ProcessTomographyFitter.fit

ProcessTomographyFitter.fit(method='auto', standard_weights=True, beta=0.5, **kwargs)[source]

Reconstruct a quantum channel using CVXPY convex optimization.

Choi matrix

The Choi matrix object is a QuantumChannel representation which may be converted to other representations using the classes SuperOp, Kraus, Stinespring, PTM, Chi from the module qiskit.quantum_info.operators. The raw matrix data for the representation may be obtained by channel.data.

Fitter method

The cvx fitter method used CVXPY convex optimization package. The lstsq method uses least-squares fitting (linear inversion). The auto method will use cvx if the CVXPY package is found on the system, otherwise it will default to lstsq.

Objective function

This fitter solves the constrained least-squares minimization: \(minimize: ||a \cdot x - b ||_2\)

subject to:

  • \(x >> 0\) (PSD)

  • \(\text{trace}(x) = \text{dim}\) (trace)

  • \(\text{partial_trace}(x) = \text{identity}\) (trace_preserving)

where:

  • a is the matrix of measurement operators \(a[i] = \text{vec}(M_i).H\)

  • b is the vector of expectation value data for each projector \(b[i] \sim \text{Tr}[M_i.H \cdot x] = (a \cdot x)[i]\)

  • x is the vectorized Choi-matrix to be fitted

PSD constraint

The PSD keyword constrains the fitted matrix to be postive-semidefinite. For the lstsq fitter method the fitted matrix is rescaled using the method proposed in Reference [1]. For the cvx fitter method the convex constraint makes the optimization problem a SDP. If PSD=False the fitted matrix will still be constrained to be Hermitian, but not PSD. In this case the optimization problem becomes a SOCP.

Trace constraint

The trace keyword constrains the trace of the fitted matrix. If trace=None there will be no trace constraint on the fitted matrix. This constraint should not be used for process tomography and the trace preserving constraint should be used instead.

Trace preserving (TP) constraint

The trace_preserving keyword constrains the fitted matrix to be TP. This should only be used for process tomography, not state tomography. Note that the TP constraint implicitly enforces the trace of the fitted matrix to be equal to the square-root of the matrix dimension. If a trace constraint is also specified that differs from this value the fit will likely fail. Note that this can only be used for the CVX method.

CVXPY Solvers:

Various solvers can be called in CVXPY using the solver keyword argument. See the CVXPY documentation for more information on solvers.

References:

[1] J Smolin, JM Gambetta, G Smith, Phys. Rev. Lett. 108, 070502

(2012). Open access: arXiv:1106.5458 [quant-ph].

Parameters

  • method (str) – (default: ‘auto’) the fitter method ‘auto’, ‘cvx’ or ‘lstsq’.

  • standard_weights (bool) – (default: True) apply weights to tomography data based on count probability

  • beta (float) – (default: 0.5) hedging parameter for converting counts to probabilities

  • **kwargs – kwargs for fitter method.

Raises

  • ValueError – In case the input data is no a valid process matrix

  • QiskitError – If the fit method is unrecognized

Returns

The fitted Choi-matrix J for the channel that maximizes \(||\text{basis_matrix} \cdot \text{vec}(J) - \text{data}||_2\). The Numpy matrix can be obtained from Choi.data.

Return type

Choi