\mathcal{B}_{\alpha k}(\omega)} {\partial u_h(t_{g'})} \right)\]
- infidelity_derivative(pulse: PulseSequence, spectrum: Sequence[float], omega: Sequence[float], control_identifiers: Optional[Sequence[str]] = None, n_oper_identifiers: Optional[Sequence[str]] = None, n_coeffs_deriv: Optional[Sequence[Sequence[float]]] = None) ndarray [source]
Calculate the entanglement infidelity derivative of the
PulseSequence
pulse.- Parameters
- pulse: PulseSequence
The
PulseSequence
instance for which to calculate the infidelity.- spectrum: array_like, shape ([[n_nops,] n_nops,] omega)
The two-sided noise power spectral density in units of inverse frequencies as an array of shape (n_omega,), (n_nops, n_omega), or (n_nops, n_nops, n_omega). In the first case, the same spectrum is taken for all noise operators, in the second, it is assumed that there are no correlations between different noise sources and thus there is one spectrum for each noise operator. In the third and most general case, there may be a spectrum for each pair of noise operators corresponding to the correlations between them. n_nops is the number of noise operators considered and should be equal to
len(n_oper_identifiers)
.- omega: array_like, shape (n_omega,)
The frequencies at which the integration is to be carried out.
- control_identifiers: Sequence[str], shape (n_ctrl,)
Sequence of strings with the control identifiers to distinguish between control and drift Hamiltonian. The default is None, in which case the derivative is computed for all known non-noise operators.
- n_oper_identifiers: Sequence[str], shape (n_nops,)
Sequence of strings with the noise identifiers for which to compute the derivative contribution. The default is None, in which case it is computed for all known noise operators.
- n_coeffs_deriv: array_like, shape (n_nops, n_ctrl, n_dt)
The derivatives of the noise susceptibilities by the control amplitudes. The rows and columns should be in the same order as the corresponding identifiers above. Defaults to None, in which case the coefficients are assumed to be constant and hence their derivative vanishing.
Warning
Internally, control and noise terms of the Hamiltonian are stored alphanumerically sorted by their identifiers. If the noise and/or control identifiers above are not explicitly given, the rows and/or columns of this parameter need to be sorted in the same fashion.
- Returns
- infid_deriv: ndarray, shape (n_nops, n_dt, n_ctrl)
Array with the derivative of the infidelity for each noise source taken for each control direction at each time step \(\frac{\partial I_e}{\partial u_h(t_{g'})}\). Sorted in the same fashion as n_coeffs_deriv or, if not given, alphanumerically by the identifiers.
- Raises
- ValueError
If the provided noise spectral density does not fit expected shape.
Notes
The infidelity’s derivative is given by
\[\frac{\partial I_e}{\partial u_h(t_{g'})} = \frac{1}{d} \int_{-\infty}^\infty \frac{d\omega}{2\pi} S_\alpha(\omega) \frac{\partial F_\alpha(\omega)} {\partial u_h(t_{g'})}\]with \(S_{\alpha}(\omega)\) the noise spectral density and \(F_{\alpha}(\omega)\) the canonical filter function for noise source \(\alpha\).
To convert to the average gate infidelity, use the following relation given by Horodecki et al. [Hor99] and Nielsen [Nie02]:
\[\big\langle\mathcal{I}_\mathrm{avg}\big\rangle = \frac{d}{d+1} \big\langle\mathcal{I}_\mathrm{e}\big\rangle.\]References
- Hor99
Horodecki, M., Horodecki, P., & Horodecki, R. (1999). General teleportation channel, singlet fraction, and quasidistillation. Physical Review A - Atomic, Molecular, and Optical Physics, 60(3), 1888–1898. https://doi.org/10.1103/PhysRevA.60.1888
- Nie02
Nielsen, M. A. (2002). A simple formula for the average gate fidelity of a quantum dynamical operation. Physics Letters, Section A: General, Atomic and Solid State Physics, 303(4), 249–252. https://doi.org/10.1016/S0375-9601(02)01272-0
2.8. filter_functions.superoperator module
This module provides some functions related to superoperators and quantum maps.
2.8.1. Functions
liouville_representation()
Calculate the Liouville representation of a unitary with respect to a basis
liouville_to_choi()
Convert from Liouville to Choi matrix representation.
liouville_is_CP()
Check if superoperator in Liouville representation is completely positive.
liouville_is_cCP()
Check if superoperator in Liouville representation is conditional CP.
- liouville_is_CP(superoperator: ndarray, basis: Basis, return_eig: Optional[bool] = False, atol: Optional[float] = None) Union[bool, Tuple[bool, Tuple[ndarray, ndarray]]] [source]
Test if a Liouville superoperator is completely positive (CP).
- Parameters
- superoperator: ndarray, shape (…, d**2, d**2)
The superoperator in Liouville representation to be checked for CPness.
- basis: Basis, shape (d**2, d, d)
The operator basis defining the Liouville representation.
- return_eig: bool, optional
Return the tuple of eigenvalues and eigenvectors of the Choi matrix. The default is False.
- atol: float, optional
Absolute tolerance for the complete positivity.
- Returns
- CP: bool, (shape (…,))
The (array, if broadcasted) of bools indicating if superoperator is CP.
- (D, V): Tuple[ndarray, ndarray]
The eigenvalues and eigenvectors of the Choi matrix (only if return_eig is True).
See also
liouville_representation
Calculate Liouville representation of a unitary.
Liouville_to_choi
Convert from Liouville to Choi matrix representation.
liouville_is_cCP
Test if a superoperator is conditional CP.
Notes
A superoperator \(\mathcal{S}\) is completely positive (CP) if and only if its Choi matrix representation is positive semidefinite:
\[\mathcal{S}\text{ is CP }\Leftrightarrow \mathrm{choi}(\mathcal{S})\geq 0.\]
- liouville_is_cCP(superoperator: ndarray, basis: Basis, return_eig: Optional[bool] = False, atol: Optional[float] = None) Union[bool, Tuple[bool, Tuple[ndarray, ndarray]]] [source]
Test if a Liouville superoperator is conditional completely positive.
- Parameters
- superoperator: ndarray, shape (…, d**2, d**2)
The superoperator in Liouville representation to be checked for cCPness
- basis: Basis, shape (d**2, d, d)
The operator basis defining the Liouville representation.
- return_eig: bool, optional
Return the tuple of eigenvalues and eigenvectors of the Choi matrix projected on the complement of the maximally entangled state. The default is False.
- atol: float, optional
Absolute tolerance for the complete positivity.
- Returns
- cCP: bool, (shape (…,))
The (array, if broadcasted) of bools indicating if superoperator is cCP
- (D, V): Tuple[ndarray, ndarray]
The eigenvalues and eigenvectors of the projected Choi matrix (only if return_eig is True).
See also
liouville_representation
Calculate Liouville representation of a unitary.
Liouville_to_choi
Convert from Liouville to Choi matrix representation.
liouville_is_CP
Test if a superoperator is CP.
Notes
A superoperator \(\mathcal{S}\) is conditional completely positive (cCP) if and only if its Choi matrix projected on the complement of the maximally entangled state is positive semidefinite:
\[\mathcal{S}\text{ is cCP }\Leftrightarrow Q\mathrm{choi}(\mathcal{S})Q\geq 0\]with \(Q = \mathbb{I} - |\Omega\rangle\langle\Omega|\).
- liouville_representation(U: ndarray, basis: Basis) ndarray [source]
Get the Liouville representaion of the unitary U with respect to the basis.
- Parameters
- U: ndarray, shape (…, d, d)
The unitary.
- basis: Basis, shape (d**2, d, d)
The basis used for the representation, e.g. a Pauli basis.
- Returns
- R: ndarray, shape (…, d**2, d**2)
The Liouville representation of U.
Notes
The Liouville representation of a unitary quantum operation \(\mathcal{U}:\rho\rightarrow U\rho U^\dagger\) is given by
\[\mathcal{U}_{ij} = \mathrm{tr}(C_i U C_j U^\dagger)\]with \(C_i\) elements of the basis spanning \(\mathbb{C}^{d\times d}\) with \(d\) the dimension of the Hilbert space.
- liouville_to_choi(superoperator: ndarray, basis: Basis) ndarray [source]
Convert from Liouville to Choi matrix representation.
- Parameters
- superoperator: ndarray, shape (…, d**2, d**2)
The Liouville representation of a superoperator.
- basis: Basis, shape (d**2, d, d)
The operator basis defining the Liouville representation.
- Returns
- choi: ndarray, shape (…, d**2, d**2)
The Choi matrix representation of the superoperator.
See also
liouville_representation
Calculate Liouville representation of a unitary.
liouville_is_CP
Test if a superoperator is completely positive (CP).
liouville_is_cCP
Test if a superoperator is conditional CP.
Notes
The Choi matrix is given by
\[\begin{split}\mathrm{choi}(\mathcal{S}) &= (\mathbb{I}\otimes\mathcal{S}) (|\Omega\rangle\langle\Omega|) \\ &= \sum_{ij} E_{ij}\otimes\mathcal{S}(E_{ij}) \\ &= \sum_{ij}\mathcal{S}_{ij} C_j^T\otimes C_i\end{split}\]where \(|\Omega\rangle\) is a maximally entangled state, \(E_{ij} = |i\rangle\langle j|\), and \(C_i\) are the basis elements that define the Liouville representation \(\mathcal{S}_{ij}\) [Mer13].
References
- Mer13
Merkel, S. T. et al. Self-consistent quantum process tomography. Physical Review A - Atomic, Molecular, and Optical Physics, 87, 062119 (2013). https://doi.org/10.1103/PhysRevA.87.062119
2.9. filter_functions.types module
Defines custom types for the package.
2.10. filter_functions.util module
This module provides various helper functions.
2.10.1. Functions
abs2()
Absolute value squared
get_indices_from_identifiers()
The the indices of a subset of identifiers within a list of identifiers.
tensor()
Fast, flexible tensor product of an arbitrary number of inputs using
einsum()
tensor_insert()
For an array that is known to be a tensor product, insert arrays at a given position in the product chain
tensor_merge()
For two arrays that are tensor products of known dimensions, merge them at arbitary positions in the product chain
tensor_transpose()
For a tensor product, transpose the order of the constituents in the product chain
mdot()
Multiple matrix product
remove_float_errors()
Set entries whose absolute value is below a certain threshold to zero
oper_equiv()
Determine if two vectors or operators are equal up to a global phase
dot_HS()
Hilbert-Schmidt inner product
get_sample_frequencies()
Get frequencies with typical infrared and ultraviolet cutoffs for a
PulseSequence
progressbar()
A progress bar for loops. Uses tqdm if available and a simple custom one if not.
hash_array_along_axis()
Return a list of hashes along a given axis
all_array_equal()
Check if all arrays in an iterable are equal
2.10.2. Exceptions
CalculationError
Exception raised if trying to fetch the pulse correlation function when it was not computed during concatenation
- abs2(x: ndarray) ndarray [source]
Fast function to calculate the absolute value squared,
\[|\cdot|^2 := \Re(\cdot)^2 + \Im(\cdot)^2\]Equivalent to:
np.abs(x)**2
- all_array_equal(it: Iterable) bool [source]
Return
True
if all array elements ofit
are equal by hashing the bytes representation of each array. Note that this is not thread-proof.
- dot_HS(U: Union[ndarray, Qobj], V: Union[ndarray, Qobj], eps: Optional[float] = None) Union[float, complex, ndarray] [source]
Return the Hilbert-Schmidt inner product of U and V,
\[\langle U, V\rangle_\mathrm{HS} := \mathrm{tr}(U^\dagger V).\]- Parameters
- U, V: qutip.Qobj or ndarray
Objects to compute the inner product of.
- eps: float
The floating point precision. The result is rounded to abs(int(np.log10(eps))) decimals if eps > 0.
- Returns
- result: float, complex
The result rounded to precision eps.
Examples
>>> U, V = paulis[1:3] >>> dot_HS(U, V) 0.0 >>> dot_HS(U, U) 2.0
- get_sample_frequencies(pulse: PulseSequence, n_samples: int = 300, spacing: str = 'log', include_quasistatic: bool = False, omega_min: Optional[float] = None, omega_max: Optional[float] = None) ndarray [source]
Get n_samples sample frequencies spaced ‘linear’ or ‘log’.
The ultraviolet cutoff is taken to be one order of magnitude larger than the timescale of the pulse tau. In the case of log spacing, the values are clipped in the infrared at two orders of magnitude below the timescale of the pulse.
- Parameters
- pulse: PulseSequence
The pulse to get frequencies for.
- n_samples: int, optional
The number of frequency samples. Default is 300.
- spacing: str, optional
The spacing of the frequencies. Either ‘log’ or ‘linear’, default is ‘log’.
- include_quasistatic: bool, optional
Include zero frequency. Default is False.
- omega_min, omega_max: float, optional
Minimum and maximum angular frequencies included (DC notwithstanding). Default to \(2\pi\times 10^{-2}/\tau\) and \(2\pi\times 10^{+1}/\Delta t_{\mathrm{min}}\).
- Returns
- omega: ndarray
The angular frequencies.
- hash_array_along_axis(arr: ndarray, axis: int = 0) List[int] [source]
Return the hashes of arr along the first axis
- oper_equiv(psi: Union[ndarray, Qobj], phi: Union[ndarray, Qobj], eps: Optional[float] = None, normalized: bool = False) Tuple[bool, float] [source]
Checks whether psi and phi are equal up to a global phase, i.e.
\[|\psi\rangle = e^{i\chi}|\phi\rangle \Leftrightarrow \langle \phi|\psi\rangle = e^{i\chi},\]and returns the phase. If the first return value is false, the second is meaningless in this context. psi and phi can also be operators.
- Parameters
- psi, phi: qutip.Qobj or array_like
Vectors or operators to be compared
- eps: float
The tolerance below which the two objects are treated as equal, i.e., the function returns
True
ifabs(1 - modulus) <= eps
.- normalized: bool
Flag indicating if psi and phi are normalized with respect to the Hilbert-Schmidt inner product
dot_HS()
.
Examples
>>> psi = paulis[1] >>> phi = paulis[1]*np.exp(1j*1.2345) >>> oper_equiv(psi, phi) (True, 1.2345)
- progressbar(iterable: Iterable, *args, **kwargs)[source]
Progress bar for loops. Uses tqdm.
Usage:
for i in progressbar(range(10)): do_something()
- remove_float_errors(arr: ndarray, eps_scale: Optional[float] = None) ndarray [source]
Clean up arr by removing floating point numbers smaller than the dtype’s precision multiplied by eps_scale. Treats real and imaginary parts separately.
Obviously only works for arrays with norm ~1.
- tensor(*args, rank: int = 2, optimize: Union[bool, str] = False) ndarray [source]
Fast, flexible tensor product using einsum. The product is taken over the last rank axes and broadcast over the remaining axes which thus need to follow numpy broadcasting rules. Note that vectors are treated as rank 2 tensors with shape (1, x) or (x, 1).
For example, the following shapes are compatible:
rank == 2
(e.g. matrices or vectors):(a, b, c, d, d), (a, b, c, e, e) -> (a, b, c, d*e, d*e) (a, b, c), (a, d, e) -> (a, b*d, c*e) (a, b), (c, d, e) -> (c, a*d, b*e) (1, a), (b, 1, c) -> (b, 1, a*c)
rank == 1
:(a, b), (a, c) -> (a, b*c) (a, b, 1), (a, c) -> (a, b, c)
- Parameters
- args: array_like
The elements of the tensor product
- rank: int, optional (default: 2)
The rank of the tensors. E.g., for a Kronecker product between two matrices
rank == 2
. The remaining axes are broadcast over.- optimize: bool|str, optional (default: False)
Optimize the tensor contraction order. Passed through to
numpy.einsum()
.
See also
numpy.kron
NumPy tensor product.
tensor_insert
Insert array at given position in tensor product chain.
tensor_merge
Merge tensor product chains.
tensor_transpose
Transpose the order of a tensor product chain.
Examples
>>> Z = np.diag([1, -1]) >>> np.array_equal(tensor(Z, Z), np.kron(Z, Z)) True
>>> A, B = np.arange(2), np.arange(2, 5) >>> tensor(A, B, rank=1) array([[0, 0, 0, 2, 3, 4]])
>>> args = np.random.randn(4, 10, 3, 2) >>> result = tensor(*args, rank=1) >>> result.shape == (10, 3, 2**4) True >>> result = tensor(*args, rank=2) >>> result.shape == (10, 3**4, 2**4) True
>>> A, B = np.random.randn(1, 3), np.random.randn(3, 4) >>> result = tensor(A, B) >>> result.shape == (1*3, 3*4) True
>>> A, B = np.random.randn(3, 1, 2), np.random.randn(2, 2, 2) >>> try: ... result = tensor(A, B, rank=2) ... except ValueError as err: # cannot broadcast over axis 0 ... print(err) Incompatible shapes (3, 1, 2) and (2, 2, 2) for tensor product of rank 2. >>> result = tensor(A, B, rank=3) >>> result.shape == (3*2, 1*2, 2*2) True
- tensor_insert(arr: ndarray, *args, pos: Union[int, Sequence[int]], arr_dims: Sequence[Sequence[int]], rank: int = 2, optimize: Union[bool, str] = False) ndarray [source]
For a tensor product arr, insert args into the product chain at pos. E.g, if \(\verb|arr|\equiv A\otimes B\otimes C\) and \(\verb|pos|\equiv 2\), the result will be the tensor product
\[A\otimes B\otimes\left[\bigotimes_{X\in\verb|args|}X\right] \otimes C.\]This function works in a similar way to
numpy.insert()
and the following would be functionally equivalent in the case that the constituent tensors of the product arr are known:>>> tensor_insert(tensor(*arrs, rank=rank), *args, pos=pos, arr_dims=..., ... rank=rank)
>>> tensor(*np.insert(arrs, pos, args, axis=0), rank=rank)
- Parameters
- arr: ndarray
The tensor product in whose chain the other args should be inserted
- *args: ndarray
The tensors to be inserted in the product chain
- pos: int|sequence of ints
The position(s) at which the args are inserted in the product chain. If an int and
len(args) > 1
, it is repeated so that all args are inserted in a row. If a sequence, it should indicate the indices in the original tensor product chain that led to arr before which args should be inserted.- arr_dims: array_like, shape (rank, n_const)
The last rank dimensions of the n_const constituent tensors of the tensor product arr as a list of lists with the list at position i containing the i-th relevant dimension of all args. Since the remaing axes are broadcast over, their shape is irrelevant.
For example, if
arr = tensor(a, b, c, rank=2)
anda,b,c
have shapes(2, 3, 4), (5, 2, 2, 1), (2, 2)
,arr_dims = [[3, 2, 2], [4, 1, 2]]
.- rank: int, optional (default: 2)
The rank of the tensors. E.g., for a Kronecker product between two vectors,
rank == 1
, and between two matricesrank == 2
. The remaining axes are broadcast over.- optimize: bool|str, optional (default: False)
Optimize the tensor contraction order. Passed through to
numpy.einsum()
.
See also
numpy.insert
NumPy array insertion with similar syntax.
numpy.kron
NumPy tensor product.
tensor_insert
Insert array at given position in tensor product chain.
tensor_merge
Merge tensor product chains.
tensor_transpose
Transpose the order of a tensor product chain.
Examples
>>> I, X, Y, Z = paulis >>> arr = tensor(X, I) >>> r = tensor_insert(arr, Y, Z, arr_dims=[[2, 2], [2, 2]], pos=0) >>> np.allclose(r, tensor(Y, Z, X, I)) True >>> r = tensor_insert(arr, Y, Z, arr_dims=[[2, 2], [2, 2]], pos=1) >>> np.allclose(r, tensor(X, Y, Z, I)) True >>> r = tensor_insert(arr, Y, Z, arr_dims=[[2, 2], [2, 2]], pos=2) >>> np.allclose(r, tensor(X, I, Y, Z)) True
Other ranks and different dimensions:
>>> from numpy.random import randn >>> A, B, C = randn(2, 3, 1, 2), randn(2, 2, 2, 2), randn(3, 2, 1) >>> arr = tensor(A, C, rank=3) >>> r = tensor_insert(arr, B, pos=1, rank=3, ... arr_dims=[[3, 3], [1, 2], [2, 1]]) >>> np.allclose(r, tensor(A, B, C, rank=3)) True
>>> arrs, args = randn(2, 2, 2), randn(2, 2, 2) >>> arr_dims = [[2, 2], [2, 2]] >>> r = tensor_insert(tensor(*arrs), *args, pos=(0, 1), arr_dims=arr_dims) >>> np.allclose(r, tensor(args[0], arrs[0], args[1], arrs[1])) True >>> r = tensor_insert(tensor(*arrs), *args, pos=(0, 0), arr_dims=arr_dims) >>> np.allclose(r, tensor(*args, *arrs)) True >>> r = tensor_insert(tensor(*arrs), *args, pos=(1, 2), arr_dims=arr_dims) >>> np.allclose(r, tensor(*np.insert(arrs, (1, 2), args, axis=0))) True
- tensor_merge(arr: ndarray, ins: ndarray, pos: Sequence[int], arr_dims: Sequence[Sequence[int]], ins_dims: Sequence[Sequence[int]], rank: int = 2, optimize: Union[bool, str] = False) ndarray [source]
For two tensor products arr and ins, merge ins into the product chain at indices pos. E.g, if \(\verb|arr|\equiv A\otimes B\otimes C\), \(\verb|ins|\equiv D\otimes E\), and \(\verb|pos|\equiv [1, 2]\), the result will be the tensor product
\[A\otimes D\otimes B\otimes E\otimes C.\]This function works in a similar way to
numpy.insert()
andtensor_insert()
.- Parameters
- arr: ndarray
The tensor product in whose chain the other args should be inserted
- ins: ndarray
The tensor product to be inserted in the product chain
- pos: sequence of ints
The positions at which the constituent tensors of ins are inserted in the product chain. Should indicate the indices in the original tensor product chain that led to arr before which the constituents of ins should be inserted.
- arr_dims: array_like, shape (rank, n_const)
The last rank dimensions of the n_const constituent tensors of the tensor product arr as a list of lists with the list at position i containing the i-th relevant dimension of all args. Since the remaing axes are broadcast over, their shape is irrelevant.
For example, if
arr = tensor(a, b, c, rank=2)
anda,b,c
have shapes(2, 3, 4), (5, 2, 2, 1), (2, 2)
,arr_dims = [[3, 2, 2], [4, 1, 2]]
.- ins_dims: array_like, shape (rank, n_const)
The last rank dimensions of the n_const constituent tensors of the tensor product ins as a list of lists with the list at position i containing the i-th relevant dimension of ins. Since the remaing axes are broadcast over, their shape is irrelevant.
- rank: int, optional (default: 2)
The rank of the tensors. E.g., for a Kronecker product between two vectors,
rank == 1
, and between two matricesrank == 2
. The remaining axes are broadcast over.- optimize: bool|str, optional (default: False)
Optimize the tensor contraction order. Passed through to
numpy.einsum()
.
See also
numpy.insert
NumPy array insertion with similar syntax.
numpy.kron
NumPy tensor product.
tensor
Fast tensor product with broadcasting.
tensor_insert
Insert array at given position in tensor product chain.
tensor_transpose
Transpose the order of a tensor product chain.
Examples
>>> I, X, Y, Z = paulis >>> arr = tensor(X, Y, Z) >>> ins = tensor(I, I) >>> r1 = tensor_merge(arr, ins, pos=[1, 2], arr_dims=[[2]*3, [2]*3], ... ins_dims=[[2]*2, [2]*2]) >>> np.allclose(r1, tensor(X, I, Y, I, Z)) True >>> r2 = tensor_merge(ins, arr, pos=[0, 1, 2], arr_dims=[[2]*2, [2]*2], ... ins_dims=[[2]*3, [2]*3]) >>> np.allclose(r1, r2) True
tensor_insert()
can provide the same functionality in some cases:>>> arr = tensor(Y, Z) >>> ins = tensor(I, X) >>> r1 = tensor_merge(arr, ins, pos=[0, 0], arr_dims=[[2]*2, [2]*2], ... ins_dims=[[2]*2, [2]*2]) >>> r2 = tensor_insert(arr, I, X, pos=[0, 0], arr_dims=[[2]*2, [2]*2]) >>> np.allclose(r1, r2) True
Also tensors of rank other than 2 and numpy broadcasting are supported:
>>> arr = np.random.randn(2, 10, 3, 4) >>> ins = np.random.randn(2, 10, 3, 2) >>> r = tensor_merge(tensor(*arr, rank=1), tensor(*ins, rank=1), [0, 1], ... arr_dims=[[4, 4]], ins_dims=[[2, 2]], rank=1) >>> np.allclose(r, tensor(ins[0], arr[0], ins[1], arr[1], rank=1)) True
- tensor_transpose(arr: ndarray, order: Sequence[int], arr_dims: Sequence[Sequence[int]], rank: int = 2) ndarray [source]
Transpose the order of a tensor product chain.
- Parameters
- arr: ndarray
The tensor product whose chain should be reordered.
- order: sequence of ints
The transposition order. If
arr == tensor(A, B)
andorder == (1, 0)
, the result will betensor(B, A)
.- arr_dims: array_like, shape (rank, n_const)
The last rank dimensions of the n_const constituent tensors of the tensor product arr as a list of lists with the list at position i containing the i-th relevant dimension of all args. Since the remaing axes are broadcast over, their shape is irrelevant.
For example, if
arr = tensor(a, b, c, rank=2)
anda,b,c
have shapes(2, 3, 4), (5, 2, 2, 1), (2, 2)
,arr_dims = [[3, 2, 2], [4, 1, 2]]
.- rank: int, optional (default: 2)
The rank of the tensors. E.g., for a Kronecker product between two vectors,
rank == 1
, and between two matricesrank == 2
. The remaining axes are broadcast over.
- Returns
- transposed_arr: ndarray
The tensor product arr with its order transposed according to order
See also
numpy.insert
NumPy array insertion with similar syntax.
numpy.kron
NumPy tensor product.
tensor
Fast tensor product with broadcasting.
tensor_insert
Insert array at given position in tensor product chain.
tensor_merge
Merge tensor product chains.
Examples
>>> I, X, Y, Z = paulis >>> arr = tensor(X, Y, Z) >>> transposed = tensor_transpose(arr, [1, 2, 0], arr_dims=[[2, 2, 2]]*2) >>> np.allclose(transposed, tensor(Y, Z, X)) True
2.11. Module contents
Package for efficient calculation of generalized filter functions
- class Basis(basis_array: Sequence, traceless: Optional[bool] = None, btype: Optional[str] = None, labels: Optional[Sequence[str]] = None)[source]
Bases:
ndarray
Class for operator bases. There are several ways to instantiate a Basis object:
by just calling this constructor with a (possibly incomplete) array of basis matrices. No checks regarding orthonormality or hermiticity are performed.
by calling one of the classes alternative constructors (classmethods):
pauli()
: Pauli operator basisggm()
: Generalized Gell-Mann basisfrom_partial()
Generate an complete basis from partial elements
These bases guarantee the following properties:
hermitian
orthonormal
[traceless] (can be controlled by a flag)
Since Basis is a subclass of NumPy’s
ndarray
, it inherits all of its attributes, e.g.shape
. The following attributes behave slightly differently to a ndarray, howeverA == B
isTrue
if all elements evaluate almost equal, i.e. equivalent tonp.allclose(A, B)
.basis.T
transposes the last two axes ofbasis
. For a full basis, this corresponds to transposing each element individually. For a basis element, it corresponds to normal transposition.
- Parameters
- basis_array: array_like, shape (n, d, d)
An array or list of square matrices that are elements of an operator basis spanning \(\mathbb{C}^{d\times d}\). n should be smaller than or equal to d**2.
- traceless: bool, optional (default: auto)
Controls whether a traceless basis is forced. Here, traceless means that the first element of the basis is the identity and the remaining elements are matrices of trace zero. If an element of
basis_array
is neither traceless nor the identity andtraceless == True
, an exception will be raised. Defaults toTrue
if basis_array is traceless andFalse
if not.- btype: str, optional (default: ``’custom’``)
A string describing the basis type. For example, a basis created by the factory method
pauli()
has btype ‘pauli’.- labels: sequence of str, optional
A list of labels for the individual basis elements. Defaults to ‘C_0’, ‘C_1’, …
- Attributes
- Other than the attributes inherited from ``ndarray``, a ``Basis``
- instance has the following attributes:
- btype: str
Basis type.
- labels: sequence of str
The labels for the basis elements.
- d: int
Dimension of the space spanned by the basis.
- H: Basis
Hermitian conjugate.
- isherm: bool
If the basis is hermitian.
- isorthonorm: bool
If the basis is orthonormal.
- istraceless: bool
If the basis is traceless except for an identity element
- iscomplete: bool
If the basis is complete, ie spans the full space.
- sparse: COO, shape (n, d, d)
Representation in the COO format supplied by the
sparse
package.- four_element_traces: COO, shape (n, n, n, n)
Traces over all possible combinations of four elements of self. This is required for the calculation of the error transfer matrix and thus cached in the Basis instance.
- Most of the attributes above are properties which are lazily
- evaluated and cached.
Methods
Other than the methods inherited from ``ndarray``, a ``Basis``
instance has the following methods:
normalize(b)
Normalizes the basis (used internally when creating a basis from elements)
tidyup(eps_scale=None)
Cleans up floating point errors in-place to make zeros actual zeros.
eps_scale
is an optional argument multiplied to the data type’seps
to get the absolute tolerance.Constructor.
- property four_element_traces: COO
Return all traces of the form \(\mathrm{tr}(C_i C_j C_k C_l)\) as a sparse COO array for \(i,j,k,l > 0\) (i.e. excluding the identity).
- classmethod from_partial(partial_basis_array: Sequence, traceless: Optional[bool] = None, btype: Optional[str] = None, labels: Optional[Sequence[str]] = None) Basis [source]
Generate complete and orthonormal basis from a partial set.
The basis is completed using singular value decomposition to determine the null space of the expansion coefficients of the partial basis with respect to another complete basis.
- Parameters
- partial_basis_array: array_like
A sequence of basis elements.
- traceless: bool, optional
If a traceless basis should be generated (i.e. the first element is the identity and all the others have trace zero).
- btype: str, optional
A custom identifier.
- labels: Sequence[str], optional
A list of custom labels for each element. If len(labels) == len(partial_basis_array), the newly created elements get labels ‘C_i’.
- Returns
- basis: Basis, shape (d**2, d, d)
The orthonormal basis.
- Raises
- ValueError
If the given elements are not orthonormal.
- ValueError
If the given elements are not traceless but traceless==True.
- ValueError
If not len(partial_basis_array) or d**2 labels were given.
- classmethod ggm(d: int) Basis [source]
Returns a generalized Gell-Mann basis in \(d\) dimensions [Bert08] where the elements \(\Lambda_i\) are normalized with respect to the Hilbert-Schmidt inner product,
\[\begin{split}\langle\Lambda_i,\Lambda_j\rangle &= \mathrm{Tr}\,\Lambda_i^\dagger\Lambda_j \\ &= \delta_{ij}.\end{split}\]- Parameters
- d: int
The dimensionality of the space spanned by the basis
- Returns
- basis: Basis
The Basis object representing the GGM.
References
- Bert08
Bertlmann, R. A., & Krammer, P. (2008). Bloch vectors for qudits. Journal of Physics A: Mathematical and Theoretical, 41(23). https://doi.org/10.1088/1751-8113/41/23/235303
- classmethod pauli(n: int) Basis [source]
Returns a Pauli basis for \(n\) qubits, i.e. the basis spans the space \(\mathbb{C}^{d\times d}\) with \(d = 2^n\):
\[\mathcal{P} = \{I, X, Y, Z\}^{\otimes n}.\]The elements \(\sigma_i\) are normalized with respect to the Hilbert-Schmidt inner product,
\[\begin{split}\langle\sigma_i,\sigma_j\rangle &= \mathrm{Tr}\,\sigma_i^\dagger\sigma_j \\ &= \delta_{ij}.\end{split}\]- Parameters
- n: int
The number of qubits.
- Returns
- basis: Basis
The Basis object representing the Pauli basis.
- property sparse: COO
Return the basis as a sparse COO array
- class PulseSequence(*args, **kwargs)[source]
Bases:
object
A class for pulse sequences and their filter functions.
The Hamiltonian is separated into a control and a noise part with
\[\begin{split}\mathcal{H}_c &= \sum_i a_i(t) A_i \\ \mathcal{H}_n &= \sum_j s_j(t) b_j(t) B_j\end{split}\]where \(A_i\) and \(B_j\) are hermitian operators and \(b_j(t)\) are classically fluctuating noise variables captured in a power spectral density and not needed at instantiation of a
PulseSequence
.- Parameters
- H_c: list of lists
A nested list of n_cops nested lists as taken by QuTiP functions (see for example
qutip.propagator.propagator()
) describing the control part of the Hamiltonian. The i-th entry of the list should be a list consisting of the i-th operator \(A_i\) making up the control Hamiltonian and a list or array \(a_i(t)\) describing the magnitude of that operator during the time intervals dt. Optionally, the list may also include operator identifiers. That is, H_c should look something like this:H = [[c_oper1, c_coeff1, c_oper_identifier1], [c_oper2, c_coeff2, c_oper_identifier2], ...]
The operators may be given either as NumPy arrays or QuTiP Qobjs and each coefficient array should have the same number of elements as dt, and should be given in units of \(\hbar\). If not every sublist (read: operator) was given a identifier, they are automatically filled up with ‘A_i’ where i is the position of the operator.
- H_n: list of lists
A nested list of n_nops nested lists as taken by QuTiP functions (see for example
qutip.propagator.propagator()
) describing the noise part of the Hamiltonian. The j-th entry of the list should be a list consisting of the j-th operator \(B_j\) making up the noise Hamiltonian and a list or array describing the sensitivity \(s_j(t)\) of the system to the noise operator during the time intervals dt. Optionally, the list may also include operator identifiers. That is, H_n should look something like this:H = [[n_oper1, n_coeff1, n_oper_identifier1], [n_oper2, n_coeff2, n_oper_identifier2], ...]
The operators may be given either as NumPy arrays or QuTiP Qobjs and each coefficient array should have the same number of elements as dt, and should be given in units of \(\hbar\). If not every sublist (read: operator) was given a identifier, they are automatically filled up with ‘B_i’ where i is the position of the operator.
- dt: array_like, shape (n_dt,)
The segment durations of the Hamiltonian (i.e. durations of constant control). Internally, the control operation is taken to start at \(t_0\equiv 0\), i.e. the edges of the constant control segments are at times
t = [0, *np.cumsum(dt)]
.- basis: Basis, shape (d**2, d, d), optional
The operator basis in which to calculate. If a Generalized Gell-Mann basis (see
ggm()
) is chosen, some calculations will be faster for large dimensions due to a simpler basis expansion. However, when extending the pulse sequence to larger qubit registers, cached filter functions cannot be retained since the GGM basis does not factor into tensor products. In this case a Pauli basis is preferable.
Notes
Due to the heavy use of NumPy’s
einsum()
function, results have a floating point error of ~1e-13.Examples
A rotation by \(\pi\) around the axis between x and y preceeded and followed by a period of free evolution with the system subject to dephasing noise.
>>> import qutip as qt; import numpy as np >>> H_c = [[qt.sigmax(), [0, np.pi, 0]], [qt.sigmay(), [0, np.pi, 0]]] >>> # Equivalent pulse: >>> # H_c = [[qt.sigmax() + qt.sigmay(), [0, np.pi, 0]]] >>> # The noise sensitivity is constant >>> H_n = [[qt.sigmaz()/np.sqrt(2), [1, 1, 1], 'Z']] >>> dt = [1, 1, 1] >>> # Free evolution between t=0 and t=1, rotation between t=1 and t=2, >>> # and free evolution again from t=2 to t=3. >>> pulse = PulseSequence(H_c, H_n, dt) >>> pulse.c_oper_identifiers ['A_0', 'A_1'] >>> pulse.n_oper_identifiers ['Z'] >>> omega = np.logspace(-1, 2, 500) >>> F = pulse.get_filter_function(omega) # shape (1, 500) >>> # Plot the resulting filter function: >>> from filter_functions import plotting >>> fig, ax, leg = plotting.plot_filter_function(pulse)
- Attributes
- c_opers: ndarray, shape (n_cops, d, d)
Control operators. Note that they are stored sorted by their corresponding identifiers.
- n_opers: ndarray, shape (n_nops, d, d)
Noise operators. Note that they are stored sorted by their corresponding identifiers.
- c_oper_identifers: sequence of str
Identifiers for the control operators of the system. Stored sorted.
- n_oper_identifers: sequence of str
Identifiers for the noise operators of the system. Stored sorted.
- c_coeffs: ndarray, shape (n_cops, n_dt)
Control parameters in units of \(\hbar\). Note that they are stored sorted by their corresponding identifiers.
- n_coeffs: ndarray, shape (n_nops, n_dt)
Noise sensitivities in units of \(\hbar\). Note that they are stored sorted by their corresponding identifiers.
- dt: ndarray, shape (n_dt,)
Time steps
- t: ndarray, shape (n_dt + 1,)
Absolute times taken to start at \(t_0\equiv 0\)
- tau: float
Total duration. Equal to t[-1].
- d: int
Dimension of the Hamiltonian
- basis: Basis, shape (d**2, d, d)
The operator basis used for calculation
- nbytes: int
An estimate of the memory consumed by the PulseSequence instance and its attributes
- If the Hamiltonian was diagonalized, the eigenvalues and -vectors as
- well as the cumulative propagators are cached:
- eigvals: ndarray, shape (n_dt, d)
Eigenvalues \(D^{(g)}\)
- eigvecs: ndarray, shape (n_dt, d, d)
Eigenvectors \(V^{(g)}\)
- propagators: ndarray, shape (n_dt+1, d, d)
Cumulative propagators \(Q_g\)
- total_propagator: ndarray, shape (d, d)
The total propagator \(Q\) of the pulse alone. That is, \(|\psi(\tau)\rangle = Q|\psi(0)\rangle\).
- total_propagator_liouville: array_like, shape (d**2, d**2)
The transfer matrix for the total propagator of the pulse. Given by
liouville_representation(pulse.total_propagator, pulse.basis)
.- Furthermore, when the filter function is calculated, the frequencies
- are cached as well as other relevant quantities.
Methods
get_filter_function_derivative
(omega[, ...])Calculate the pulse sequence's filter function derivative.
cleanup(method=’conservative’)
Delete cached attributes
is_cached(attr)
Checks if a given attribute of the
PulseSequence
is cacheddiagonalize()
Diagonalize the Hamiltonian of the pulse sequence, computing eigenvalues and -vectors as well as cumulative propagators
get_control_matrix(omega, show_progressbar=False)
Calculate the control matrix for frequencies omega
get_filter_function(omega, which=’fidelity’, show_progressbar=False)
Calculate the filter function for frequencies omega
get_pulse_correlation_filter_function(which=’fidelity’)
Get the pulse correlation filter function (only possible if computed during concatenation)
propagator_at_arb_t(t)
Calculate the propagator at arbitrary times
Initialize a PulseSequence instance.
- cache_control_matrix(omega: Sequence[float], control_matrix: Optional[ndarray] = None, show_progressbar: bool = False, cache_intermediates: bool = False) None [source]
Cache the control matrix and the frequencies it was calculated for.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies for which to cache the filter function.
- control_matrix: array_like, shape ([n_nops,] n_nops, d**2, n_omega), optional
The control matrix for the frequencies omega. If
None
, it is computed.- show_progressbar: bool
Show a progress bar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep intermediate terms of the calculation that are also required by other computations. Only applies if control_matrix is not supplied.
- cache_filter_function(omega: Sequence[float], control_matrix: Optional[ndarray] = None, filter_function: Optional[ndarray] = None, which: str = 'fidelity', order: int = 1, show_progressbar: bool = False, cache_intermediates: bool = False) None [source]
Cache the filter function. If control_matrix.ndim == 4, it is taken to be the ‘pulse correlation control matrix’ and summed along the first axis. In that case, also the pulse correlation filter function is calculated and cached. Total phase factors and transfer matrices of the the cumulative propagator are also cached so they can be reused during concatenation.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies for which to cache the filter function.
- control_matrix: array_like, shape ([n_nops,] n_nops, d**2, n_omega), optional
The control matrix for the frequencies omega. If
None
, it is computed and the filter function derived from it.- filter_function: array_like, shape (n_nops, n_nops, [d**2, d**2,] n_omega), optional
The filter function for the frequencies omega. If
None
, it is computed from R in the caseorder == 1
and from scratch else.- which: str, optional
Which filter function to cache. Either ‘fidelity’ (default) or ‘generalized’.
- order: int, optional
First or second order filter function.
- show_progressbar: bool, optional
Show a progress bar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep intermediate terms of the calculation that are also required by other computations.
See also
PulseSequence.get_filter_function
Getter method
- cache_total_phases(omega: Sequence[float], total_phases: Optional[ndarray] = None) None [source]
Cache the total phase factors for this pulse and omega.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies for which to cache the phase factors.
- total_phases: array_like, shape (n_omega,), optional
The total phase factors for the frequencies omega. If
None
, they are computed.
- cleanup(method: str = 'conservative') None [source]
Delete cached byproducts of the calculation of the filter function that are not necessarily needed anymore in order to free up memory.
- Parameters
- method: optional
If set to ‘conservative’ (the default), only the following attributes are deleted:
_eigvals
_eigvecs
_propagators
If set to ‘greedy’, all of the above as well as the following attributes are deleted:
_total_propagator
_total_propagator_liouville
_total_phases
_control_matrix
_control_matrix_pc
If set to ‘all’, all of the above as well as the following attributes are deleted:
omega
_filter_function
_filter_function_gen
_filter_function_pc
_filter_function_pc_gen
_filter_function_2
_intermediates[‘control_matrix_step’]
If set to ‘frequency dependent’ only attributes that are functions of frequency are initalized to
None
.Note that if this
PulseSequence
is concatenated with another one, some of the attributes might need to be calculated again, resulting in slower execution of the concatenation.
- get_control_matrix(omega: Sequence[float], show_progressbar: bool = False, cache_intermediates: bool = False) ndarray [source]
Get the control matrix for the frequencies omega. If it has been cached for the same frequencies, the cached version is returned, otherwise it is calculated from scratch.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies at which to evaluate the control matrix.
- show_progressbar: bool
Show a progress bar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep intermediate terms of the calculation that are also required by other computations.
- Returns
- control_matrix: ndarray, shape (n_nops, d**2, n_omega)
The control matrix for the noise operators.
- get_filter_function(omega: Sequence[float], which: str = 'fidelity', order: int = 1, show_progressbar: bool = False, cache_intermediates: bool = False) ndarray [source]
Get the first or second order filter function.
The filter function is cached so it doesn’t need to be calculated twice for the same frequencies.
- Parameters
- omega: array_like, shape (n_omega,)
The frequencies at which to evaluate the filter function.
- which: str, optional
Which filter function to return. Either ‘fidelity’ (default) or ‘generalized’ (see Notes). Only if
order == 1
.- order: int, optional
First or second order filter function.
- show_progressbar: bool, optional
Show a progress bar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep intermediate terms of the calculation that are also required by other computations.
- Returns
- filter_function: ndarray, shape (n_nops, n_nops, [d**2, d**2,] n_omega)
The filter function for each combination of noise operators as a function of omega.
Notes
The first-order generalized filter function is given by
\[F_{\alpha\beta,kl}(\omega) = \tilde{\mathcal{B}}_{\alpha k}^\ast(\omega) \tilde{\mathcal{B}}_{\beta l}(\omega),\]where \(\alpha,\beta\) are indices counting the noise operators \(B_\alpha\) and \(k,l\) indices counting the basis elements \(C_k\).
The fidelity filter function is obtained by tracing over the basis indices:
\[F_{\alpha\beta}(\omega) = \sum_{k} F_{\alpha\beta,kk}(\omega).\]
- get_filter_function_derivative(omega: Sequence[float], control_identifiers: Optional[Sequence[str]] = None, n_oper_identifiers: Optional[Sequence[str]] = None, n_coeffs_deriv: Optional[Sequence[Sequence[float]]] = None) ndarray [source]
Calculate the pulse sequence’s filter function derivative.
- Parameters
- omega: array_like, shape (n_omega,)
Frequencies at which the pulse control matrix is to be evaluated.
- control_identifiers: Sequence[str], shape (n_ctrl,)
Sequence of strings with the control identifiers to distinguish between control and drift Hamiltonian. The default is None, in which case the derivative is computed for all known non-noise operators.
- n_oper_identifiers: Sequence[str], shape (n_nops,)
Sequence of strings with the noise identifiers for which to compute the derivative contribution. The default is None, in which case it is computed for all known noise operators.
- n_coeffs_deriv: array_like, shape (n_nops, n_ctrl, n_dt)
The derivatives of the noise susceptibilities by the control amplitudes. The rows and columns should be in the same order as the corresponding identifiers above. Defaults to None, in which case the coefficients are assumed to be constant and hence their derivative vanishing.
Warning
Internally, control and noise terms of the Hamiltonian are stored alphanumerically sorted by their identifiers. If the noise and/or control identifiers above are not explicitly given, the rows and/or columns of this parameter need to be sorted in the same fashion.
- Returns
- filter_function_deriv: ndarray, shape (n_nops, n_t, n_ctrl, n_omega)
The regular filter functions’ derivatives for variation in each control contribution. Sorted in the same fashion as n_coeffs_deriv or, if not given, alphanumerically by the identifiers.
- get_pulse_correlation_control_matrix() ndarray [source]
Get the pulse correlation control matrix if it was cached.
- get_pulse_correlation_filter_function(which: str = 'fidelity') ndarray [source]
Get the pulse correlation filter function given by
\[F_{\alpha\beta}^{(gg')}(\omega) = e^{i\omega(t_{g-1} - t_{g'-1})} \tilde{\mathcal{B}}^{(g)}(\omega)\mathcal{Q}^{(g-1)} \mathcal{Q}^{(g'-1)\dagger} \tilde{\mathcal{B}}^{(g')\dagger}(\omega),\]where \(g,g'\) index the pulse in the sequence and \(\alpha,\beta\) index the noise operators, if it was computed during concatenation. Since the calculation requires the individual pulse’s control matrices and phase factors, which are not retained after concatenation, the pulse correlation filter function cannot be computed afterwards.
Note that the frequencies for which the filter function was calculated are not stored.
- Returns
- filter_function_pc: ndarray, shape (n_pls, n_pls, n_nops, n_nops, n_omega)
The pulse correlation filter function for each noise operator as a function of omega. The first two axes correspond to the pulses in the sequence, i.e. if the concatenated pulse sequence is \(C\circ B\circ A\), the first two axes are arranged like
\[\begin{split}F_{\alpha\beta}^{(gg')} &= \begin{pmatrix} F_{\alpha\beta}^{(AA)} & F_{\alpha\beta}^{(AB)} & F_{\alpha\beta}^{(AC)} \\ F_{\alpha\beta}^{(BA)} & F_{\alpha\beta}^{(BB)} & F_{\alpha\beta}^{(BC)} \\ F_{\alpha\beta}^{(CA)} & F_{\alpha\beta}^{(CB)} & F_{\alpha\beta}^{(CC)} \end{pmatrix}\end{split}\]for \(g,g'\in\{A, B, C\}\).
- get_total_phases(omega: Sequence[float]) ndarray [source]
Get the (cached) total phase factors for this pulse and omega.
- property nbytes: int
Return an estimate of the amount of memory consumed by this object (or, more precisely, the array attributes of this object).
- concatenate(pulses: Iterable[PulseSequence], calc_pulse_correlation_FF: bool = False, calc_filter_function: Optional[bool] = None, which: str = 'fidelity', omega: Optional[Sequence[float]] = None, show_progressbar: bool = False) PulseSequence [source]
Concatenate an arbitrary number of pulses. Note that pulses are concatenated left-to-right, that is,
\[\mathtt{concatenate((A, B))} \equiv B \circ A\]so that \(A\) is executed before \(B\) when applying the concatenated pulse.
- Parameters
- pulses: sequence of PulseSequences
The PulseSequence instances to be concatenated. If any of the instances have a cached filter function, the filter function for the composite pulse will also be calculated in order to make use of the speedup gained from concatenating the filter functions. If omega is given, calculation of the composite filter function is forced.
- calc_pulse_correlation_FF: bool, optional
Switch to control whether the pulse correlation filter function (see
PulseSequence.get_pulse_correlation_filter_function()
) is calculated. If omega is not given, the cached frequencies of all pulses need to be equal.- calc_filter_function: bool, optional
Switch to force the calculation of the filter function to be carried out or not. Overrides the automatic behavior of calculating it if at least one pulse has a cached control matrix. If
True
and no pulse has a cached control matrix, a list of frequencies must be supplied as omega.- which: str, optional
Which filter function to compute. Either ‘fidelity’ (default) or ‘generalized’ (see
PulseSequence.get_filter_function()
andPulseSequence.get_pulse_correlation_filter_function()
).- omega: array_like, optional
Frequencies at which to evaluate the (pulse correlation) filter functions. If
None
, an attempt is made to use cached frequencies.- show_progressbar: bool
Show a progress bar for the calculation of the control matrix.
- Returns
- pulse: PulseSequence
The concatenated pulse.
- concatenate_periodic(pulse: PulseSequence, repeats: int) PulseSequence [source]
Concatenate a pulse sequence pulse whose Hamiltonian is periodic repeats times. Although performing the same task, this function is much faster for concatenating many identical pulses with filter functions than
concatenate()
.Note that for large dimensions, the calculation of the control matrix using this function might be very memory intensive.
- Parameters
- pulse: PulseSequence
The
PulseSequence
instance to be repeated. If it has a cached filter function, the filter function for the new pulse will also be computed.- repeats: int
The number of repetitions
- Returns
- newpulse: PulseSequence
The concatenated
PulseSequence
See also
concatenate
Concatenate arbitrary PulseSequences.
Notes
The total control matrix is given by
\[\begin{split}\tilde{\mathcal{B}}(\omega) &= \tilde{\mathcal{B}}^{(1)}(\omega) \sum_{g=0}^{G-1} \left(e^{i\omega T}\right)^g \\ &= \tilde{\mathcal{B}}^{(1)}(\omega)\bigl( \mathbb{I} - e^{i\omega T} \mathcal{Q}^{(1)}\bigr)^{-1}\bigl( \mathbb{I} - \bigl(e^{i\omega T} \mathcal{Q}^{(1)}\bigr)^G\bigr).\end{split}\]with \(T\) the period of the control Hamiltonian and \(G\) the number of periods. The last equality is valid only if \(\mathbb{I} - e^{i\omega T}\mathcal{Q}^{(1)}\) is invertible.
- error_transfer_matrix(pulse: Optional[PulseSequence] = None, spectrum: Optional[ndarray] = None, omega: Optional[Sequence[float]] = None, n_oper_identifiers: Optional[Sequence[str]] = None, second_order: bool = False, cumulant_function: Optional[ndarray] = None, show_progressbar: bool = False, memory_parsimonious: bool = False, cache_intermediates: bool = False) ndarray [source]
Compute the error transfer matrix up to unitary rotations.
- Parameters
- pulse: PulseSequence
The
PulseSequence
instance for which to compute the error transfer matrix.- spectrum: array_like, shape ([[n_nops,] n_nops,] n_omega)
The two-sided noise power spectral density in units of inverse frequencies as an array of shape (n_omega,), (n_nops, n_omega), or (n_nops, n_nops, n_omega). In the first case, the same spectrum is taken for all noise operators, in the second, it is assumed that there are no correlations between different noise sources and thus there is one spectrum for each noise operator. In the third and most general case, there may be a spectrum for each pair of noise operators corresponding to the correlations between them. n_nops is the number of noise operators considered and should be equal to
len(n_oper_identifiers)
.- omega: array_like, shape (n_omega,)
The frequencies at which to calculate the filter functions.
- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which to evaluate the error transfer matrix. The default is all. Note that, since in general contributions from different noise operators won’t commute, not selecting all noise operators results in neglecting terms of order \(\xi^4\).
- second_order: bool, optional
Also take into account the frequency shifts \(\Delta\) that correspond to second order Magnus expansion and constitute unitary terms. Default
False
.- cumulant_function: ndarray, shape ([[n_pls, n_pls,] n_nops,] n_nops, d**2, d**2)
A precomputed cumulant function. If given, pulse, spectrum, omega are not required.
- show_progressbar: bool, optional
Show a progress bar for the calculation of the control matrix.
- memory_parsimonious: bool, optional
Trade memory footprint for performance. See
calculate_decay_amplitudes()
. The default isFalse
.- cache_intermediates: bool, optional
Keep and return intermediate terms of the calculation of the control matrix (if it is not already cached) that can be reused for second order or gradients. Can consume large amount of memory, but speed up the calculation.
- Returns
- error_transfer_matrix: ndarray, shape (d**2, d**2)
The error transfer matrix. The individual noise operator contributions are summed up before exponentiating as they might not commute.
See also
calculate_cumulant_function
Calculate the cumulant function \(\mathcal{K}\)
calculate_decay_amplitudes
Calculate the \(\Gamma_{\alpha\beta,kl}\)
infidelity
Calculate only infidelity of a pulse.
Notes
The error transfer matrix is given by
\[\tilde{\mathcal{U}} = \exp\mathcal{K}(\tau)\]with \(\mathcal{K}(\tau)\) the cumulant function (see
calculate_cumulant_function()
). For Gaussian noise this expression is exact when taking into account the decay amplitudes \(\Gamma\) and frequency shifts \(\Delta\). As the latter effects coherent errors it can be neglected if we assume that the experimenter has calibrated their pulse.For non-Gaussian noise the expression above is perturbative and includes noise up to order \(\xi^2\) and hence \(\tilde{\mathcal{U}} = \mathbb{1} + \mathcal{K}(\tau) + \mathcal{O}(\xi^4)\) (although it is evaluated as a matrix exponential in any case).
Given the above expression of the error transfer matrix, the entanglement fidelity is given by
\[\mathcal{F}_\mathrm{e} = \frac{1}{d^2}\mathrm{tr}\,\tilde{\mathcal{U}}.\]
- extend(pulse_to_qubit_mapping: Sequence[Sequence[Optional[Union[PulseSequence, Sequence[int], int, Mapping[str, str]]]]], N: Optional[int] = None, d_per_qubit: int = 2, additional_noise_Hamiltonian: Optional[Sequence[Sequence[Union[ndarray, Qobj, Sequence[float]]]]] = None, cache_diagonalization: Optional[bool] = None, cache_filter_function: Optional[bool] = None, omega: Optional[Sequence[float]] = None, show_progressbar: bool = False) PulseSequence [source]
Map one or more pulse sequences to different qubits.
- Parameters
- pulse_to_qubit_mapping: sequence of mapping tuples
A sequence of tuples with the first entry a
PulseSequence
instance and the second anint
or tuple ofint
s indicating the qubits that thePulseSequence
should be mapped to. A mapping of operator identifiers may optionally be given as a third element of each tuple. By default, the index of the qubit the operator is mapped to is appended to its identifier.Pulse sequences defined for multiple qubits may also be extended to non-neighboring qubits. Note that for multi-qubit pulses the order of the qubits is respected, i.e. mapping a pulse to (1, 0) is different from mapping it to (0, 1).
- N: int
The total number of qubits the new
PulseSequence
should be defined for. By default, this is inferred frompulse_to_qubit_mapping
.- d_per_qubit: int
The size of the Hilbert space a single qubit requires.
- additional_noise_Hamiltonian: list of lists
Additional noise operators and corresponding sensitivities for the new pulse sequence.
- cache_diagonalization: bool
Force diagonalizing the new pulse sequence. By default, diagonalization is cached if all pulses in
pulse_to_qubit_mapping
have been diagonalized since it is much cheaper to get the relevant quantities as tensor products from the mapped pulses instead of diagonalizing the new pulse.- cache_filter_function: bool
Force computing the filter functions for the new pulse sequence. Noise operators of individual pulses will be extended to the new Hilbert space. By default, this is done if all pulses in
pulse_to_qubit_mapping
have their filter functions cached.Note that extending the filter functions is only possible if they the mapped pulses are using a separable basis like the Pauli basis.
- omega: array_like
Frequencies for which to compute the filter functions if
cache_filter_function == True
. Defaults toNone
, in which case the cached frequencies of the individual pulses need to be the same.- show_progressbar: bool
Show a progress bar for the calculation of the control matrix.
- Returns
- newpulse: PulseSequence
The new pulse sequence on the larger qubit register. The noise operators (and possibly filter functions) are stored in the following order: first those of the multi-qubit pulses in the order they appeared in
pulse_to_qubit_mapping
, then those of the single-qubit pulses, and lastly any additional ones that may be given byadditional_noise_Hamiltonian
.
See also
remap
Map PulseSequence to a different qubit.
concatenate
Concatenate PulseSequences (in time).
concatenate_periodic
Periodically concatenate a PulseSequence.
Examples
>>> import filter_functions as ff >>> I, X, Y, Z = ff.util.paulis >>> X_pulse = ff.PulseSequence([[X, [np.pi/2], 'X']], ... [[X, [1], 'X'], [Z, [1], 'Z']], ... [1], basis=ff.Basis.pauli(1)) >>> XX_pulse = ff.extend([(X_pulse, 0), (X_pulse, 1)]) >>> XX_pulse.d 4 >>> XIX_pulse_1 = ff.extend([(X_pulse, 0), (X_pulse, 2)]) >>> XIX_pulse_1.d 8 >>> XXI_pulse = ff.extend([(X_pulse, 0), (X_pulse, 1)], N=3) >>> XXI_pulse.d 8
Filter functions are automatically cached if they are for mapped pulses:
>>> omega = ff.util.get_sample_frequencies(X_pulse) >>> X_pulse.cache_filter_function(omega) >>> XX_pulse = ff.extend([(X_pulse, 0), (X_pulse, 1)]) >>> XX_pulse.is_cached('filter_function') True
This behavior can also be overriden manually:
>>> XX_pulse = ff.extend([(X_pulse, 0), (X_pulse, 1)], ... cache_filter_function=False) >>> XX_pulse.is_cached('filter_function') False
Mapping pulses to non-neighboring qubits is also possible:
>>> Y_pulse = ff.PulseSequence([[Y, [np.pi/2], 'Y']], ... [[Y, [1], 'Y'], [Z, [1], 'Z']], ... [1], basis=ff.Basis.pauli(1)) >>> XXY_pulse = ff.extend([(XX_pulse, (0, 1)), (Y_pulse, 2)]) >>> XYX_pulse = ff.extend([(XX_pulse, (0, 2)), (Y_pulse, 1)])
Additionally, pulses can have the order of the qubits they are defined for permuted (see
remap()
):>>> Z_pulse = ff.PulseSequence([[Z, [np.pi/2], 'Z']], [[Z, [1], 'Z']], ... [1], basis=ff.Basis.pauli(1)) >>> XY_pulse = ff.extend([(X_pulse, 0), (Y_pulse, 1)]) >>> YZX_pulse = ff.extend([(XY_pulse, (2, 0)), (Z_pulse, 1)])
Control and noise operator identifiers can be mapped according to a specified mapping:
>>> YX_pulse = ff.extend([(X_pulse, 1, {'X': 'IX', 'Z': 'IZ'}), ... (Y_pulse, 0, {'Y': 'YI', 'Z': 'ZI'})]) >>> YX_pulse.c_oper_identifiers array(['IX', 'YI'], dtype='<U2') >>> YX_pulse.n_oper_identifiers array(['IX', 'IZ', 'YI', 'ZI'], dtype='<U2')
We can also add an additional noise Hamiltonian:
>>> H_n = [[ff.util.tensor(Z, Z, Z), [1], 'ZZZ']] >>> XYX_pulse = ff.extend([(XX_pulse, (0, 2)), (Y_pulse, 1)], ... additional_noise_Hamiltonian=H_n) >>> 'ZZZ' in XYX_pulse.n_oper_identifiers True
- infidelity(pulse: PulseSequence, spectrum: Union[Sequence[float], Callable], omega: Union[Sequence[float], Dict[str, Union[int, str]]], n_oper_identifiers: Optional[Sequence[str]] = None, which: str = 'total', show_progressbar: bool = False, cache_intermediates: bool = False, return_smallness: bool = False, test_convergence: bool = False) Union[ndarray, Any] [source]
Calculate the leading order entanglement infidelity.
This function calculates the infidelity approximately from the leading peturbation (see Notes). To compute it exactly for Gaussian noise and vanishing coherent errors (second order Magnus terms), use
error_transfer_matrix()
to obtain it from the full process matrix.- Parameters
- pulse: PulseSequence
The
PulseSequence
instance for which to calculate the infidelity for.- spectrum: array_like, shape ([[n_nops,] n_nops,] omega) or callable
The two-sided noise power spectral density in units of inverse frequencies as an array of shape (n_omega,), (n_nops, n_omega), or (n_nops, n_nops, n_omega). In the first case, the same spectrum is taken for all noise operators, in the second, it is assumed that there are no correlations between different noise sources and thus there is one spectrum for each noise operator. In the third and most general case, there may be a spectrum for each pair of noise operators corresponding to the correlations between them. n_nops is the number of noise operators considered and should be equal to
len(n_oper_identifiers)
.If test_convergence is
True
, a function handle to compute the power spectral density from a sequence of frequencies is expected.- omega: array_like or dict
The frequencies at which the integration is to be carried out. If test_convergence is
True
, a dict with possible keys (‘omega_IR’, ‘omega_UV’, ‘spacing’, ‘n_min’, ‘n_max’, ‘n_points’), where all entries are integers except forspacing
which should be a string, either ‘linear’ or ‘log’. ‘n_points’ controls how many steps are taken.- n_oper_identifiers: array_like, optional
The identifiers of the noise operators for which to calculate the infidelity contribution. If given, the infidelities for each noise operator will be returned. Otherwise, all noise operators will be taken into account.
- which: str, optional
Which infidelities should be calculated, may be either ‘total’ (default) or ‘correlations’. In the former case, one value is returned for each noise operator, corresponding to the total infidelity of the pulse (or pulse sequence). In the latter, an array of infidelities is returned where element (i,j) corresponds to the infidelity contribution of the correlations between pulses i and j (see Notes). Note that this option is only available if the pulse correlation filter functions have been computed during concatenation (see
calculate_pulse_correlation_filter_function()
andconcatenate()
).- show_progressbar: bool, optional
Show a progressbar for the calculation of the control matrix.
- cache_intermediates: bool, optional
Keep and return intermediate terms of the calculation of the control matrix (if it is not already cached) that can be reused for second order or gradients. The default is False.
- return_smallness: bool, optional
Return the smallness parameter \(\xi\) for the given spectrum.
- test_convergence: bool, optional
Test the convergence of the integral with respect to the number of frequency samples. Returns the number of frequency samples and the corresponding fidelities. See spectrum and omega for more information.
- Returns
- infid: ndarray, shape ([[n_pls, n_pls,], n_nops,] n_nops)
Array with the infidelity contributions for each spectrum spectrum on the last axis or axes, depending on the shape of spectrum and which. If
which
iscorrelations
, the first two axes are the individual pulse contributions. If spectrum is 2-d (3-d), the last axis (two axes) are the individual spectral contributions. Only if test_convergence isFalse
.- n_samples: array_like
Array with number of frequency samples used for convergence test. Only if test_convergence is
True
.- convergence_infids: array_like
Array with infidelities calculated in convergence test. Only if test_convergence is
True
.
See also
calculate_decay_amplitudes
Calculate the full matrix of first order terms.
error_transfer_matrix
Calculate the full process matrix.
plotting.plot_infidelity_convergence
Convenience function to plot results.
pulse_sequence.concatenate
Concatenate
PulseSequence
objects.calculate_pulse_correlation_filter_function.
Notes
The infidelity is given by
\[\begin{split}\mathcal{I}_{\alpha\beta} &= 1 - \frac{1}{d^2}\mathrm{tr}\:\tilde{\mathcal{U}} \\ &= \frac{1}{d}\int_{-\infty}^{\infty} \frac{\mathrm{d}\omega}{2\pi}S_{\alpha\beta}(\omega) F_{\alpha\beta}(\omega) + \mathcal{O}\big(\xi^4\big) \\ &= \sum_{g,g'=1}^G \mathcal{I}_{\alpha\beta}^{(gg')}\end{split}\]with \(S_{\alpha\beta}(\omega)\) the two-sided noise spectral density and \(F_{\alpha\beta}(\omega)\) the first-order filter function for noise sources \(\alpha,\beta\). The noise spectrum may include correlated noise sources, that is, its entry at \((\alpha,\beta)\) corresponds to the correlations between sources \(\alpha\) and \(\beta\). \(\mathcal{I}_{\alpha\beta}^{(gg')}\) are the correlation infidelities that can be computed by setting
which='correlations'
.To convert to the average gate infidelity, use the following relation given by Horodecki et al. [Hor99] and Nielsen [Nie02]:
\[\mathcal{I}_\mathrm{avg} = \frac{d}{d+1}\mathcal{I}.\]The smallness parameter is given by
\[\xi^2 = \sum_\alpha\left[ \lvert\lvert B_\alpha\rvert\rvert^2 \int_{-\infty}^\infty\frac{\mathrm{d}\omega}{2\pi} S_\alpha(\omega)\left(\sum_gs_\alpha^{(g)}\Delta t_g \right)^2 \right].\]Note that in practice, the integral is only evaluated on the interval \(\omega\in[\omega_\mathrm{min},\omega_\mathrm{max}]\).
References
- Hor99
Horodecki, M., Horodecki, P., & Horodecki, R. (1999). General teleportation channel, singlet fraction, and quasidistillation. Physical Review A - Atomic, Molecular, and Optical Physics, 60(3), 1888–1898. https://doi.org/10.1103/PhysRevA.60.1888
- Nie02
Nielsen, M. A. (2002). A simple formula for the average gate fidelity of a quantum dynamical operation. Physics Letters, Section A: General, Atomic and Solid State Physics, 303(4), 249–252. https://doi.org/10.1016/S0375-9601(02)01272-0
- infidelity_derivative(pulse: PulseSequence, spectrum: Sequence[float], omega: Sequence[float], control_identifiers: Optional[Sequence[str]] = None, n_oper_identifiers: Optional[Sequence[str]] = None, n_coeffs_deriv: Optional[Sequence[Sequence[float]]] = None) ndarray [source]
Calculate the entanglement infidelity derivative of the
PulseSequence
pulse.- Parameters
- pulse: PulseSequence
The
PulseSequence
instance for which to calculate the infidelity.- spectrum: array_like, shape ([[n_nops,] n_nops,] omega)
The two-sided noise power spectral density in units of inverse frequencies as an array of shape (n_omega,), (n_nops, n_omega), or (n_nops, n_nops, n_omega). In the first case, the same spectrum is taken for all noise operators, in the second, it is assumed that there are no correlations between different noise sources and thus there is one spectrum for each noise operator. In the third and most general case, there may be a spectrum for each pair of noise operators corresponding to the correlations between them. n_nops is the number of noise operators considered and should be equal to
len(n_oper_identifiers)
.- omega: array_like, shape (n_omega,)
The frequencies at which the integration is to be carried out.
- control_identifiers: Sequence[str], shape (n_ctrl,)
Sequence of strings with the control identifiers to distinguish between control and drift Hamiltonian. The default is None, in which case the derivative is computed for all known non-noise operators.
- n_oper_identifiers: Sequence[str], shape (n_nops,)
Sequence of strings with the noise identifiers for which to compute the derivative contribution. The default is None, in which case it is computed for all known noise operators.
- n_coeffs_deriv: array_like, shape (n_nops, n_ctrl, n_dt)
The derivatives of the noise susceptibilities by the control amplitudes. The rows and columns should be in the same order as the corresponding identifiers above. Defaults to None, in which case the coefficients are assumed to be constant and hence their derivative vanishing.
Warning
Internally, control and noise terms of the Hamiltonian are stored alphanumerically sorted by their identifiers. If the noise and/or control identifiers above are not explicitly given, the rows and/or columns of this parameter need to be sorted in the same fashion.
- Returns
- infid_deriv: ndarray, shape (n_nops, n_dt, n_ctrl)
Array with the derivative of the infidelity for each noise source taken for each control direction at each time step \(\frac{\partial I_e}{\partial u_h(t_{g'})}\). Sorted in the same fashion as n_coeffs_deriv or, if not given, alphanumerically by the identifiers.
- Raises
- ValueError
If the provided noise spectral density does not fit expected shape.
Notes
The infidelity’s derivative is given by
\[\frac{\partial I_e}{\partial u_h(t_{g'})} = \frac{1}{d} \int_{-\infty}^\infty \frac{d\omega}{2\pi} S_\alpha(\omega) \frac{\partial F_\alpha(\omega)} {\partial u_h(t_{g'})}\]with \(S_{\alpha}(\omega)\) the noise spectral density and \(F_{\alpha}(\omega)\) the canonical filter function for noise source \(\alpha\).
To convert to the average gate infidelity, use the following relation given by Horodecki et al. [Hor99] and Nielsen [Nie02]:
\[\big\langle\mathcal{I}_\mathrm{avg}\big\rangle = \frac{d}{d+1} \big\langle\mathcal{I}_\mathrm{e}\big\rangle.\]References
- Hor99
Horodecki, M., Horodecki, P., & Horodecki, R. (1999). General teleportation channel, singlet fraction, and quasidistillation. Physical Review A - Atomic, Molecular, and Optical Physics, 60(3), 1888–1898. https://doi.org/10.1103/PhysRevA.60.1888
- Nie02
Nielsen, M. A. (2002). A simple formula for the average gate fidelity of a quantum dynamical operation. Physics Letters, Section A: General, Atomic and Solid State Physics, 303(4), 249–252. https://doi.org/10.1016/S0375-9601(02)01272-0
- liouville_representation(U: ndarray, basis: Basis) ndarray [source]
Get the Liouville representaion of the unitary U with respect to the basis.
- Parameters
- U: ndarray, shape (…, d, d)
The unitary.
- basis: Basis, shape (d**2, d, d)
The basis used for the representation, e.g. a Pauli basis.
- Returns
- R: ndarray, shape (…, d**2, d**2)
The Liouville representation of U.
Notes
The Liouville representation of a unitary quantum operation \(\mathcal{U}:\rho\rightarrow U\rho U^\dagger\) is given by
\[\mathcal{U}_{ij} = \mathrm{tr}(C_i U C_j U^\dagger)\]with \(C_i\) elements of the basis spanning \(\mathbb{C}^{d\times d}\) with \(d\) the dimension of the Hilbert space.
- remap(pulse: PulseSequence, order: Sequence[int], d_per_qubit: int = 2, oper_identifier_mapping: Optional[Mapping[str, str]] = None) PulseSequence [source]
Remap a PulseSequence by changing the order of qubits in the register. Cached attributes are automatically attempted to be retained.
Caution
This function simply permutes the order of the tensor product elements of control and noise operators. Thus, the resultant pulse will have its filter functions defined for different noise operators than the original one.
- Parameters
- pulse: PulseSequence
The pulse whose qubit order should be permuted.
- order: sequence of ints
A list of permutation indices. E.g., if pulse is defined for two qubits,
order == [1, 0]
will reverse the order of qubits.- d_per_qubit: int (default: 2)
The size of the Hilbert space a single qubit inhabitates.
- oper_identifier_mapping: dict_like
A mapping that maps operator identifiers from the old pulse to the remapped pulse. The default is the identity mapping.
- Returns
- remapped_pulse: PulseSequence
A new
PulseSequence
instance with the order of the qubits permuted according to order.
See also
extend
Map PulseSequences to composite Hilbert spaces.
util.tensor_transpose
Transpose the order of a tensor product.
Examples
>>> X, Y = util.paulis[1:3] >>> XY, YX = util.tensor(X, Y), util.tensor(Y, X) >>> pulse = PulseSequence([[XY, [np.pi/2], 'XY']], [[YX, [1], 'YX']], [1], ... Basis.pauli(2)) >>> mapping = {'XY': 'YX', 'YX': 'XY'} >>> remapped_pulse = remap(pulse, (1, 0), oper_identifier_mapping=mapping) >>> target_pulse = PulseSequence([[YX, [np.pi/2], 'YX']], ... [[XY, [1], 'XY']], [1], Basis.pauli(2)) >>> remapped_pulse == target_pulse True
Caching of attributes is automatically handled >>> remapped_pulse.is_cached(‘filter_function’) False >>> pulse.cache_filter_function(util.get_sample_frequencies(pulse)) >>> remapped_pulse = remap(pulse, (1, 0)) >>> remapped_pulse.is_cached(‘filter_function’) True