Source code for thewalrus.quantum.adjacency_matrices

# Copyright 2019-2020 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.
"""
Functions for rescaling adjacency matrices as well as calculating the Qmat
covariance matrix from an adjacency matrix.
"""

import numpy as np
from scipy.optimize import root_scalar

from .conversions import Xmat, Covmat
from .means_and_variances import mean_clicks


[docs] def adj_scaling_torontonian(A, c_mean): r"""Returns the scaling parameter by which the adjacency matrix A should be rescaled so that the Gaussian state that encodes it has give a mean number of clicks equal to ``c_mean`` when measured with threshold detectors. Args: A (array): adjacency matrix c_mean (float): mean photon number of the Gaussian state Returns: float: scaling parameter """ n, _ = A.shape if c_mean < 0 or c_mean > n: raise ValueError("The mean number of clicks should be smaller than the number of modes") vals = np.linalg.svd(A, compute_uv=False) localA = A / vals[0] # rescale the matrix so that the singular values are between 0 and 1. def cost(x): r"""Cost function giving the difference between the wanted number of clicks and the number of clicks at a given scaling value. It assumes that the adjacency matrix has been rescaled so that it has singular values between 0 and 1. Args: x (float): scaling value Return: float: difference between desired and obtained mean number of clicks """ if x >= 1.0: return c_mean - n if x <= 0: return c_mean return c_mean - _mean_clicks_adj(x * localA) res = root_scalar(cost, x0=0.5, bracket=(0.0, 1.0)) # Do the optimization if not res.converged: raise ValueError("The search for a scaling value failed") return res.root / vals[0]
def _mean_clicks_adj(A): r"""Given an adjacency matrix this function calculates the mean number of clicks. For this to make sense the user must provide a matrix with singular values less than or equal to one. See Appendix A.3 of <https://arxiv.org/abs/1902.00462>`_ by Banchi et al. Args: A (array): rescaled adjacency matrix Returns: float: mean number of clicks """ n, _ = A.shape idn = np.identity(n) X = np.block([[0 * idn, idn], [idn, 0 * idn]]) B = np.block([[A, 0 * A], [0 * A, np.conj(A)]]) Q = np.linalg.inv(np.identity(2 * n) - X @ B) return mean_clicks(Covmat(Q))
[docs] def adj_scaling(A, n_mean): r"""Returns the scaling parameter by which the adjacency matrix A should be rescaled so that the Gaussian state that endodes it has a total mean photon number n_mean. Args: A (array): Adjacency matrix n_mean (float): Mean photon number of the Gaussian state Returns: float: Scaling parameter """ eps = 1e-10 ls = np.linalg.svd(A, compute_uv=False) max_sv = ls[0] a_lim = 0.0 b_lim = 1.0 / (eps + max_sv) x_init = 0.5 * b_lim if 1000 * eps >= max_sv: raise ValueError("The singular values of the matrix A are too small.") 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 singular values of A Args: x (float): Scaling parameter vals (array): Singular values of the matrix A Returns: n_mean: Mean photon number in the Gaussian state """ vals2 = (x * vals) ** 2 n = np.sum(vals2 / (1.0 - vals2)) return n # The following function is implicitly tested in test_adj_scaling def grad_mean_photon_number(x, vals): # pragma: no cover r"""Returns the gradient od the mean number of photons in the Gaussian state that encodes the adjacency matrix x*A with respect to x. vals are the singular values of A Args: x (float): Scaling parameter vals (array): Singular values of the matrix A Returns: d_n_mean: Derivative of the mean photon number in the Gaussian state with respect to x """ vals1 = vals * x dn = (2.0 / x) * np.sum((vals1 / (1 - vals1**2)) ** 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)) if not res.converged: raise ValueError("The search for a scaling value failed") return res.root
[docs] def adj_to_qmat(A, n_mean): r"""Returns the Qmat xp-covariance matrix associated to a graph with adjacency matrix :math:`A` and with mean photon number :math:`n_{mean}`. Args: A (array): a :math:`N\times N` ``np.float64`` (symmetric) adjacency matrix n_mean (float): mean photon number of the Gaussian state Returns: array: the :math:`2N\times 2N` Q matrix. """ n, m = A.shape if n != m: raise ValueError("Matrix must be square.") sc = adj_scaling(A, n_mean) Asc = sc * A A = np.block([[Asc, 0 * Asc], [0 * Asc, Asc.conj()]]) I = np.identity(2 * n) X = Xmat(n) Q = np.linalg.inv(I - X @ A) return Q

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