pygsti.tools.rbtheory¶

RB-related functions of gates and models

Module Contents¶

Functions¶

 predicted_rb_number(model, target_model, weights=None, d=None, rtype='EI') Predicts the RB error rate from a model. predicted_rb_decay_parameter(model, target_model, weights=None) Computes the second largest eigenvalue of the 'L matrix' (see the L_matrix function). rb_gauge(model, target_model, weights=None, mx_basis=None, eigenvector_weighting=1.0) Computes the gauge transformation required so that the RB number matches the average model infidelity. transform_to_rb_gauge(model, target_model, weights=None, mx_basis=None, eigenvector_weighting=1.0) Transforms a Model into the "RB gauge" (see the RB_gauge function). L_matrix(model, target_model, weights=None) Constructs a generalization of the 'L-matrix' linear operator on superoperators. R_matrix_predicted_rb_decay_parameter(model, group, group_to_model=None, weights=None) Returns the second largest eigenvalue of a generalization of the 'R-matrix' [see the R_matrix function]. R_matrix(model, group, group_to_model=None, weights=None) Constructs a generalization of the 'R-matrix' of Proctor et al Phys. Rev. Lett. 119, 130502 (2017). errormaps(model, target_model) Computes the 'left-multiplied' error maps associated with a noisy gate set, along with the average error map. gate_dependence_of_errormaps(model, target_model, norm='diamond', mx_basis=None) Computes the "gate-dependence of errors maps" parameter defined by
pygsti.tools.rbtheory.predicted_rb_number(model, target_model, weights=None, d=None, rtype='EI')

Predicts the RB error rate from a model.

Uses the “L-matrix” theory from Proctor et al Phys. Rev. Lett. 119, 130502 (2017). Note that this gives the same predictions as the theory in Wallman Quantum 2, 47 (2018).

This theory is valid for various types of RB, including standard Clifford RB – i.e., it will accurately predict the per-Clifford error rate reported by standard Clifford RB. It is also valid for “direct RB” under broad circumstances.

For this function to be valid the model should be trace preserving and completely positive in some representation, but the particular representation of the model used is irrelevant, as the predicted RB error rate is a gauge-invariant quantity. The function is likely reliable when complete positivity is slightly violated, although the theory on which it is based assumes complete positivity.

Parameters
• model (Model) – The model to calculate the RB number of. This model is the model randomly sampled over, so this is not necessarily the set of physical primitives. In Clifford RB this is a set of Clifford gates; in “direct RB” this normally would be the physical primitives.

• target_model (Model) – The target model, corresponding to model. This function is not invariant under swapping model and target_model: this Model must be the target model, and should consistent of perfect gates.

• weights (dict, optional) – If not None, a dictionary of floats, whereby the keys are the gates in model and the values are the unnormalized probabilities to apply each gate at each stage of the RB protocol. If not None, the values in weights must all be non-negative, and they must not all be zero. Because, when divided by their sum, they must be a valid probability distribution. If None, the weighting defaults to an equal weighting on all gates, as this is used in many RB protocols (e.g., Clifford RB). But, this weighting is flexible in the “direct RB” protocol.

• d (int, optional) – The Hilbert space dimension. If None, then sqrt(model.dim) is used.

• rtype (str, optional) –

The type of RB error rate, either “EI” or “AGI”, corresponding to different dimension-dependent rescalings of the RB decay constant p obtained from fitting to Pm = A + Bp^m. “EI” corresponds to an RB error rate that is associated with entanglement infidelity, which is the probability of error for a gate with stochastic errors. This is the RB error rate defined in the “direct RB” protocol, and is given by:

r = (d^2 - 1)(1 - p)/d^2,

The AGI-type r is given by

r = (d - 1)(1 - p)/d,

which is the conventional r definition in Clifford RB. This r is associated with (gate-averaged) average gate infidelity.

Returns

r (float.) – The predicted RB number.

pygsti.tools.rbtheory.predicted_rb_decay_parameter(model, target_model, weights=None)

Computes the second largest eigenvalue of the ‘L matrix’ (see the L_matrix function).

For standard Clifford RB and direct RB, this corresponds to the RB decay parameter p in Pm = A + Bp^m for “reasonably low error” trace preserving and completely positive gates. See also the predicted_rb_number function.

Parameters
• model (Model) – The model to calculate the RB decay parameter of. This model is the model randomly sampled over, so this is not necessarily the set of physical primitives. In Clifford RB this is a set of Clifford gates; in “direct RB” this normally would be the physical primitives.

• target_model (Model) – The target model corresponding to model. This function is not invariant under swapping model and target_model: this Model must be the target model, and should consistent of perfect gates.

• weights (dict, optional) – If not None, a dictionary of floats, whereby the keys are the gates in model and the values are the unnormalized probabilities to apply each gate at each stage of the RB protocol. If not None, the values in weights must all be non-negative, and they must not all be zero. Because, when divided by their sum, they must be a valid probability distribution. If None, the weighting defaults to an equal weighting on all gates, as this is used in many RB protocols (e.g., Clifford RB). But, this weighting is flexible in the “direct RB” protocol.

Returns

p (float.) – The second largest eigenvalue of L. This is the RB decay parameter for various types of RB.

pygsti.tools.rbtheory.rb_gauge(model, target_model, weights=None, mx_basis=None, eigenvector_weighting=1.0)

Computes the gauge transformation required so that the RB number matches the average model infidelity.

This function computes the gauge transformation required so that, when the model is transformed via this gauge-transformation, the RB number – as predicted by the function predicted_rb_number – is the average model infidelity between the transformed model model and the target model target_model. This transformation is defined Proctor et al Phys. Rev. Lett. 119, 130502 (2017), and see also Wallman Quantum 2, 47 (2018).

Parameters
• model (Model) – The RB model. This is not necessarily the set of physical primitives – it is the model randomly sampled over in the RB protocol (e.g., the Cliffords).

• target_model (Model) – The target model corresponding to model. This function is not invariant under swapping model and target_model: this Model must be the target model, and should consistent of perfect gates.

• weights (dict, optional) – If not None, a dictionary of floats, whereby the keys are the gates in model and the values are the unnormalized probabilities to apply each gate at each stage of the RB protocol. If not None, the values in weights must all be non-negative, and they must not all be zero. Because, when divided by their sum, they must be a valid probability distribution. If None, the weighting defaults to an equal weighting on all gates, as this is used in many RB protocols (e.g., Clifford RB). But, this weighting is flexible in the “direct RB” protocol.

• mx_basis ({"std","gm","pp"}, optional) – The basis of the models. If None, the basis is obtained from the model.

• eigenvector_weighting (float, optional) – Must be non-zero. A weighting on the eigenvector with eigenvalue that is the RB decay parameter, in the sum of this eigenvector and the eigenvector with eigenvalue of 1 that defines the returned matrix l_operator. The value of this factor does not change whether this l_operator transforms into a gauge in which r = AGsI, but it may impact on other properties of the gates in that gauge. It is irrelevant if the gates are unital.

Returns

l_operator (array) – The matrix defining the gauge-transformation.

pygsti.tools.rbtheory.transform_to_rb_gauge(model, target_model, weights=None, mx_basis=None, eigenvector_weighting=1.0)

Transforms a Model into the “RB gauge” (see the RB_gauge function).

This notion was introduced in Proctor et al Phys. Rev. Lett. 119, 130502 (2017). This gauge is a function of both the model and its target. These may be input in any gauge, for the purposes of obtaining “r = average model infidelity” between the output Model and target_model.

Parameters
• model (Model) – The RB model. This is not necessarily the set of physical primitives – it is the model randomly sampled over in the RB protocol (e.g., the Cliffords).

• target_model (Model) – The target model corresponding to model. This function is not invariant under swapping model and target_model: this Model must be the target model, and should consistent of perfect gates.

• weights (dict, optional) – If not None, a dictionary of floats, whereby the keys are the gates in model and the values are the unnormalized probabilities to apply each gate at each stage of the RB protocol. If not None, the values in weights must all be non-negative, and they must not all be zero. Because, when divided by their sum, they must be a valid probability distribution. If None, the weighting defaults to an equal weighting on all gates, as this is used in many RB protocols (e.g., Clifford RB). But, this weighting is flexible in the “direct RB” protocol.

• mx_basis ({"std","gm","pp"}, optional) – The basis of the models. If None, the basis is obtained from the model.

• eigenvector_weighting (float, optional) – Must be non-zero. A weighting on the eigenvector with eigenvalue that is the RB decay parameter, in the sum of this eigenvector and the eigenvector with eigenvalue of 1 that defines the returned matrix l_operator. The value of this factor does not change whether this l_operator transforms into a gauge in which r = AGsI, but it may impact on other properties of the gates in that gauge. It is irrelevant if the gates are unital.

Returns

model_in_RB_gauge (Model) – The model model transformed into the “RB gauge”.

pygsti.tools.rbtheory.L_matrix(model, target_model, weights=None)

Constructs a generalization of the ‘L-matrix’ linear operator on superoperators.

From Proctor et al Phys. Rev. Lett. 119, 130502 (2017), the ‘L-matrix’ is represented as a matrix via the “stack” operation. This eigenvalues of this matrix describe the decay constant (or constants) in an RB decay curve for an RB protocol whereby random elements of the provided model are sampled according to the weights probability distribution over the model. So, this facilitates predictions of Clifford RB and direct RB decay curves.

Parameters
• model (Model) – The RB model. This is not necessarily the set of physical primitives – it is the model randomly sampled over in the RB protocol (e.g., the Cliffords).

• target_model (Model) – The target model corresponding to model. This function is not invariant under swapping model and target_model: this Model must be the target model, and should consistent of perfect gates.

• weights (dict, optional) – If not None, a dictionary of floats, whereby the keys are the gates in model and the values are the unnormalized probabilities to apply each gate at each stage of the RB protocol. If not None, the values in weights must all be non-negative, and they must not all be zero. Because, when divided by their sum, they must be a valid probability distribution. If None, the weighting defaults to an equal weighting on all gates, as this is used in many RB protocols (e.g., Clifford RB). But, this weighting is flexible in the “direct RB” protocol.

Returns

L (float) – A weighted version of the L operator from Proctor et al Phys. Rev. Lett. 119, 130502 (2017), represented as a matrix using the ‘stacking’ convention.

pygsti.tools.rbtheory.R_matrix_predicted_rb_decay_parameter(model, group, group_to_model=None, weights=None)

Returns the second largest eigenvalue of a generalization of the ‘R-matrix’ [see the R_matrix function].

Introduced in Proctor et al Phys. Rev. Lett. 119, 130502 (2017). This number is a prediction of the RB decay parameter for trace-preserving gates and a variety of forms of RB, including Clifford and direct RB. This function creates a matrix which scales super-exponentially in the number of qubits.

Parameters
• model (Model) – The model to predict the RB decay paramter for. If group_to_model is None, the labels of the gates in model should be the same as the labels of the group elements in group. For Clifford RB this would be the clifford model, for direct RB it would be the primitive gates.

• group (MatrixGroup) – The group that the model model contains gates from (model does not need to be the full group, and could be a subset of group). For Clifford RB and direct RB, this would be the Clifford group.

• group_to_model (dict, optional) – If not None, a dictionary that maps labels of group elements to labels of model. If model and group elements have the same labels, this dictionary is not required. Otherwise it is necessary.

• weights (dict, optional) – If not None, a dictionary of floats, whereby the keys are the gates in model and the values are the unnormalized probabilities to apply each gate at each stage of the RB protocol. If not None, the values in weights must all be positive or zero, and they must not all be zero (because, when divided by their sum, they must be a valid probability distribution). If None, the weighting defaults to an equal weighting on all gates, as used in most RB protocols.

Returns

p (float) – The predicted RB decay parameter. Valid for standard Clifford RB or direct RB with trace-preserving gates, and in a range of other circumstances.

pygsti.tools.rbtheory.R_matrix(model, group, group_to_model=None, weights=None)

Constructs a generalization of the ‘R-matrix’ of Proctor et al Phys. Rev. Lett. 119, 130502 (2017).

This matrix described the exact behaviour of the average success probablities of RB sequences. This matrix is super-exponentially large in the number of qubits, but can be constructed for 1-qubit models.

Parameters
• model (Model) – The noisy model (e.g., the Cliffords) to calculate the R matrix of. The correpsonding target model (not required in this function) must be equal to or a subset of (a faithful rep of) the group group. If group_to_model `is None, the labels of the gates in model should be the same as the labels of the corresponding group elements in `group. For Clifford RB model should be the clifford model; for direct RB this should be the native model.

• group (MatrixGroup) – The group that the model model contains gates from. For Clifford RB or direct RB, this would be the Clifford group.

• group_to_model (dict, optional) – If not None, a dictionary that maps labels of group elements to labels of model. This is required if the labels of the gates in model are different from the labels of the corresponding group elements in group.

• weights (dict, optional) – If not None, a dictionary of floats, whereby the keys are the gates in model and the values are the unnormalized probabilities to apply each gate at for each layer of the RB protocol. If None, the weighting defaults to an equal weighting on all gates, as used in most RB protocols (e.g., Clifford RB).

Returns

R (float) – A weighted, a subset-sampling generalization of the ‘R-matrix’ from Proctor et al Phys. Rev. Lett. 119, 130502 (2017).

pygsti.tools.rbtheory.errormaps(model, target_model)

Computes the ‘left-multiplied’ error maps associated with a noisy gate set, along with the average error map.

This is the model [E_1,…] such that

G_i = E_iT_i,

where T_i is the gate which G_i is a noisy implementation of. There is an additional gate in the set, that has the key ‘Gavg’. This is the average of the error maps.

Parameters
• model (Model) – The imperfect model.

• target_model (Model) – The target model.

Returns

errormaps (Model) – The left multplied error gates, along with the average error map, with the key ‘Gavg’.

pygsti.tools.rbtheory.gate_dependence_of_errormaps(model, target_model, norm='diamond', mx_basis=None)

Computes the “gate-dependence of errors maps” parameter defined by

delta_avg = avg_i|| E_i - avg_i(E_i) ||,

where E_i are the error maps, and the norm is either the diamond norm or the 1-to-1 norm. This quantity is defined in Magesan et al PRA 85 042311 2012.

Parameters
• model (Model) – The actual model

• target_model (Model) – The target model.

• norm (str, optional) – The norm used in the calculation. Can be either ‘diamond’ for the diamond norm, or ‘1to1’ for the Hermitian 1 to 1 norm.

• mx_basis ({"std","gm","pp"}, optional) – The basis of the models. If None, the basis is obtained from the model.

Returns

delta_avg (float) – The value of the parameter defined above.