Source code for qutip.core.metrics

# -*- coding: utf-8 -*-

"""
This module contains a collection of functions for calculating metrics
(distance measures) between states and operators.
"""

__all__ = ['fidelity', 'tracedist', 'bures_dist', 'bures_angle',
           'hellinger_dist', 'hilbert_dist', 'average_gate_fidelity',
           'process_fidelity', 'unitarity', 'dnorm']

from .numpy_backend import np
from scipy import linalg as la
import scipy.sparse as sp
from .superop_reps import to_choi, _to_superpauli, to_super, kraus_to_choi
from .superoperator import operator_to_vector, vector_to_operator
from .operators import qeye, qeye_like
from .states import ket2dm
from .semidefinite import dnorm_problem, dnorm_sparse_problem
from . import data as _data

try:
    import cvxpy
except ImportError:
    cvxpy = None


[docs] def fidelity(A, B): """ Calculates the fidelity (pseudo-metric) between two density matrices. Notes ----- Uses the definition from Nielsen & Chuang, "Quantum Computation and Quantum Information". It is the square root of the fidelity defined in R. Jozsa, Journal of Modern Optics, 41:12, 2315 (1994), used in :func:`qutip.core.metrics.process_fidelity`. Parameters ---------- A : qobj Density matrix or state vector. B : qobj Density matrix or state vector with same dimensions as A. Returns ------- fid : float Fidelity pseudo-metric between A and B. Examples -------- >>> x = fock_dm(5,3) >>> y = coherent_dm(5,1) >>> np.testing.assert_almost_equal(fidelity(x,y), 0.24104350624628332) """ if A.isket or A.isbra: if B.isket or B.isbra: # The fidelity for pure states reduces to the modulus of their # inner product. return np.abs(A.overlap(B)) # Take advantage of the fact that the density operator for A # is a projector to avoid a sqrtm call. sqrtmA = ket2dm(A) else: if B.isket or B.isbra: # Swap the order so that we can take a more numerically # stable square root of B. return fidelity(B, A) # If we made it here, both A and B are operators, so # we have to take the sqrtm of one of them. sqrtmA = A.sqrtm() if sqrtmA.dims != B.dims: raise TypeError('Density matrices do not have same dimensions.') # We don't actually need the whole matrix here, just the trace # of its square root, so let's just get its eigenenergies instead. # We also truncate negative eigenvalues to avoid nan propagation; # even for positive semidefinite matrices, small negative eigenvalues # can be reported. eig_vals = (sqrtmA * B * sqrtmA).eigenenergies() eig_vals_non_neg = np.where(eig_vals > 0, eig_vals, 0) return np.real(np.sqrt(eig_vals_non_neg).sum())
def _hilbert_space_dims(oper): """ For a quantum channel `oper`, return the dimensions `[dims_out, dims_in]` of the output Hilbert space and the input Hilbert space. - If oper is a unitary, then `oper.dims == [dims_out, dims_in]`. - If oper is a list of Kraus operators, then `oper[0].dims == [dims_out, dims_in]`. - If oper is a superoperator with `oper.superrep == 'super'`: `oper.dims == [[dims_out, dims_out], [dims_in, dims_in]]` - If oper is a superoperator with `oper.superrep == 'choi'`: `oper.dims == [[dims_in, dims_out], [dims_in, dims_out]]` - If oper is a superoperator with `oper.superrep == 'chi', then `dims_out == dims_in` and `oper.dims == [[dims_out, dims_out], [dims_out, dims_out]]`. :param oper: A quantum channel, represented by a unitary, a list of Kraus operators, or a superoperator :return: `[dims_out, dims_in]`, where `dims_out` and `dims_in` are lists of integers """ if isinstance(oper, list): return oper[0].dims elif oper.type == 'oper': # interpret as unitary quantum channel return oper.dims elif oper.type == 'super' and oper.superrep in ['choi', 'chi', 'super']: return [oper.dims[0][1], oper.dims[1][0]] else: raise TypeError('oper is not a valid quantum channel!') def _process_fidelity_to_id(oper): """ Internal function returning the process fidelity of a quantum channel to the identity quantum channel. Parameters ---------- oper : :class:`.Qobj`/list A unitary operator, or a superoperator in supermatrix, Choi or chi-matrix form, or a list of Kraus operators Returns ------- fid : float """ dims_out, dims_in = _hilbert_space_dims(oper) if dims_out != dims_in: raise TypeError('The process fidelity to identity is only defined ' 'for dimension preserving channels.') d = np.prod(dims_in) if isinstance(oper, list): # oper is a list of Kraus operators return np.sum([np.abs(k.tr()) ** 2 for k in oper]) / d ** 2 elif oper.type == 'oper': # interpret as unitary return np.abs(oper.tr()) ** 2 / d ** 2 elif oper.type == 'super': if oper.superrep == 'chi': return oper[0, 0].real / d ** 2 else: # oper.superrep is either 'super' or 'choi': return to_super(oper).tr().real / d ** 2 def _kraus_or_qobj_to_choi(oper): if isinstance(oper, list): return kraus_to_choi(oper) else: return to_choi(oper)
[docs] def process_fidelity(oper, target=None): """ Returns the process fidelity of a quantum channel to the target channel, or to the identity channel if no target is given. The process fidelity between two channels is defined as the state fidelity between their normalized Choi matrices. Parameters ---------- oper : :class:`.Qobj`/list A unitary operator, or a superoperator in supermatrix, Choi or chi-matrix form, or a list of Kraus operators target : :class:`.Qobj`/list, optional A unitary operator, or a superoperator in supermatrix, Choi or chi-matrix form, or a list of Kraus operators Returns ------- fid : float Process fidelity between oper and target, or between oper and identity. Notes ----- Since Qutip 5.0, this function computes the process fidelity as defined for example in: A. Gilchrist, N.K. Langford, M.A. Nielsen, Phys. Rev. A 71, 062310 (2005). Previously, it computed a function that is now implemented as ``get_fidelity`` in qutip-qtrl. The definition of state fidelity that the process fidelity is based on is the one from R. Jozsa, Journal of Modern Optics, 41:12, 2315 (1994). It is the square of the one implemented in :func:`qutip.core.metrics.fidelity` which follows Nielsen & Chuang, "Quantum Computation and Quantum Information" """ if target is None: return _process_fidelity_to_id(oper) dims_out, dims_in = _hilbert_space_dims(oper) if dims_out != dims_in: raise NotImplementedError('Process fidelity only implemented for ' 'dimension-preserving operators.') dims_out_target, dims_in_target = _hilbert_space_dims(target) if [dims_out, dims_in] != [dims_out_target, dims_in_target]: raise TypeError('Dimensions of oper and target do not match') if not isinstance(target, list) and target.type == 'oper': # interpret target as unitary. if isinstance(oper, list): # oper is a list of Kraus operators return _process_fidelity_to_id([k * target.dag() for k in oper]) elif oper.type == 'oper': return _process_fidelity_to_id(oper * target.dag()) elif oper.type == 'super': oper_super = to_super(oper) target_dag_super = to_super(target.dag()) return _process_fidelity_to_id(oper_super * target_dag_super) else: # target is a list of Kraus operators or a superoperator if not isinstance(oper, list) and oper.type == 'oper': return process_fidelity(target, oper) # reverse order oper_choi = _kraus_or_qobj_to_choi(oper) target_choi = _kraus_or_qobj_to_choi(target) d = np.prod(dims_in) return (fidelity(oper_choi, target_choi)/d)**2
[docs] def average_gate_fidelity(oper, target=None): """ Returns the average gate fidelity of a quantum channel to the target channel, or to the identity channel if no target is given. Parameters ---------- oper : :class:`.Qobj`/list A unitary operator, or a superoperator in supermatrix, Choi or chi-matrix form, or a list of Kraus operators target : :class:`.Qobj` A unitary operator Returns ------- fid : float Average gate fidelity between oper and target, or between oper and identity. Notes ----- The average gate fidelity is defined for example in: A. Gilchrist, N.K. Langford, M.A. Nielsen, Phys. Rev. A 71, 062310 (2005). The definition of state fidelity that the average gate fidelity is based on is the one from R. Jozsa, Journal of Modern Optics, 41:12, 2315 (1994). It is the square of the fidelity implemented in :func:`qutip.core.metrics.fidelity` which follows Nielsen & Chuang, "Quantum Computation and Quantum Information" """ dims_out, dims_in = _hilbert_space_dims(oper) if not (target is None or target.type == 'oper'): raise TypeError( 'target must be None or a Qobj representing a unitary.') d = np.prod(dims_in) return (d * process_fidelity(oper, target) + 1) / (d + 1)
[docs] def tracedist(A, B, sparse=False, tol=0): """ Calculates the trace distance between two density matrices.. See: Nielsen & Chuang, "Quantum Computation and Quantum Information" Parameters ----------!= A : qobj Density matrix or state vector. B : qobj Density matrix or state vector with same dimensions as A. tol : float, default: 0 Tolerance used by sparse eigensolver, if used. (0 = Machine precision) sparse : bool, default: False Use sparse eigensolver. Returns ------- tracedist : float Trace distance between A and B. Examples -------- >>> x=fock_dm(5,3) >>> y=coherent_dm(5,1) >>> np.testing.assert_almost_equal(tracedist(x,y), 0.9705143161472971) """ if A.isket or A.isbra: A = A.proj() if B.isket or B.isbra: B = B.proj() if A.dims != B.dims: raise TypeError("A and B do not have same dimensions.") diff = A - B diff = diff.dag() * diff vals = diff.eigenenergies(sparse=sparse, tol=tol) return np.real(0.5 * np.sum(np.sqrt(np.abs(vals))))
[docs] def hilbert_dist(A, B): """ Returns the Hilbert-Schmidt distance between two density matrices A & B. Parameters ---------- A : qobj Density matrix or state vector. B : qobj Density matrix or state vector with same dimensions as A. Returns ------- dist : float Hilbert-Schmidt distance between density matrices. Notes ----- See V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998). """ if A.isket or A.isbra: A = A.proj() if B.isket or B.isbra: B = B.proj() if A.dims != B.dims: raise TypeError('A and B do not have same dimensions.') return ((A - B)**2).tr()
[docs] def bures_dist(A, B): """ Returns the Bures distance between two density matrices A & B. The Bures distance ranges from 0, for states with unit fidelity, to sqrt(2). Parameters ---------- A : qobj Density matrix or state vector. B : qobj Density matrix or state vector with same dimensions as A. Returns ------- dist : float Bures distance between density matrices. """ if A.isket or A.isbra: A = A.proj() if B.isket or B.isbra: B = B.proj() if A.dims != B.dims: raise TypeError('A and B do not have same dimensions.') dist = np.sqrt(2 * (1 - fidelity(A, B))) return dist
[docs] def bures_angle(A, B): """ Returns the Bures Angle between two density matrices A & B. The Bures angle ranges from 0, for states with unit fidelity, to pi/2. Parameters ---------- A : qobj Density matrix or state vector. B : qobj Density matrix or state vector with same dimensions as A. Returns ------- angle : float Bures angle between density matrices. """ if A.isket or A.isbra: A = A.proj() if B.isket or B.isbra: B = B.proj() if A.dims != B.dims: raise TypeError('A and B do not have same dimensions.') return np.arccos(fidelity(A, B))
[docs] def hellinger_dist(A, B, sparse=False, tol=0): """ Calculates the quantum Hellinger distance between two density matrices. Formula: ``hellinger_dist(A, B) = sqrt(2 - 2 * tr(sqrt(A) * sqrt(B)))`` See: D. Spehner, F. Illuminati, M. Orszag, and W. Roga, "Geometric measures of quantum correlations with Bures and Hellinger distances" arXiv:1611.03449 Parameters ---------- A : :class:`.Qobj` Density matrix or state vector. B : :class:`.Qobj` Density matrix or state vector with same dimensions as A. tol : float, default: 0 Tolerance used by sparse eigensolver, if used. (0 = Machine precision) sparse : bool, default: False Use sparse eigensolver. Returns ------- hellinger_dist : float Quantum Hellinger distance between A and B. Ranges from 0 to sqrt(2). Examples -------- >>> x = fock_dm(5,3) >>> y = coherent_dm(5,1) >>> np.allclose(hellinger_dist(x, y), 1.3725145002591095) True """ if A.isket or A.isbra: sqrtmA = ket2dm(A) else: sqrtmA = A.sqrtm(sparse=sparse, tol=tol) if B.isket or B.isbra: sqrtmB = ket2dm(B) else: sqrtmB = B.sqrtm(sparse=sparse, tol=tol) if sqrtmA.dims != sqrtmB.dims: raise TypeError("A and B do not have compatible dimensions.") product = sqrtmA*sqrtmB eigs = product.eigenenergies(sparse=sparse, tol=tol) # np.maximum() is to avoid nan appearing sometimes due to numerical # instabilities causing np.sum(eigs) slightly (~1e-8) larger than 1 when # hellinger_dist(A, B) is called for A=B return np.sqrt(2.0 * np.maximum(0, 1 - np.real(np.sum(eigs))))
[docs] def dnorm(A, B=None, solver="CVXOPT", verbose=False, force_solve=False, sparse=True): r""" Calculates the diamond norm of the quantum map q_oper, using the simplified semidefinite program of [Wat13]_. The diamond norm SDP is solved by using `CVXPY <https://www.cvxpy.org/>`_. If B is provided and both A and B are unitary, then the diamond norm of the difference is calculated more efficiently using the following geometric interpretation: :math:`\|A - B\|_{\diamond}` equals :math:`2 \sqrt(1 - d^2)`, where :math:`d`is the distance between the origin and the convex hull of the eigenvalues of :math:`A B^{\dagger}`. See [AKN98]_ page 18, in the paragraph immediately below the proof of 12.6, as a reference. Parameters ---------- A : Qobj Quantum map to take the diamond norm of. B : Qobj or None If provided, the diamond norm of :math:`A - B` is taken instead. solver : str {"CVXOPT", "SCS"}, default: "CVXOPT" Solver to use with CVXPY. "SCS" tends to be significantly faster, but somewhat less accurate. verbose : bool, default: False If True, prints additional information about the solution. force_solve : bool, default: False If True, forces dnorm to solve the associated SDP, even if a special case is known for the argument. sparse : bool, default: True Whether to use sparse matrices in the convex optimisation problem. Default True. Returns ------- dn : float Diamond norm of q_oper. Raises ------ ImportError If CVXPY cannot be imported. """ if cvxpy is None: # pragma: no cover raise ImportError("dnorm() requires CVXPY to be installed.") if B is not None and A.dims != B.dims: raise TypeError("A and B do not have the same dimensions.") # We follow the strategy of using Watrous' simpler semidefinite # program in its primal form. This is the same strategy used, # for instance, by both pyGSTi and SchattenNorms.jl. (By contrast, # QETLAB uses the dual problem.) # Check if A and B are both unitaries. If so we can use the geometric # interpretation mentioned in D. Aharonov, A. Kitaev, and N. Nisan. (1998). # We find the eigenvalues of AB⁺ and the distance d between the origin # and the complex hull of these. Plugging this into 2√1-d² gives the # diamond norm. if ( not force_solve and A.isunitary and B is not None and B.isunitary ): # Special optimisation for a difference of unitaries. U = A * B.dag() eigs = U.eigenenergies() d = _find_poly_distance(eigs) return 2 * np.sqrt(1 - d**2) # plug d into formula J = to_choi(A) if B is not None: # If B is provided, calculate difference J -= to_choi(B) if not force_solve and J.iscptp: # diamond norm of a CPTP map is 1 (Prop 3.44 Watrous 2018) return 1.0 # Watrous 2012 also points out that the diamond norm of Lambda # is the same as the completely-bounded operator-norm (∞-norm) # of the dual map of Lambda. We can evaluate that norm much more # easily if Lambda is completely positive, since then the largest # eigenvalue is the same as the largest singular value. if not force_solve and J.iscp: S_dual = to_super(J.dual_chan()) vec_eye = operator_to_vector(qeye(S_dual.dims[1][1])) op = vector_to_operator(S_dual * vec_eye) # The 2-norm was not implemented for sparse matrices as of the time # of this writing. Thus, we must yet again go dense. return la.norm(op.full(), 2) # If we're still here, we need to actually solve the problem. # Assume square... dim = int(np.prod(J.dims[0][0])) # Load the parameters with the Choi matrix passed in. J_dat = _data.to('csr', J.data).as_scipy() if not sparse: problem, Jr, Ji = dnorm_problem(dim) # Load the parameters with the Choi matrix passed in. Jr.value = sp.csr_matrix((J_dat.data.real, J_dat.indices, J_dat.indptr), shape=J_dat.shape).toarray() Ji.value = sp.csr_matrix((J_dat.data.imag, J_dat.indices, J_dat.indptr), shape=J_dat.shape).toarray() else: problem = dnorm_sparse_problem(dim, J_dat) problem.solve(solver=solver, verbose=verbose) return problem.value
[docs] def unitarity(oper): """ Returns the unitarity of a quantum map, defined as the Frobenius norm of the unital block of that map's superoperator representation. Parameters ---------- oper : Qobj Quantum map under consideration. Returns ------- u : float Unitarity of ``oper``. """ Eu = _to_superpauli(oper).full()[1:, 1:] return np.linalg.norm(Eu, 'fro')**2 / len(Eu)
def _find_poly_distance(eigenvals) -> float: """ Returns the distance between the origin and the convex hull of eigenvalues. The complex eigenvalues must have unit length (i.e. lie on the circle about the origin). """ phases = np.angle(eigenvals) phase_max = phases.max() phase_min = phases.min() if phase_min > 0: # all eigenvals have pos phase: hull is above x axis return np.cos((phase_max - phase_min) / 2) if phase_max <= 0: # all eigenvals have neg phase: hull is below x axis return np.cos((np.abs(phase_min) - np.abs(phase_max)) / 2) pos_phase_min = np.where(phases > 0, phases, np.inf).min() neg_phase_max = np.where(phases <= 0, phases, -np.inf).max() big_angle = phase_max - phase_min small_angle = pos_phase_min - neg_phase_max if big_angle >= np.pi: if small_angle <= np.pi: # hull contains the origin return 0 else: # hull is left of y axis return np.cos((2 * np.pi - small_angle) / 2) else: # hull is right of y axis return np.cos(big_angle / 2)