# Source code for qiskit.ignis.verification.tomography.fitters.process_fitter

```
# -*- coding: utf-8 -*-
# This code is part of Qiskit.
#
# (C) Copyright IBM 2019.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""
Maximum-Likelihood estimation quantum process tomography fitter
"""
import numpy as np
from qiskit import QiskitError
from qiskit.quantum_info.operators import Choi
from .base_fitter import TomographyFitter
from .cvx_fit import cvx_fit
from .lstsq_fit import lstsq_fit
[docs]class ProcessTomographyFitter(TomographyFitter):
"""Maximum-Likelihood estimation process tomography fitter."""
[docs] def fit(self, # pylint: disable=arguments-differ
method: str = 'auto',
standard_weights: bool = True,
beta: float = 0.5,
**kwargs) -> Choi:
r"""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: :math:`minimize: ||a \cdot x - b ||_2`
subject to:
* :math:`x >> 0` (PSD)
* :math:`\text{trace}(x) = \text{dim}` (trace)
* :math:`\text{partial_trace}(x) = \text{identity}` (trace_preserving)
where:
* a is the matrix of measurement operators
:math:`a[i] = \text{vec}(M_i).H`
* b is the vector of expectation value data for each projector
:math:`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
<https://www.cvxpy.org/tutorial/advanced/index.html#solve-method-options>`_
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].
Args:
method: (default: 'auto') the fitter method 'auto', 'cvx' or 'lstsq'.
standard_weights: (default: True) apply weights
to tomography data based on count probability
beta: (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:
Choi: The fitted Choi-matrix J for the channel that maximizes
:math:`||\text{basis_matrix} \cdot
\text{vec}(J) - \text{data}||_2`.
The Numpy matrix can be obtained from `Choi.data`.
"""
# Get fitter data
data, basis_matrix, weights = self._fitter_data(standard_weights,
beta)
# Calculate trace of Choi-matrix from projector length
_, cols = np.shape(basis_matrix)
dim = int(np.sqrt(np.sqrt(cols)))
if dim ** 4 != cols:
raise ValueError("Input data does not correspond "
"to a process matrix.")
# Choose automatic method
if method == 'auto':
self._check_for_sdp_solver()
if self._HAS_SDP_SOLVER:
method = 'cvx'
else:
method = 'lstsq'
if method == 'lstsq':
return Choi(lstsq_fit(data, basis_matrix, weights=weights,
trace=dim, **kwargs))
if method == 'cvx':
return Choi(cvx_fit(data, basis_matrix, weights=weights, trace=dim,
trace_preserving=True, **kwargs))
raise QiskitError('Unrecognized fit method {}'.format(method))
```