# Source code for qiskit.ignis.verification.tomography.fitters.base_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 tomography fitter
"""
import logging
import itertools as it
from typing import List, Union, Optional, Dict, Tuple, Callable
from ast import literal_eval
import numpy as np
from qiskit import QiskitError
from qiskit import QuantumCircuit
from qiskit.result import Result
from ..basis import TomographyBasis, default_basis
from ..data import marginal_counts, combine_counts, count_keys
from .lstsq_fit import lstsq_fit
from .cvx_fit import cvx_fit, _HAS_CVX
# Create logger
logger = logging.getLogger(__name__)
[docs]class TomographyFitter:
"""Base maximum-likelihood estimate tomography fitter class"""
_HAS_SDP_SOLVER = None
_HAS_SDP_SOLVER_NOT_SCS = False
def __init__(self,
result: Union[Result, List[Result]],
circuits: Union[List[QuantumCircuit], List[str]],
meas_basis: Union[TomographyBasis, str] = 'Pauli',
prep_basis: Union[TomographyBasis, str] = 'Pauli'):
"""Initialize tomography fitter with experimental data.
Args:
result: a Qiskit Result object obtained from executing
tomography circuits.
circuits: a list of circuits or circuit names to extract
count information from the result object.
meas_basis: (default: 'Pauli') A function to return
measurement operators corresponding to measurement
outcomes. See Additional Information.
prep_basis: (default: 'Pauli') A function to return
preparation operators. See Additional
Information
"""
# Set the measure and prep basis
self._meas_basis = None
self._prep_basis = None
self.set_measure_basis(meas_basis)
self.set_preparation_basis(prep_basis)
# Add initial data
self._data = {}
if isinstance(result, Result):
result = [result] # unify results handling
self.add_data(result, circuits)
[docs] def set_measure_basis(self, basis: Union[TomographyBasis, str]):
"""Set the measurement basis
Args:
basis: measurement basis
Raises:
QiskitError: In case of invalid measurement or preparation basis.
"""
self._meas_basis = default_basis(basis)
if isinstance(self._meas_basis, TomographyBasis):
if self._meas_basis.measurement is not True:
raise QiskitError("Invalid measurement basis")
[docs] def set_preparation_basis(self, basis: Union[TomographyBasis, str]):
"""Set the preparation basis function
Args:
basis: preparation basis
Raises:
QiskitError: in case the basis has no preperation data
"""
self._prep_basis = default_basis(basis)
if isinstance(self._prep_basis, TomographyBasis):
if self._prep_basis.preparation is not True:
raise QiskitError("Invalid preparation basis")
@property
def measure_basis(self):
"""Return the tomography measurement basis."""
return self._meas_basis
@property
def preparation_basis(self):
"""Return the tomography preparation basis."""
return self._prep_basis
[docs] def fit(self,
method: str = 'auto',
standard_weights: bool = True,
beta: float = 0.5,
psd: bool = True,
trace: Optional[int] = None,
trace_preserving: bool = False,
**kwargs) -> np.array:
r"""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: :math:`||a \cdot x - b ||_2`
subject to:
* :math:`x \succeq 0`
* :math:`\text{trace}(x) = 1`
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 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
<https://www.cvxpy.org/tutorial/advanced/index.html#solve-method-options>`_
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 <https://arxiv.org/abs/1106.5458>`_ [quant-ph].
Args:
method: The fitter method 'auto', 'cvx' or 'lstsq'.
standard_weights: (default: True) Apply weights to
tomography data based on count probability
beta: hedging parameter for converting counts
to probabilities
psd: Enforced the fitted matrix to be positive semidefinite.
trace: trace constraint for the fitted matrix.
trace_preserving: 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.
Returns:
The fitted matrix rho that minimizes
:math:`||\text{basis_matrix} * \text{vec(rho)} - \text{data}||_2`.
"""
# Get fitter data
data, basis_matrix, weights = self._fitter_data(standard_weights,
beta)
# Choose automatic method
if method == 'auto':
self._check_for_sdp_solver()
if self._HAS_SDP_SOLVER_NOT_SCS:
# We don't use the SCS solver for automatic method as it has
# lower accuracy than the other supported SDP solvers which
# typically results in the returned matrix not being
# completely positive.
method = 'cvx'
else:
method = 'lstsq'
if method == 'lstsq':
return lstsq_fit(data, basis_matrix,
weights=weights,
psd=psd,
trace=trace,
**kwargs)
if method == 'cvx':
return cvx_fit(data, basis_matrix,
weights=weights,
psd=psd,
trace=trace,
trace_preserving=trace_preserving,
**kwargs)
raise QiskitError('Unrecognized fit method {}'.format(method))
@property
def data(self):
"""
Return tomography data
"""
return self._data
[docs] def add_data(self,
results: List[Result],
circuits: List[Union[QuantumCircuit, str]]
):
"""Add tomography data from a Qiskit Result object.
Args:
results: The results obtained from executing tomography circuits.
circuits: circuits or circuit names to extract
count information from the result object.
Raises:
QiskitError: In case some of the tomography data is not found
in the results
"""
if len(circuits) == 0:
raise QiskitError("No circuit data given")
if isinstance(circuits[0], str) or len(circuits[0].cregs) == 1:
marginalize = False
else:
marginalize = True
# Process measurement counts into probabilities
for circ in circuits:
counts = None
for result in results:
try:
counts = result.get_counts(circ)
except QiskitError:
pass
if counts is None:
raise QiskitError("Result for {} not found".format(circ.name))
if isinstance(circ, str):
tup = literal_eval(circ)
elif isinstance(circ, QuantumCircuit):
tup = literal_eval(circ.name)
else:
tup = circ
if marginalize:
counts = marginal_counts(counts, range(len(tup[0])))
if tup in self._data:
self._data[tup] = combine_counts(self._data[tup], counts)
else:
self._data[tup] = counts
def _fitter_data(self, standard_weights, beta):
"""Generate tomography fitter data from a tomography data dictionary.
Args:
standard_weights (bool, optional): Apply weights to basis matrix
and data based on count probability (default: True)
beta (float): hedging parameter for 0, 1
probabilities (default: 0.5)
Returns:
tuple: (data, basis_matrix, weights) where `data`
is a vector of the
probability values, and `basis_matrix`
is a matrix of the preparation
and measurement operator, and `weights`
is a vector of weights for the
given probabilities.
Additional Information
----------------------
standard_weights:
Weights are calculated from from binomial distribution standard
deviation
"""
# Get basis matrix functions
if self._meas_basis:
measurement = self._meas_basis.measurement_matrix
else:
measurement = None
if self._prep_basis:
preparation = self._prep_basis.preparation_matrix
else:
preparation = None
data = []
basis_blocks = []
if standard_weights:
weights = []
else:
weights = None
# Check if input data is state or process tomography data based
# on the label tuples
label = next(iter(self._data))
is_qpt = (isinstance(label, tuple) and len(label) == 2 and
isinstance(label[0], tuple) and isinstance(label[1], tuple))
# Generate counts keys for converting to np array
if is_qpt:
ctkeys = count_keys(len(label[1]))
else:
ctkeys = count_keys(len(label))
for label, cts in self._data.items():
# Convert counts dict to numpy array
if isinstance(cts, dict):
cts = np.array([cts.get(key, 0) for key in ctkeys])
# Get probabilities
shots = np.sum(cts)
probs = np.array(cts) / shots
data += list(probs)
# Compute binomial weights
if standard_weights is True:
wts = self._binomial_weights(cts, beta)
weights += list(wts)
# Get reconstruction basis operators
if is_qpt:
prep_label = label[0]
meas_label = label[1]
else:
prep_label = None
meas_label = label
prep_op = self._preparation_op(prep_label, preparation)
meas_ops = self._measurement_ops(meas_label, measurement)
block = self._basis_operator_matrix(
[np.kron(prep_op.T, mop) for mop in meas_ops])
basis_blocks.append(block)
return data, np.vstack(basis_blocks), weights
def _binomial_weights(self, counts: Dict[str, int],
beta: float = 0.5
) -> np.array:
"""
Compute binomial weights for list or dictionary of counts.
Args:
counts: A set of measurement counts for
all outcomes of a given measurement configuration.
beta: (default: 0.5) A nonnegative hedging parameter used to bias
probabilities computed from input counts away from 0 or 1.
Returns:
The binomial weights for the input counts and beta parameter.
Raises:
ValueError: In case beta is negative.
Additional Information:
The weights are determined by
w[i] = sqrt(shots / p[i] * (1 - p[i]))
p[i] = (counts[i] + beta) / (shots + K * beta)
where
`shots` is the sum of all counts in the input
`p` is the hedged probability computed for a count
`K` is the total number of possible measurement outcomes.
"""
# Sort counts if input is a dictionary
if isinstance(counts, dict):
mcts = marginal_counts(counts, pad_zeros=True)
ordered_keys = sorted(list(mcts))
counts = np.array([mcts[k] for k in ordered_keys])
# Assume counts are already sorted if a list
else:
counts = np.array(counts)
shots = np.sum(counts)
# If beta is 0 check if we would be dividing by zero
# If so change beta value and log warning.
if beta < 0:
raise ValueError('beta = {} must be non-negative.'.format(beta))
if beta == 0 and (shots in counts or 0 in counts):
beta = 0.5
msg = ("Counts result in probabilities of 0 or 1 "
"in binomial weights "
"calculation. Setting hedging "
"parameter beta={} to prevent "
"dividing by zero.".format(beta))
logger.warning(msg)
outcomes_num = len(counts)
# Compute hedged frequencies which are shifted to never be 0 or 1.
freqs_hedged = (counts + beta) / (shots + outcomes_num * beta)
# Return gaussian weights for 2-outcome measurements.
return np.sqrt(shots / (freqs_hedged * (1 - freqs_hedged)))
def _basis_operator_matrix(self, basis: List[np.array]) -> np.array:
"""Return a basis measurement matrix of the input basis.
Args:
basis: a list of basis matrices.
Returns:
A numpy array of shape (n, col * row) where n is the number
of operators of shape (row, col) in `basis`.
"""
# Dimensions
num_ops = len(basis)
nrows, ncols = basis[0].shape
size = nrows * ncols
ret = np.zeros((num_ops, size), dtype=complex)
for j, b in enumerate(basis):
ret[j] = np.array(b).reshape((1, size), order='F').conj()
return ret
def _preparation_op(self,
label: Tuple[str],
prep_matrix_fn: Callable[[str], np.array]
) -> np.array:
"""
Return the multi-qubit matrix for a state preparation label.
Args:
label: a preparation configuration label for a
tomography circuit.
prep_matrix_fn: a function that returns the matrix
corresponding to a single qubit preparation label.
The functions should have signature:
``prep_matrix_fn(str) -> np.array``
Returns:
A Numpy array for the multi-qubit preparation operator specified
by label.
Additional Information:
See the Pauli and SIC-POVM preparation functions
`pauli_preparation_matrix` and `sicpovm_preparation_matrix` for
examples.
"""
# Trivial case
if label is None:
return np.eye(1, dtype=complex)
# Construct preparation matrix
op = np.eye(1, dtype=complex)
for label_inst in label:
op = np.kron(prep_matrix_fn(label_inst), op)
return op
def _measurement_ops(self,
label: Tuple[str],
meas_matrix_fn: Callable[[str, int], np.array]
) -> List[np.array]:
"""
Return a list multi-qubit matrices for a measurement label.
Args:
label: a measurement configuration label for a
tomography circuit.
meas_matrix_fn: a function that returns the matrix
corresponding to a single qubit measurement label
for a given outcome. The functions should have
signature meas_matrix_fn(str, int) -> np.array
Returns:
A list of Numpy array for the multi-qubit measurement operators
for all measurement outcomes for the measurement basis specified
by the label. These are ordered in increasing binary order. Eg for
2-qubits the returned matrices correspond to outcomes
[00, 01, 10, 11]
Additional Information:
See the Pauli measurement function `pauli_measurement_matrix`
for an example.
"""
num_qubits = len(label)
meas_ops = []
# Construct measurement POVM for all measurement outcomes for a given
# measurement label. This will be a list of 2 ** n operators.
for outcomes in sorted(it.product((0, 1), repeat=num_qubits)):
op = np.eye(1, dtype=complex)
# Reverse label to correspond to QISKit bit ordering
for m, outcome in zip(reversed(label), outcomes):
op = np.kron(op, meas_matrix_fn(m, outcome))
meas_ops.append(op)
return meas_ops
@classmethod
def _check_for_sdp_solver(cls):
"""Check if CVXPY solver is available"""
if cls._HAS_SDP_SOLVER is None:
if _HAS_CVX:
# pylint:disable=import-error
import cvxpy
solvers = cvxpy.installed_solvers()
# Check for other SDP solvers cvxpy supports
if 'CVXOPT' in solvers or 'MOSEK' in solvers:
cls._HAS_SDP_SOLVER_NOT_SCS = True
cls._HAS_SDP_SOLVER = True
return
if 'SCS' in solvers:
# Try example problem to see if built with BLAS
# SCS solver cannot solver larger than 2x2 matrix
# problems without BLAS
try:
var = cvxpy.Variable((5, 5), PSD=True)
obj = cvxpy.Minimize(cvxpy.norm(var))
cvxpy.Problem(obj).solve(solver='SCS')
cls._HAS_SDP_SOLVER = True
return
except cvxpy.error.SolverError:
pass
cls._HAS_SDP_SOLVER = False
```