Table of Contents

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.

scipy_linear_lstsq

scipy_linear_lstsq(outcome_data, shot_data, measurement_data, preparation_data, measurement_basis=None, preparation_basis=None, measurement_qubits=None, preparation_qubits=None, weights=None, **kwargs)[source]

Weighted linear least-squares tomography fitter.

Overview

This fitter reconstructs the maximum-likelihood estimate by using scipy.linalg.lstsq() to minimize the least-squares negative log likelihood function

\[\begin{split}\hat{\rho} &= -\mbox{argmin }\log\mathcal{L}{\rho} \\ &= \mbox{argmin }\sum_i w_i^2(\mbox{Tr}[E_j\rho] - \hat{p}_i)^2 \\ &= \mbox{argmin }\|W(Ax - y) \|_2^2\end{split}\]

where

  • \(A = \sum_j |j \rangle\!\langle\!\langle E_j|\) is the matrix of measured basis elements.

  • \(W = \sum_j w_j|j\rangle\!\langle j|\) is an optional diagonal weights matrix if an optional weights vector is supplied.

  • \(y = \sum_j \hat{p}_j |j\langle\) is the vector of estimated measurement outcome probabilites for each basis element.

  • \(x = |\rho\rangle\!\rangle\) is the vectorized density matrix.

Note

Linear least-squares constructs the full basis matrix \(A\) as a dense numpy array so should not be used for than 5 or 6 qubits. For larger number of qubits try the linear_inversion() fitter function.

Deprecated since version unknown: The scipy lstsq tomography fitters are deprecated as of 0.4 and will be removed after the 0.5 release. Use the linear_lstsq, cvxpy_linear_lstsq, or cvxpy_gaussian_lstsq fitters instead.

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.

  • weights (Optional[ndarray]) – Optional array of weights for least squares objective.

  • kwargs – additional kwargs for scipy.linalg.lstsq().

Raises:

AnalysisError – If the fitted vector is not a square matrix

Return type:

Tuple[ndarray, Dict]

Returns:

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