qiskit.ignis.verification.TomographyFitter.fit

TomographyFitter.fit(method='auto', standard_weights=True, beta=0.5, psd=True, trace=None, trace_preserving=False, **kwargs)

Reconstruct a quantum state using CVXPY convex optimization.

Fitter method

The 'cvx' fitter method uses the CVXPY convex optimization package with a SDP solver. The 'lstsq' method uses least-squares fitting. The 'auto' method will use 'cvx' if the both the CVXPY and a suitable SDP solver packages are 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:

• \(x \succeq 0\)

• \(\text{trace}(x) = 1\)

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. If psd=True an SDP solver is required other an SOCP solver is required. See the CVXPY documentation for more information on solvers. Note that the default SDP solver (‘SCS’) distributed with CVXPY will not be used for the 'auto' method due its reduced accuracy compared to other solvers. When using the 'cvx' method we strongly recommend installing one of the other supported SDP 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) – hedging parameter for converting counts to probabilities

• psd (bool) – Enforced the fitted matrix to be positive semidefinite.

• trace (Optional[int]) – trace constraint for the fitted matrix.

• trace_preserving (bool) – Enforce the fitted matrix to be trace preserving when fitting a Choi-matrix in quantum process tomography. Note this method does not apply for ‘lstsq’ fitter method.

• **kwargs – kwargs for fitter method.

Raises

QiskitError – In case the fitting method is unrecognized.

Return type

array

Returns

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