qutip/core/metrics.py
# -*- 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']
import numpy as 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
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()
return float(np.real(np.sqrt(eig_vals[eig_vals > 0]).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)
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
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)
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 float(np.real(0.5 * np.sum(np.sqrt(np.abs(vals)))))
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()
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
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))
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))))
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
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: np.ndarray) -> 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)