Source code for thewalrus.csamples
# Copyright 2019 Xanadu Quantum Technologies Inc.
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
# http://www.apache.org/licenses/LICENSE-2.0
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
r"""
Classical sampling algorithms
=============================
**Module name:** :mod:`thewalrus.csamples`
.. currentmodule:: thewalrus.csamples
This submodule provides access to classical sampling algorithms for thermal states going through
an interferometer specified by a real orthogonal matrix. The quantum state to be sampled is
specified by a positive semidefinite real matrix and a mean photon number. The algorithm implemented
here was first derived in
* Saleh Rahimi-Keshari, Austin P Lund, and Timothy C Ralph.
"What can quantum optics say about computational complexity theory?" `Physical Review Letters, 114(6):060501, (2015).
<https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.060501>`_
For more precise details of the implementation see
* Soran Jahangiri, Juan Miguel Arrazola, Nicolás Quesada, and Nathan Killoran.
"Point processes with Gaussian boson sampling" `Phys. Rev. E 101, 022134, (2020).
<https://journals.aps.org/pre/abstract/10.1103/PhysRevE.101.022134>`_.
Summary
-------
.. autosummary::
rescale_adjacency_matrix_thermal
rescale_adjacency_matrix
generate_thermal_samples
Code details
------------
"""
# pylint: disable=too-many-arguments
import numpy as np
from scipy.optimize import root_scalar
[docs]
def rescale_adjacency_matrix_thermal(
A, n_mean, check_positivity=True, check_symmetry=True, rtol=1e-05, atol=1e-08
):
r"""Returns the scaling parameter by which the adjacency matrix A
should be rescaled so that the Gaussian state that encodes it has
a total mean photon number n_mean for thermal sampling.
Args:
A (array): Adjacency matrix, assumed to be positive semi-definite and real
n_mean (float): Mean photon number of the Gaussian state
check_positivity (bool): Checks if the matrix A is positive semidefinite
check_symmetry (bool): Checks if the matrix is symmetric
rtol: relative tolerance for the checks
atol: absolute tolerance for the checks
Returns:
tuple(array,array): rescaled eigenvalues and eigenvectors of the matrix A
"""
return rescale_adjacency_matrix(
A,
n_mean,
1.0,
check_positivity=check_positivity,
check_symmetry=check_symmetry,
rtol=rtol,
atol=atol,
)
[docs]
def rescale_adjacency_matrix(
A, n_mean, scale, check_positivity=True, check_symmetry=True, rtol=1e-05, atol=1e-08
):
r"""Returns the scaling parameter by which the adjacency matrix A
should be rescaled so that the Gaussian state that encodes it has
a total mean photon number n_mean.
Args:
A (array): Adjacency matrix, assumed to be positive semi-definite and real
n_mean (float): Mean photon number of the Gaussian state
scale (float): Determines whether to rescale the matrix for thermal sampling (scale = 1.0)
or for squashed sampling (scale = 2.0)
check_positivity (bool): Checks if the matrix A is positive semidefinite
check_symmetry (bool): Checks if the matrix is symmetric
rtol: relative tolerance for the checks
atol: absolute tolerance for the checks
Returns:
tuple(array,array): rescaled eigenvalues and eigenvectors of the matrix A
"""
ls, O = np.linalg.eigh(A)
ls[np.abs(ls) < atol] = 0.0
if check_symmetry is True:
assert np.allclose(A, A.T, rtol=rtol, atol=atol)
if check_positivity is True:
assert np.all(ls >= 0)
max_sv = ls[-1]
a_lim = 0.0
b_lim = 1 / (atol + scale * max_sv)
x_init = 0.5 * b_lim
def mean_photon_number(x, vals):
r"""Returns the mean number of photons in the Gaussian state that
encodes the adjacency matrix x*A where vals are the eigenvalues of positive semidefinite A
Args:
x (float): Scaling parameter
vals (array): Eigenvalues of the matrix A
Returns:
n_mean: Mean photon number in the Gaussian state
"""
vals2 = scale * x * vals
n = np.sum(vals2 / (1.0 - vals2))
return n
# The following function is implicitly tested in test_rescaling_thermal
def grad_mean_photon_number(x, vals): # pragma: no cover
r"""Returns the gradient of the mean number of photons in the Gaussian state that
encodes the adjacency matrix x*A with respect to x.
vals are the eigenvalues of A
Args:
x (float): Scaling parameter
vals (array): Eigenvalues of the matrix A
Returns:
d_n_mean: Derivative of the mean photon number in the Gaussian state
with respect to x
"""
vals1 = scale * x * vals
dn = np.sum((scale * vals) / (1 - vals1) ** 2)
return dn
f = lambda x: mean_photon_number(x, ls) - n_mean
df = lambda x: grad_mean_photon_number(x, ls)
res = root_scalar(f, fprime=df, x0=x_init, bracket=(a_lim, b_lim))
assert res.converged
return res.root * ls, O
[docs]
def generate_thermal_samples(ls, O, num_samples=1):
r"""Generates samples of the Gaussian state in terms of the mean photon number parameter ls and the interferometer O.
Args:
ls (array): squashing parameters
O (array): Orthogonal matrix representing the interferometer
num_samples: Number of samples to generate
Returns:
list(array: samples
"""
rs = 0.5 * ls / (1 - ls)
return [
np.random.poisson(
np.abs(O @ (np.random.normal(0, np.sqrt(rs)) + 1j * np.random.normal(0, np.sqrt(rs))))
** 2
)
for _ in range(num_samples)
]
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