Source code for qiskit_experiments.library.tomography.fitters.cvxpy_lstsq

# 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.
"""
Constrained convex least-squares tomography fitter.
"""

from typing import Optional, Dict, Tuple, Union
import time
import numpy as np

from qiskit_experiments.library.tomography.basis import (
    MeasurementBasis,
    PreparationBasis,
)
from . import cvxpy_utils
from .cvxpy_utils import cvxpy
from . import lstsq_utils
from .fitter_data import _basis_dimensions


[docs] @cvxpy_utils.requires_cvxpy def cvxpy_linear_lstsq( 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[Tuple[int, ...]] = None, conditional_preparation_indices: Optional[Tuple[int, ...]] = None, trace: Union[None, float, str] = "auto", psd: bool = True, trace_preserving: Union[None, bool, str] = "auto", partial_trace: Optional[np.ndarray] = None, weights: Optional[np.ndarray] = None, **kwargs, ) -> Tuple[np.ndarray, Dict]: r"""Constrained weighted linear least-squares tomography fitter. Overview This fitter reconstructs the maximum-likelihood estimate by using ``cvxpy`` to minimize the constrained least-squares negative log likelihood function .. math:: \hat{\rho} &= -\mbox{argmin }\log\mathcal{L}{\rho} \\ &= \mbox{argmin }\sum_i w_i^2(\mbox{Tr}[E_j\rho] - \hat{p}_i)^2 \\ &= \mbox{argmin }\|W(Ax - y) \|_2^2 subject to - *Positive-semidefinite* (``psd=True``): :math:`\rho \gg 0` is constrained to be a positive-semidefinite matrix. - *Trace* (``trace=t``): :math:`\mbox{Tr}(\rho) = t` is constrained to have the specified trace. - *Trace preserving* (``trace_preserving=True``): When performing process tomography the Choi-state :math:`\rho` represents is constrained to be trace preserving. where - :math:`A` is the matrix of measurement operators :math:`A = \sum_i |i\rangle\!\langle\!\langle M_i|` - :math:`y` is the vector of expectation value data for each projector corresponding to estimates of :math:`b_i = Tr[M_i \cdot x]`. - :math:`x` is the vectorized density matrix (or Choi-matrix) to be fitted :math:`x = |\rho\rangle\\!\rangle`. .. note: Various solvers can be called in CVXPY using the `solver` keyword argument. When ``psd=True`` the optimization problem is a case of a *semidefinite program* (SDP) and requires a SDP compatible solver for CVXPY. CVXPY includes an SDP compatible solver `SCS`` but it is recommended to install the the open-source ``CVXOPT`` solver or one of the supported commercial solvers. See the `CVXPY documentation <https://www.cvxpy.org/tutorial/advanced/index.html#solve-method-options>`_ for more information on solvers. .. note:: Linear least-squares constructs the full basis matrix :math:`A` as a dense numpy array so should not be used for than 5 or 6 qubits. For larger number of qubits try the :func:`~qiskit_experiments.library.tomography.fitters.linear_inversion` fitter function. Args: outcome_data: measurement outcome frequency data. shot_data: basis measurement total shot data. measurement_data: measurement basis index data. preparation_data: preparation basis index data. measurement_basis: Optional, measurement matrix basis. preparation_basis: Optional, preparation matrix 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]`` for N 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. trace: trace constraint for the fitted matrix. If "auto" this will be set to 1 for QST or the input dimension for QST (default: "auto"). psd: If True rescale the eigenvalues of fitted matrix to be positive semidefinite (default: True) trace_preserving: Enforce the fitted matrix to be trace preserving when fitting a Choi-matrix in quantum process tomography. If "auto" this will be set to True for QPT and False for QST (default: "auto"). partial_trace: Enforce conditional fitted Choi matrices to partial trace to POVM matrices. weights: Optional array of weights for least squares objective. Weights should be the same shape as the outcome_data. kwargs: kwargs for cvxpy solver. Raises: QiskitError: If CVXPy is not installed on the current system. AnalysisError: If analysis fails. Returns: The fitted matrix rho that maximizes the least-squares likelihood function. """ 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, ) if trace_preserving == "auto" and preparation_data.shape[1] > 0: trace_preserving = True if trace == "auto" and output_dims is not None: trace = np.prod(output_dims) metadata = { "fitter": "cvxpy_linear_lstsq", "input_dims": input_dims, "output_dims": output_dims, } # Conditional measurement indices if conditional_measurement_indices: cond_meas_data = measurement_data[:, np.array(conditional_measurement_indices, dtype=int)] cond_meas_indices = np.unique(cond_meas_data, axis=0) num_meas_cond = len(cond_meas_indices) metadata["conditional_measurement_index"] = [] metadata["conditional_measurement_outcome"] = [] else: num_meas_cond = 0 cond_meas_data = np.zeros((measurement_data.shape[0], 0), dtype=int) cond_meas_indices = np.zeros((1, 0), dtype=int) if conditional_preparation_indices: cond_prep_data = preparation_data[:, np.array(conditional_preparation_indices, dtype=int)] cond_prep_indices = np.unique(cond_prep_data, axis=0) num_prep_cond = len(cond_meas_indices) metadata["conditional_preparation_index"] = [] else: num_prep_cond = 0 cond_prep_data = np.zeros((preparation_data.shape[0], 0), dtype=int) cond_prep_indices = np.zeros((1, 0), dtype=int) if outcome_data.shape[0] > 1: metadata["conditional_circuit_outcome"] = [] fits = [] for cond_prep_idx in cond_prep_indices: for cond_meas_idx in cond_meas_indices: # Mask for specified conditional indices cond_mask = np.all(cond_meas_data == cond_meas_idx, axis=1) & np.all( cond_prep_data == cond_prep_idx, axis=1 ) if weights is None: cond_weights = None else: cond_weights = weights[:, cond_mask] basis_matrix, probability_data, probability_weights = lstsq_utils.lstsq_data( outcome_data[:, cond_mask], shot_data[cond_mask], measurement_data[cond_mask], preparation_data[cond_mask], measurement_basis=measurement_basis, preparation_basis=preparation_basis, measurement_qubits=measurement_qubits, preparation_qubits=preparation_qubits, weights=cond_weights, conditional_measurement_indices=conditional_measurement_indices, conditional_preparation_indices=conditional_preparation_indices, ) # Since CVXPY only works with real variables we must specify the real # and imaginary parts of matrices separately: rho = rho_r + 1j * rho_i num_circ_components, num_tomo_components, _ = probability_data.shape dim = int(np.sqrt(basis_matrix.shape[1])) # Generate list of conditional components for block diagonal matrix # rho = sum_k |k><k| \otimes rho(k) rhos_r = [] rhos_i = [] cons = [] for i in range(num_circ_components): for j in range(num_tomo_components): rho_r, rho_i, cons_i = cvxpy_utils.complex_matrix_variable( dim, hermitian=True, psd=psd ) rhos_r.append(rho_r) rhos_i.append(rho_i) cons.append(cons_i) if num_circ_components > 1: metadata["conditional_circuit_outcome"].append(i) if num_meas_cond: metadata["conditional_measurement_index"].append(tuple(cond_meas_idx)) metadata["conditional_measurement_outcome"].append(j) if num_prep_cond: metadata["conditional_preparation_index"].append(tuple(cond_prep_idx)) # Partial trace when fitting Choi-matrices for quantum process tomography. # This applied to the sum of conditional components # Note that this adds an implicitly # trace preserving is a specific partial trace constraint ptrace(rho) = I # Note: partial trace constraints implicitly define a trace constraint, # so if a different trace constraint is specified it will be ignored joint_cons = None if partial_trace is not None: for rho_r, rho_i, povm in zip(rhos_r, rhos_i, partial_trace): joint_cons = cvxpy_utils.partial_trace_constaint(rho_r, rho_i, povm) elif trace_preserving: input_dim = np.prod(input_dims) joint_cons = cvxpy_utils.trace_preserving_constaint( rhos_r, rhos_i, input_dim=input_dim, hermitian=True, ) elif trace is not None: joint_cons = cvxpy_utils.trace_constraint( rhos_r, rhos_i, trace=trace, hermitian=True ) # OBJECTIVE FUNCTION # The function we wish to minimize is || arg ||_2 where # arg = bm * vec(rho) - data # Since we are working with real matrices in CVXPY we expand this as # bm * vec(rho) = (bm_r + 1j * bm_i) * vec(rho_r + 1j * rho_i) # = bm_r * vec(rho_r) - bm_i * vec(rho_i) # + 1j * (bm_r * vec(rho_i) + bm_i * vec(rho_r)) # = bm_r * vec(rho_r) - bm_i * vec(rho_i) # where we drop the imaginary part since the expectation value is real # Construct block diagonal fit variable from conditional components # Construct objective function if probability_weights is not None: probability_data = probability_weights * probability_data bms_r = [] bms_i = [] for i in range(num_circ_components): for j in range(num_tomo_components): weighted_mat = probability_weights[i, j][:, None] * basis_matrix bms_r.append(np.real(weighted_mat)) bms_i.append(np.imag(weighted_mat)) else: bm_r = np.real(basis_matrix) bm_i = np.imag(basis_matrix) bms_r = [bm_r] * num_circ_components * num_tomo_components bms_i = [bm_i] * num_circ_components * num_tomo_components # Stack lstsq objective from sum of components args = [] idx = 0 for i in range(num_circ_components): for j in range(num_tomo_components): model = bms_r[idx] @ cvxpy.vec(rhos_r[idx], order="F") - bms_i[idx] @ cvxpy.vec( rhos_i[idx], order="F" ) data = probability_data[i, j] args.append(model - data) idx += 1 # Combine all variables and constraints into a joint optimization problem # if there is a joint constraint if joint_cons: args = [cvxpy.hstack(args)] for cons_i in cons: joint_cons += cons_i cons = [joint_cons] # Solve each component separately metadata["cvxpy_solver"] = None metadata["cvxpy_status"] = [] for arg, con in zip(args, cons): # Optimization problem obj = cvxpy.Minimize(cvxpy.norm(arg, p=2)) prob = cvxpy.Problem(obj, con) # Solve SDP cvxpy_utils.solve_iteratively(prob, 5000, **kwargs) # Return optimal values and problem metadata metadata["cvxpy_solver"] = prob.solver_stats.solver_name metadata["cvxpy_status"].append(prob.status) fits += [rho_r.value + 1j * rho_i.value for rho_r, rho_i in zip(rhos_r, rhos_i)] # Add additional metadata if psd: metadata["psd_constraint"] = True if partial_trace is not None: metadata["partial_trace"] = partial_trace elif trace_preserving: metadata["trace_preserving"] = True elif trace is not None: metadata["trace"] = trace t_stop = time.time() metadata["fitter_time"] = t_stop - t_start if len(fits) == 1: return fits[0], metadata return fits, metadata
[docs] @cvxpy_utils.requires_cvxpy def cvxpy_gaussian_lstsq( 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[Tuple[int, ...]] = None, trace: Union[None, float, str] = "auto", psd: bool = True, trace_preserving: Union[None, bool, str] = "auto", partial_trace: Optional[np.ndarray] = None, outcome_prior: Union[np.ndarray, int] = 0.5, max_weight: float = 1e10, **kwargs, ) -> Dict: r"""Constrained Gaussian linear least-squares tomography fitter. .. note:: This function calls :func:`cvxpy_linear_lstsq` with a Gaussian weights vector. Refer to its documentation for additional details. Overview This fitter reconstructs the maximum-likelihood estimate by using ``cvxpy`` to minimize the constrained least-squares negative log likelihood function .. math:: \hat{\rho} &= \mbox{argmin} (-\log\mathcal{L}{\rho}) \\ &= \mbox{argmin }\|W(Ax - y) \|_2^2 \\ -\log\mathcal{L}(\rho) &= |W(Ax -y) \|_2^2 \\ &= \sum_i \frac{1}{\sigma_i^2}(\mbox{Tr}[E_j\rho] - \hat{p}_i)^2 Additional Details The Gaussian weights are estimated from the observed frequency and shot data via a Bayesian update of a Dirichlet distribution with observed outcome data frequencies :math:`f_i(s)`, and Dirichlet prior :math:`\alpha_i(s)` for tomography basis index `i` and measurement outcome `s`. The mean posterior probabilities are computed as .. math: p_i(s) &= \frac{f_i(s) + \alpha_i(s)}{\bar{\alpha}_i + N_i} \\ Var[p_i(s)] &= \frac{p_i(s)(1-p_i(s))}{\bar{\alpha}_i + N_i + 1} w_i(s) = \sqrt{Var[p_i(s)]}^{-1} where :math:`N_i = \sum_s f_i(s)` is the total number of shots, and :math:`\bar{\alpha}_i = \sum_s \alpha_i(s)` is the norm of the prior. Args: outcome_data: measurement outcome frequency data. shot_data: basis measurement total shot data. measurement_data: measurement basis index data. preparation_data: preparation basis index data. measurement_basis: Optional, measurement matrix basis. preparation_basis: Optional, preparation matrix 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]`` for N prepared qubits. conditional_measurement_indices: Optional, conditional measurement data indices. If set this will return a list of conditional fitted states conditioned on a fixed basis measurement of these qubits. trace: trace constraint for the fitted matrix. If "auto" this will be set to 1 for QST or the input dimension for QST (default: "auto"). psd: If True rescale the eigenvalues of fitted matrix to be positive semidefinite (default: True) trace_preserving: Enforce the fitted matrix to be trace preserving when fitting a Choi-matrix in quantum process tomography. If "auto" this will be set to True for QPT and False for QST (default: "auto"). partial_trace: Enforce conditional fitted Choi matrices to partial trace to POVM matrices. outcome_prior: The Bayesian prior :math:`\alpha` to use computing Gaussian weights. See additional information. max_weight: Set the maximum value allowed for weights vector computed from tomography data variance. kwargs: kwargs for cvxpy solver. Raises: QiskitError: If CVXPY is not installed on the current system. AnalysisError: If analysis fails. Returns: The fitted matrix rho that maximizes the least-squares likelihood function. """ t_start = time.time() weights = lstsq_utils.binomial_weights( outcome_data, shot_data=shot_data, outcome_prior=outcome_prior, max_weight=max_weight, ) fits, metadata = cvxpy_linear_lstsq( outcome_data, shot_data, measurement_data, preparation_data, measurement_basis=measurement_basis, preparation_basis=preparation_basis, measurement_qubits=measurement_qubits, preparation_qubits=preparation_qubits, conditional_measurement_indices=conditional_measurement_indices, trace=trace, psd=psd, trace_preserving=trace_preserving, partial_trace=partial_trace, weights=weights, **kwargs, ) t_stop = time.time() # Update metadata metadata["fitter"] = "cvxpy_gaussian_lstsq" metadata["fitter_time"] = t_stop - t_start return fits, metadata