StateTomographyFitter.fit

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

Reconstruct a quantum state using CVXPY convex optimization.

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:

  • \(x >> 0\)

  • \(\text{trace}(x) = 1\)

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 density 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.

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) – 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

QiskitError – In case the fitting method is unrecognized.

Return type

array

Returns

The fitted matrix rho that minimizes \(||\text{basis_matrix} \cdot \text{vec}(\text{rho}) - \text{data}||_2\).