Note

This is the documentation for the current state of the development branch of Qiskit Experiments. The documentation or APIs here can change prior to being released.

cvxpy_gaussian_lstsq

cvxpy_gaussian_lstsq(outcome_data, shot_data, measurement_data, preparation_data, measurement_basis=None, preparation_basis=None, measurement_qubits=None, preparation_qubits=None, conditional_measurement_indices=None, trace='auto', psd=True, trace_preserving='auto', partial_trace=None, outcome_prior=0.5, max_weight=10000000000.0, **kwargs)

Constrained Gaussian linear least-squares tomography fitter.

Note

This function calls cvxpy_linear_lstsq() with a Gaussian weights vector. Refer to its documentation for additional details.

Overview

This fitter reconstructs the maximum-likelihood estimate by using cvxpy to minimize the constrained least-squares negative log likelihood function

$$\begin{array}{rl}\hat{\rho }& =\text{argmin}(-\log \mathcal{L}\rho )\\ & =\text{argmin }\Vert W(Ax-y){\Vert }_2^2\\ -\log \mathcal{L}(\rho )& =|W(Ax-y){\Vert }_2^2\\ & =\sum _i\frac{1}{{\sigma }_i^2}(\text{Tr}[E_j\rho ]-{\hat{p}}_i{)}^2\end{array}$$

Additional Details

The Gaussian weights are estimated from the observed frequency and shot data via a Bayesian update of a Dirichlet distribution with observed outcome data frequences $f_i(s)$, and Dirichlet prior ${\alpha }_i(s)$ for tomography basis index i and measurement outcome s.

The mean posterior probabilities are computed as

where $N_i=\sum _s{f}_i(s)$ is the total number of shots, and ${\overline{\alpha }}_i=\sum _s{\alpha }_i(s)$ is the norm of the prior.

Parameters:

• outcome_data (ndarray) – measurement outcome frequency data.

• shot_data (ndarray) – basis measurement total shot data.

• measurement_data (ndarray) – measurement basis indice data.

• preparation_data (ndarray) – preparation basis indice data.

• measurement_basis (Optional[MeasurementBasis]) – Optional, measurement matrix basis.

• preparation_basis (Optional[PreparationBasis]) – Optional, preparation matrix basis.

• measurement_qubits (Optional[Tuple[int, ...]]) – Optional, the physical qubits that were measured. If None they are assumed to be [0, ..., M-1] for M measured qubits.

• preparation_qubits (Optional[Tuple[int, ...]]) – Optional, the physical qubits that were prepared. If None they are assumed to be [0, ..., N-1] for N preparated qubits.

• conditional_measurement_indices (Optional[Tuple[int, ...]]) – Optional, conditional measurement data indices. If set this will return a list of conditional fitted states conditioned on a fixed basis measurement of these qubits.

• trace (Union[None, float, str]) – trace constraint for the fitted matrix. If “auto” this will be set to 1 for QST or the input dimension for QST (default: “auto”).

• psd (bool) – If True rescale the eigenvalues of fitted matrix to be positive semidefinite (default: True)

• trace_preserving (Union[None, bool, str]) – Enforce the fitted matrix to be trace preserving when fitting a Choi-matrix in quantum process tomography. If “auto” this will be set to True for QPT and False for QST (default: “auto”).

• partial_trace (Optional[ndarray]) – Enforce conditional fitted Choi matrices to partial trace to POVM matrices.

• outcome_prior (Union[ndarray, int]) – The Baysian prior $\alpha$ to use computing Gaussian weights. See additional information.

• max_weight (float) – Set the maximum value allowed for weights vector computed from tomography data variance.

• kwargs – kwargs for cvxpy solver.

Raises:

• QiskitError – If CVXPY is not installed on the current system.

• AnalysisError – If analysis fails.

Return type:

Dict

Returns:

The fitted matrix rho that maximizes the least-squares likelihood function.