linear_inversion

linear_inversion(outcome_data, shot_data, measurement_data, preparation_data, measurement_basis=None, preparation_basis=None, measurement_qubits=None, preparation_qubits=None, conditional_measurement_indices=None, conditional_preparation_indices=None, atol=1e-08)

Linear inversion tomography fitter.

Overview

This fitter uses linear inversion to reconstructs the maximum-likelihood estimate of the least-squares log-likelihood function

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

where

• $A=\sum _j|j⟩⟨⟨E_j|$ is the matrix of measured basis elements.

• $y=\sum _j{\hat{p}}_j|j⟩$ is the vector of estimated measurement outcome probabilities for each basis element.

• $x=|\rho ⟩⟩$ is the vectorized density matrix.

Additional Details

The linear inversion solution is given by

$$\hat{\rho }=\sum _i{\hat{p}}_i{D}_i$$

where measurement probabilities ${\hat{p}}_i=f_i/n_i$ are estimated from the observed count frequencies $f_i$ in $n_i$ shots for each basis element $i$, and $D_i$ is the dual basis element constructed from basis $\{E_i\}$ via:

Note

The Linear inversion fitter treats the input measurement and preparation bases as local bases and constructs separate 1-qubit dual basis for each individual qubit.

Linear inversion is only possible if the input bases are local and a spanning set for the vector space of the reconstructed matrix (tomographically complete). If the basis is not tomographically complete the scipy_linear_lstsq() or cvxpy_linear_lstsq() function can be used to solve the same objective function via least-squares optimization.

Parameters:

• outcome_data (ndarray) – basis outcome frequency data.

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

• measurement_data (ndarray) – measurement basis index data.

• preparation_data (ndarray) – preparation basis index data.

• measurement_basis (MeasurementBasis | None) – the tomography measurement basis.

• preparation_basis (PreparationBasis | None) – the tomography preparation basis.

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

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

• conditional_measurement_indices (ndarray | None) – Optional, conditional measurement data indices. If set this will return a list of fitted states conditioned on a fixed basis measurement of these qubits.

• conditional_preparation_indices (ndarray | None) – Optional, conditional preparation data indices. If set this will return a list of fitted states conditioned on a fixed basis preparation of these qubits.

• atol (float) – truncate any probabilities below this value to zero.

Raises:

AnalysisError – If the fitted vector is not a square matrix

Returns:

The fitted matrix rho.

Return type:

Tuple[ndarray, Dict]