# This code is part of Qiskit.
#
# (C) Copyright IBM 2021.
#
# 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.
"""
Linear inversion MLEtomography fitter.
"""
from typing import Dict, Tuple, Optional, Sequence, List
from functools import lru_cache
import time
import numpy as np
from qiskit_experiments.library.tomography.basis import (
MeasurementBasis,
PreparationBasis,
LocalMeasurementBasis,
LocalPreparationBasis,
)
from .lstsq_utils import _partial_outcome_function
from .fitter_data import _basis_dimensions
[docs]
def linear_inversion(
outcome_data: np.ndarray,
shot_data: np.ndarray,
measurement_data: np.ndarray,
preparation_data: np.ndarray,
measurement_basis: Optional[MeasurementBasis] = None,
preparation_basis: Optional[PreparationBasis] = None,
measurement_qubits: Optional[Tuple[int, ...]] = None,
preparation_qubits: Optional[Tuple[int, ...]] = None,
conditional_measurement_indices: Optional[np.ndarray] = None,
conditional_preparation_indices: Optional[np.ndarray] = None,
atol: float = 1e-8,
) -> Tuple[np.ndarray, Dict]:
r"""Linear inversion tomography fitter.
Overview
This fitter uses linear inversion to reconstructs the maximum-likelihood
estimate of the least-squares log-likelihood function
.. math::
\hat{\rho}
&= -\mbox{argmin }\log\mathcal{L}{\rho} \\
&= \mbox{argmin }\sum_i (\mbox{Tr}[E_j\rho] - \hat{p}_i)^2 \\
&= \mbox{argmin }\|Ax - y \|_2^2
where
* :math:`A = \sum_j |j \rangle\!\langle\!\langle E_j|` is the matrix of measured
basis elements.
* :math:`y = \sum_j \hat{p}_j |j\rangle` is the vector of estimated measurement
outcome probabilities for each basis element.
* :math:`x = |\rho\rangle\!\rangle` is the vectorized density matrix.
Additional Details
The linear inversion solution is given by
.. math::
\hat{\rho} = \sum_i \hat{p}_i D_i
where measurement probabilities :math:`\hat{p}_i = f_i / n_i` are estimated
from the observed count frequencies :math:`f_i` in :math:`n_i` shots for each
basis element :math:`i`, and :math:`D_i` is the *dual basis* element constructed
from basis :math:`\{E_i\}` via:
.. math:
|D_i\rangle\!\rangle = M^{-1}|E_i \rangle\!\rangle \\
M = \sum_j |E_j\rangle\!\rangle\!\langle\!\langle E_j|
.. note::
The Linear inversion fitter treats the input measurement and preparation
bases as local bases and constructs separate 1-qubit dual basis for each
individual qubit.
Linear inversion is only possible if the input bases are local and a spanning
set for the vector space of the reconstructed matrix
(*tomographically complete*). If the basis is not tomographically complete
the :func:`~qiskit_experiments.library.tomography.fitters.scipy_linear_lstsq`
or :func:`~qiskit_experiments.library.tomography.fitters.cvxpy_linear_lstsq`
function can be used to solve the same objective function via
least-squares optimization.
Args:
outcome_data: basis outcome frequency data.
shot_data: basis outcome total shot data.
measurement_data: measurement basis index data.
preparation_data: preparation basis index data.
measurement_basis: the tomography measurement basis.
preparation_basis: the tomography preparation basis.
measurement_qubits: Optional, the physical qubits that were measured.
If None they are assumed to be [0, ..., M-1] for M measured qubits.
preparation_qubits: Optional, the physical qubits that were prepared.
If None they are assumed to be [0, ..., N-1] forN prepared qubits.
conditional_measurement_indices: Optional, conditional measurement data
indices. If set this will return a list of fitted states conditioned
on a fixed basis measurement of these qubits.
conditional_preparation_indices: Optional, conditional preparation data
indices. If set this will return a list of fitted states conditioned
on a fixed basis preparation of these qubits.
atol: truncate any probabilities below this value to zero.
Raises:
AnalysisError: If the fitted vector is not a square matrix
Returns:
The fitted matrix rho.
"""
t_start = time.time()
if measurement_basis and measurement_qubits is None:
measurement_qubits = tuple(range(measurement_data.shape[1]))
if preparation_basis and preparation_qubits is None:
preparation_qubits = tuple(range(preparation_data.shape[1]))
input_dims = _basis_dimensions(
basis=preparation_basis,
qubits=preparation_qubits,
conditional_indices=conditional_preparation_indices,
)
output_dims = _basis_dimensions(
basis=measurement_basis,
qubits=measurement_qubits,
conditional_indices=conditional_measurement_indices,
)
metadata = {
"fitter": "linear_inversion",
"input_dims": input_dims,
"output_dims": output_dims,
}
if conditional_preparation_indices:
# Split measurement qubits into conditional and non-conditional qubits
f_prep_qubits = []
prep_indices = []
for i, qubit in enumerate(preparation_qubits):
if i not in conditional_preparation_indices:
f_prep_qubits.append(qubit)
prep_indices.append(i)
# Indexing array for fully tomo measured qubits
f_prep_indices = np.array(prep_indices, dtype=int)
f_cond_prep_indices = np.array(conditional_preparation_indices, dtype=int)
else:
f_prep_qubits = preparation_qubits
f_prep_indices = slice(None)
f_cond_prep_indices = slice(0, 0)
if conditional_measurement_indices:
# Split measurement qubits into conditional and non-conditional qubits
f_cond_meas_qubits = []
f_meas_qubits = []
meas_indices = []
for i, qubit in enumerate(measurement_qubits):
if i in conditional_measurement_indices:
f_cond_meas_qubits.append(qubit)
else:
f_meas_qubits.append(qubit)
meas_indices.append(i)
cond_meas_size = np.prod(measurement_basis.outcome_shape(f_cond_meas_qubits), dtype=int)
# Indexing array for fully tomo measured qubits
f_meas_indices = np.array(meas_indices, dtype=int)
f_cond_meas_indices = np.array(conditional_measurement_indices, dtype=int)
# Reduced outcome functions
f_meas_outcome = _partial_outcome_function(tuple(f_meas_indices))
f_cond_meas_outcome = _partial_outcome_function(tuple(conditional_measurement_indices))
else:
cond_meas_size = 1
f_meas_qubits = measurement_qubits
f_meas_indices = slice(None)
f_cond_meas_indices = slice(0, 0)
def f_meas_outcome(x):
return x
def f_cond_meas_outcome(_):
return 0
# Construct dual bases
meas_dual_basis = None
if measurement_basis and f_meas_qubits:
meas_duals = {i: _dual_povms(measurement_basis, i) for i in f_meas_qubits}
meas_dual_basis = LocalMeasurementBasis(
f"Dual_{measurement_basis.name}", qubit_povms=meas_duals
)
prep_dual_basis = None
if preparation_basis and f_prep_qubits:
prep_duals = {i: _dual_states(preparation_basis, i) for i in f_prep_qubits}
prep_dual_basis = LocalPreparationBasis(
f"Dual_{preparation_basis.name}", qubit_states=prep_duals
)
if shot_data is None:
# Define shots by sum of all outcome frequencies for each basis
shot_data = np.sum(outcome_data, axis=(0, -1))
# Calculate shape of matrix to be fitted
if prep_dual_basis:
pdim = np.prod(prep_dual_basis.matrix_shape(f_prep_qubits), dtype=int)
else:
pdim = 1
if meas_dual_basis:
mdim = np.prod(meas_dual_basis.matrix_shape(f_meas_qubits), dtype=int)
else:
mdim = 1
shape = (pdim * mdim, pdim * mdim)
# Construct linear inversion matrix
cond_circ_size = outcome_data.shape[0]
cond_fits = []
if cond_circ_size > 1:
metadata["conditional_circuit_outcome"] = []
if conditional_measurement_indices:
metadata["conditional_measurement_index"] = []
metadata["conditional_measurement_outcome"] = []
if conditional_preparation_indices:
metadata["conditional_preparation_index"] = []
for circ_idx in range(cond_circ_size):
fits = {}
for i, outcomes in enumerate(outcome_data[circ_idx]):
shots = shot_data[i]
pidx = preparation_data[i][f_prep_indices]
midx = measurement_data[i][f_meas_indices]
cond_prep_key = tuple(preparation_data[i][f_cond_prep_indices])
cond_meas_key = tuple(measurement_data[i][f_cond_meas_indices])
key = (cond_prep_key, cond_meas_key)
if key not in fits:
fits[key] = [np.zeros(shape, dtype=complex) for _ in range(cond_meas_size)]
# Get prep basis component
if prep_dual_basis:
p_mat = np.transpose(prep_dual_basis.matrix(pidx, f_prep_qubits))
else:
p_mat = 1
# Get probabilities and optional measurement basis component
for outcome, freq in enumerate(outcomes):
prob = freq / shots
if np.isclose(prob, 0, atol=atol):
# Skip component with zero probability
continue
# Get component on non-conditional bits
outcome_meas = f_meas_outcome(outcome)
if meas_dual_basis:
dual_op = meas_dual_basis.matrix(midx, outcome_meas, f_meas_qubits)
if prep_dual_basis:
dual_op = np.kron(p_mat, dual_op)
else:
dual_op = p_mat
# Add component to correct conditional
outcome_cond = f_cond_meas_outcome(outcome)
fits[key][outcome_cond] += prob * dual_op
# Append conditional fit metadata
for (prep_key, meas_key), c_fits in fits.items():
cond_fits += c_fits
if cond_circ_size > 1:
metadata["conditional_circuit_outcome"] += len(c_fits) * [circ_idx]
if conditional_measurement_indices:
metadata["conditional_measurement_index"] += len(c_fits) * [meas_key]
metadata["conditional_measurement_outcome"] += list(range(cond_meas_size))
if conditional_preparation_indices:
metadata["conditional_preparation_index"] += len(c_fits) * [prep_key]
t_stop = time.time()
metadata["fitter_time"] = t_stop - t_start
if len(cond_fits) == 1:
return cond_fits[0], metadata
return cond_fits, metadata
@lru_cache(None)
def _dual_states(basis: PreparationBasis, qubit: int) -> np.ndarray:
"""Construct a dual preparation basis for linear inversion"""
size = basis.index_shape((qubit,))[0]
states = np.asarray([basis.matrix((i,), (qubit,)) for i in range(size)])
return _construct_dual_states(states)
@lru_cache(None)
def _dual_povms(basis: MeasurementBasis, qubit: int) -> List[List[np.ndarray]]:
"""Construct dual POVM states for linear inversion"""
size = basis.index_shape((qubit,))[0]
num_outcomes = basis.outcome_shape((qubit,))[0]
# Concatenate all POVM effects to treat as states for linear inversion
states = []
for index in range(size):
for outcome in range(num_outcomes):
states.append(basis.matrix((index,), outcome, (qubit,)))
dual_basis = _construct_dual_states(states)
# Organize back into nested lists of dual POVM effects
dual_povms = []
idx = 0
for _ in range(size):
dual_povms.append([dual_basis[idx + i] for i in range(num_outcomes)])
idx += num_outcomes
return dual_povms
def _construct_dual_states(states: Sequence[np.ndarray]):
"""Construct a dual preparation basis for linear inversion"""
mats = np.asarray(states)
size, dim1, dim2 = np.shape(mats)
vec_basis = np.reshape(mats, (size, dim1 * dim2))
basis_mat = np.sum([np.outer(i, np.conj(i)) for i in vec_basis], axis=0)
try:
inv_mat = np.linalg.inv(basis_mat)
except np.linalg.LinAlgError as ex:
raise ValueError(
"Cannot construct dual basis states. Input states are not tomographically complete"
) from ex
vec_dual = np.tensordot(inv_mat, vec_basis, axes=([1], [1])).T
dual_mats = np.reshape(vec_dual, (size, dim1, dim2)).round(15)
return dual_mats