Source code for strawberryfields.decompositions
# 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.
"""
This module implements common shared matrix decompositions that are
used to perform gate decompositions.
"""
from itertools import groupby
from collections import defaultdict
import numpy as np
from scipy.linalg import block_diag, sqrtm, polar, schur
from thewalrus.quantum import adj_scaling
from thewalrus.symplectic import sympmat, xpxp_to_xxpp
[docs]def takagi(N, tol=1e-13, rounding=13):
r"""Autonne-Takagi decomposition of a complex symmetric (not Hermitian!) matrix.
Note that singular values of N are considered equal if they are equal after np.round(values, tol).
See :cite:`cariolaro2016` and references therein for a derivation.
Args:
N (array[complex]): square, symmetric matrix N
rounding (int): the number of decimal places to use when rounding the singular values of N
tol (float): the tolerance used when checking if the input matrix is symmetric: :math:`|N-N^T| <` tol
Returns:
tuple[array, array]: (rl, U), where rl are the (rounded) singular values,
and U is the Takagi unitary, such that :math:`N = U \diag(rl) U^T`.
"""
(n, m) = N.shape
if n != m:
raise ValueError("The input matrix must be square")
if np.linalg.norm(N - np.transpose(N)) >= tol:
raise ValueError("The input matrix is not symmetric")
N = np.real_if_close(N)
if np.allclose(N, 0):
return np.zeros(n), np.eye(n)
if np.isrealobj(N):
# If the matrix N is real one can be more clever and use its eigendecomposition
l, U = np.linalg.eigh(N)
vals = np.abs(l) # These are the Takagi eigenvalues
phases = np.sqrt(np.complex128([1 if i > 0 else -1 for i in l]))
Uc = U @ np.diag(phases) # One needs to readjust the phases
list_vals = [(vals[i], i) for i in range(len(vals))]
list_vals.sort(reverse=True)
sorted_l, permutation = zip(*list_vals)
permutation = np.array(permutation)
Uc = Uc[:, permutation]
# And also rearrange the unitary and values so that they are decreasingly ordered
return np.array(sorted_l), Uc
v, l, ws = np.linalg.svd(N)
w = np.transpose(np.conjugate(ws))
rl = np.round(l, rounding)
# Generate list with degenerancies
result = []
for k, g in groupby(rl):
result.append(list(g))
# Generate lists containing the columns that correspond to degenerancies
kk = 0
for k in result:
for ind, j in enumerate(k): # pylint: disable=unused-variable
k[ind] = kk
kk = kk + 1
# Generate the lists with the degenerate column subspaces
vas = []
was = []
for i in result:
vas.append(v[:, i])
was.append(w[:, i])
# Generate the matrices qs of the degenerate subspaces
qs = []
for i in range(len(result)):
qs.append(sqrtm(np.transpose(vas[i]) @ was[i]))
# Construct the Takagi unitary
qb = block_diag(*qs)
U = v @ np.conj(qb)
return rl, U
[docs]def graph_embed_deprecated(A, max_mean_photon=1.0, make_traceless=False, rtol=1e-05, atol=1e-08):
r"""Embed a graph into a Gaussian state.
Note: The default behaviour of graph embedding has been changed; see :func:`~.graph_embed`. This version is deprecated, but has been kept for consistency.
Given a graph in terms of a symmetric adjacency matrix
(in general with arbitrary complex off-diagonal and real diagonal entries),
returns the squeezing parameters and interferometer necessary for
creating the Gaussian state whose off-diagonal parts are proportional to that matrix.
Uses :func:`~.takagi`.
Args:
A (array[complex]): square, symmetric (weighted) adjacency matrix of the graph
max_mean_photon (float): Threshold value. It guarantees that the mode with
the largest squeezing has ``max_mean_photon`` as the mean photon number
i.e., :math:`sinh(r_{max})^2 ==` :code:``max_mean_photon``.
make_traceless (bool): Removes the trace of the input matrix, by performing the transformation
:math:`\tilde{A} = A-\mathrm{tr}(A) \I/n`. This may reduce the amount of squeezing needed to encode
the graph but will lead to different photon number statistics for events with more than
one photon in any mode.
rtol (float): relative tolerance used when checking if the input matrix is symmetric
atol (float): absolute tolerance used when checking if the input matrix is symmetric
Returns:
tuple[array, array]: squeezing parameters of the input
state to the interferometer, and the unitary matrix representing the interferometer
"""
(m, n) = A.shape
if m != n:
raise ValueError("The matrix is not square.")
if not np.allclose(A, np.transpose(A), rtol=rtol, atol=atol):
raise ValueError("The matrix is not symmetric.")
if make_traceless:
A = A - np.trace(A) * np.identity(n) / n
s, U = takagi(A, tol=atol)
sc = np.sqrt(1.0 + 1.0 / max_mean_photon)
vals = -np.arctanh(s / (s[0] * sc))
return vals, U
[docs]def graph_embed(A, mean_photon_per_mode=1.0, make_traceless=False, rtol=1e-05, atol=1e-08):
r"""Embed a graph into a Gaussian state.
Given a graph in terms of a symmetric adjacency matrix
(in general with arbitrary complex entries),
returns the squeezing parameters and interferometer necessary for
creating the Gaussian state whose off-diagonal parts are proportional to that matrix.
Uses :func:`~.takagi`.
Args:
A (array[complex]): square, symmetric (weighted) adjacency matrix of the graph
mean_photon_per_mode (float): guarantees that the mean photon number in the pure Gaussian state
representing the graph satisfies :math:`\frac{1}{N}\sum_{i=1}^N sinh(r_{i})^2 ==` :code:``mean_photon``
make_traceless (bool): Removes the trace of the input matrix, by performing the transformation
:math:`\tilde{A} = A-\mathrm{tr}(A) \I/n`. This may reduce the amount of squeezing needed to encode
the graph but will lead to different photon number statistics for events with more than
one photon in any mode.
rtol (float): relative tolerance used when checking if the input matrix is symmetric
atol (float): absolute tolerance used when checking if the input matrix is symmetric
Returns:
tuple[array, array]: squeezing parameters of the input
state to the interferometer, and the unitary matrix representing the interferometer
"""
(m, n) = A.shape
if m != n:
raise ValueError("The matrix is not square.")
if not np.allclose(A, np.transpose(A), rtol=rtol, atol=atol):
raise ValueError("The matrix is not symmetric.")
if make_traceless:
A = A - np.trace(A) * np.identity(n) / n
scale = adj_scaling(A, n * mean_photon_per_mode)
A = scale * A
s, U = takagi(A, tol=atol)
vals = -np.arctanh(s)
return vals, U
[docs]def bipartite_graph_embed(A, mean_photon_per_mode=1.0, rtol=1e-05, atol=1e-08):
r"""Embed a bipartite graph into a Gaussian state.
Given a bipartite graph in terms of an adjacency matrix
(in general with arbitrary complex entries),
returns the two-mode squeezing parameters and interferometers necessary for
creating the Gaussian state that encodes such adjacency matrix
Uses :func:`~.takagi`.
Args:
A (array[complex]): square, (weighted) adjacency matrix of the bipartite graph
mean_photon_per_mode (float): guarantees that the mean photon number in the pure Gaussian state
representing the graph satisfies :math:`\frac{1}{N}\sum_{i=1}^N sinh(r_{i})^2 ==` :code:``mean_photon``
rtol (float): relative tolerance used when checking if the input matrix is symmetric
atol (float): absolute tolerance used when checking if the input matrix is symmetric
Returns:
tuple[array, array, array]: squeezing parameters of the input
state to the interferometer, and the unitaries matrix representing the interferometer
"""
(m, n) = A.shape
if m != n:
raise ValueError("The matrix is not square.")
B = np.block([[0 * A, A], [A.T, 0 * A]])
scale = adj_scaling(B, 2 * n * mean_photon_per_mode)
A = scale * A
if np.allclose(A, A.T, rtol=rtol, atol=atol):
s, u = takagi(A, tol=atol)
v = u
else:
u, s, v = np.linalg.svd(A)
v = v.T
vals = -np.arctanh(s)
return vals, u, v
[docs]def T(m, n, theta, phi, nmax):
r"""The Clements T matrix from Eq 1 of the paper"""
mat = np.identity(nmax, dtype=np.complex128)
mat[m, m] = np.exp(1j * phi) * np.cos(theta)
mat[m, n] = -np.sin(theta)
mat[n, m] = np.exp(1j * phi) * np.sin(theta)
mat[n, n] = np.cos(theta)
return mat
[docs]def Ti(m, n, theta, phi, nmax):
r"""The inverse Clements T matrix"""
return np.transpose(T(m, n, theta, -phi, nmax))
[docs]def nullTi(m, n, U):
r"""Nullifies element m,n of U using Ti"""
(nmax, mmax) = U.shape
if nmax != mmax:
raise ValueError("U must be a square matrix")
if U[m, n] == 0:
# no swaps for the identity-like case
thetar = 0
phir = 0
elif U[m, n + 1] == 0:
# swap in the divide-by-zero case
thetar = np.pi / 2
phir = 0
else:
r = U[m, n] / U[m, n + 1]
thetar = np.arctan(np.abs(r))
phir = np.angle(r)
return [n, n + 1, thetar, phir, nmax]
[docs]def nullT(n, m, U):
r"""Nullifies element n,m of U using T"""
(nmax, mmax) = U.shape
if nmax != mmax:
raise ValueError("U must be a square matrix")
if U[n, m] == 0:
# no swaps for the identity-like case
thetar = 0
phir = 0
elif U[n - 1, m] == 0:
# swap in the divide-by-zero case
thetar = np.pi / 2
phir = 0
else:
r = -U[n, m] / U[n - 1, m]
thetar = np.arctan(np.abs(r))
phir = np.angle(r)
return [n - 1, n, thetar, phir, nmax]
[docs]def rectangular(V, tol=1e-11):
r"""Rectangular decomposition of a unitary matrix, with local
phase shifts applied between two interferometers.
See :ref:`rectangular` or :cite:`clements2016` for more details.
This function returns a circuit corresponding to an intermediate step in
the decomposition as described in Eq. 4 of the article. In this form,
the circuit comprises some T matrices (as in Eq. 1), then phases on all modes,
and more T matrices.
The procedure to construct these matrices is detailed in the supplementary
material of the article.
Args:
V (array[complex]): unitary matrix of size n_size
tol (float): the tolerance used when checking if the matrix is unitary:
:math:`|VV^\dagger-I| \leq` tol
Returns:
tuple[array]: tuple of the form ``(tilist,np.diag(localV),tlist)``
where:
* ``tilist``: list containing ``[n,m,theta,phi,n_size]`` of the Ti unitaries needed
* ``tlist``: list containing ``[n,m,theta,phi,n_size]`` of the T unitaries needed
* ``localV``: Diagonal unitary sitting sandwiched by Ti's and the T's
"""
localV = V
(nsize, _) = localV.shape
if not np.allclose(V @ V.conj().T, np.identity(nsize), atol=tol, rtol=0):
raise ValueError("The input matrix is not unitary")
tilist = []
tlist = []
for k, i in enumerate(range(nsize - 2, -1, -1)):
if k % 2 == 0:
for j in reversed(range(nsize - 1 - i)):
tilist.append(nullTi(i + j + 1, j, localV))
localV = localV @ Ti(*tilist[-1])
else:
for j in range(nsize - 1 - i):
tlist.append(nullT(i + j + 1, j, localV))
localV = T(*tlist[-1]) @ localV
return tilist, np.diag(localV), tlist
[docs]def rectangular_phase_end(V, tol=1e-11):
r"""Rectangular decomposition of a unitary matrix, with all
local phase shifts placed after the interferometers.
See :cite:`clements2016` for more details.
Final step in the decomposition of a given discrete unitary matrix.
The output is of the form given in Eq. 5.
Args:
V (array[complex]): unitary matrix of size n_size
tol (float): the tolerance used when checking if the matrix is unitary
Returns:
tuple[array]: returns a tuple of the form ``(tlist, np.diag(localV), None)``
where:
* ``tlist``: list containing ``[n,m,theta,phi,n_size]`` of the T unitaries needed
* ``localV``: Diagonal unitary matrix to be applied at the end of circuit
"""
tilist, diags, tlist = rectangular(V, tol)
new_tlist, new_diags = tilist.copy(), diags.copy()
# Push each beamsplitter through the diagonal unitary
for i in reversed(tlist):
em, en = int(i[0]), int(i[1])
alpha, beta = np.angle(new_diags[em]), np.angle(new_diags[en])
theta, phi = i[2], i[3]
# The new parameters required for D',T' st. T^(-1)D = D'T'
new_theta = theta
new_phi = (alpha - beta + np.pi) % (2 * np.pi)
new_alpha = beta - phi + np.pi
new_beta = beta
new_i = [i[0], i[1], new_theta, new_phi, i[4]]
new_diags[em], new_diags[en] = np.exp(1j * new_alpha), np.exp(1j * new_beta)
new_tlist = new_tlist + [new_i]
return new_tlist, new_diags, None
[docs]def mach_zehnder(m, n, internal_phase, external_phase, nmax):
r"""A two-mode Mach-Zehnder interferometer section.
This section is constructed by an external phase shifter on the input mode
m, a symmetric beamsplitter combining modes m and n, an internal phase
shifter on mode m, and another symmetric beamsplitter combining modes m
and n.
The resulting matrix is
.. math::
M = i e^{i \phi_{i}/2} \left[\begin{matrix}\sin \left( \phi_{i}/2 \right) e^{i \phi_{e}} & \cos \left( \phi_{i}/2 \right) \\
\cos \left( \phi_{i}/2 \right) e^{i \phi_{e}} & - \sin \left( \phi_{i}/2 \right) \end{matrix}\right]
Args:
m (int): mode number on which the phase shifters act
n (int): mode number which is combined with mode m by the beamsplitters
internal_phase (float): phase in between the symmetric beamsplitters
external_phase (float): phase acting before the first beamsplitter
nmax (int): maximum number of modes in the circuit
Returns:
array: unitary matrix of the effective transformation the series of phaseshifters
and beamsplitters.
"""
Rexternal = np.identity(nmax, dtype=np.complex128)
Rexternal[m, m] = np.exp(1j * external_phase)
Rinternal = np.identity(nmax, dtype=np.complex128)
Rinternal[m, m] = np.exp(1j * internal_phase)
BS = np.identity(nmax, dtype=np.complex128)
BS[m, m] = 1.0 / np.sqrt(2)
BS[m, n] = 1.0j / np.sqrt(2)
BS[n, m] = 1.0j / np.sqrt(2)
BS[n, n] = 1.0 / np.sqrt(2)
return np.round(BS @ Rinternal @ BS @ Rexternal, 14)
[docs]def mach_zehnder_inv(m, n, phi_int, phi_ext, nmax):
r"""The inverse of the Mach-Zehnder unitary matrix.
See :func:`~.mach_zehnder` for more details on the Mach-Zehnder unitary.
"""
return mach_zehnder(m, n, phi_int, phi_ext, nmax).conj().T
[docs]def nullMZi(m, n, U):
r"""Nullifies element m,n of U using mach_zehnder_inv.
Args:
m (int): row index of element to be nullified
n (int): column index of element to be nullified
U (array): matrix whose m,n element is to be nullified
Returns:
list: list containing ``[m, n, internal_phase, external_phase, nmax]`` of the
mach_zehnder_inv unitaries needed
"""
(nmax, mmax) = U.shape
if nmax != mmax:
raise ValueError("U must be a square matrix")
if U[m, n] == 0:
# no swaps for the identity-like case
phi_i = np.pi
phi_e = 0
elif U[m, n + 1] == 0:
# swap in the divide-by-zero case
phi_i = 0
phi_e = 0
else:
r = -U[m, n + 1] / U[m, n]
phi_i = 2 * np.arctan(np.abs(r))
phi_e = -np.angle(r)
return [n, n + 1, phi_i, phi_e, nmax]
[docs]def nullMZ(n, m, U):
r"""Nullifies element n,m of U using mach_zehnder.
Args:
n (int): row index of element to be nullified
m (int): column index of element to be nullified
U (array): matrix whose m,n element is to be nullified
Returns:
list: list containing ``[m, n, internal_phase, external_phase, nmax]`` of the
mach_zehnder unitaries needed
"""
(nmax, mmax) = U.shape
if nmax != mmax:
raise ValueError("U must be a square matrix")
if U[n, m] == 0:
# no swaps for the identity-like case
phi_i = np.pi
phi_e = 0
elif U[n - 1, m] == 0:
# swap in the divide-by-zero case
phi_i = 0
phi_e = 0
else:
r = U[n - 1, m] / U[n, m]
phi_i = 2 * np.arctan(np.abs(r))
phi_e = -np.angle(r)
return [n - 1, n, phi_i, phi_e, nmax]
[docs]def rectangular_MZ(V, tol=1e-11):
r"""Rectangular decomposition of a unitary matrix, with local
phase shifts applied between two interferometers.
Is similar to :func:`~.rectangular` except that it uses Mach Zehnder matrices to null elements of V
using the :func:`~.null_MZ` and :func:`~.null_MZi` instead of :func:`~.T` matrices and corresponding :func:`~.nullT`
and :func:`~.nullTi` functions.
Args:
V (array[complex]): unitary matrix of size n_size
tol (float): the tolerance used when checking if the matrix is unitary
Returns:
tuple[array]: tuple of the form ``(tilist, np.diag(localV), tlist)``
where:
* ``tilist``: list containing ``[n,m,phi_int,phi_ext,n_size]`` of the ``mach_zehnder_inv`` unitaries needed
* ``tlist``: list containing ``[n,m,phi_int,phi_ext,n_size]`` of the ``mach_zehnder`` unitaries needed
* ``localV``: Diagonal unitary sitting sandwiched by ``mach_zehnder_inv``'s and the ``mach_zehnder``'s
"""
localV = V
(nsize, _) = localV.shape
if not np.allclose(V @ V.conj().T, np.identity(nsize), atol=tol, rtol=0):
raise ValueError("The input matrix is not unitary")
tilist = []
tlist = []
for k, i in enumerate(range(nsize - 2, -1, -1)):
if k % 2 == 0:
for j in reversed(range(nsize - 1 - i)):
tilist.append(nullMZi(i + j + 1, j, localV))
tilist[-1][2] %= 2 * np.pi
tilist[-1][3] %= 2 * np.pi
# repeat modulo operations, otherwise the input unitary
# numpy.identity(20) yields an external_phase of exactly 2 * pi
tilist[-1][2] %= 2 * np.pi
tilist[-1][3] %= 2 * np.pi
localV = localV @ mach_zehnder_inv(*tilist[-1])
else:
for j in range(nsize - 1 - i):
tlist.append(nullMZ(i + j + 1, j, localV))
tlist[-1][2] %= 2 * np.pi
tlist[-1][3] %= 2 * np.pi
# repeat modulo operations, otherwise the input unitary
# numpy.identity(20) yields an external_phase of exactly 2 * pi
tlist[-1][2] %= 2 * np.pi
tlist[-1][3] %= 2 * np.pi
localV = mach_zehnder(*tlist[-1]) @ localV
return tilist, np.diag(localV), tlist
[docs]def rectangular_symmetric(V, tol=1e-11):
r"""Decomposition of a unitary into an array of symmetric beamsplitters.
This decomposition starts with the output from :func:`~.rectangular_MZ`
and performs the equivalent of :func:`~.rectangular_phase_end` by placing all the
local phase shifts after the interferometers.
If the Mach-Zehnder unitaries are represented as M and the local phase shifts as D, the new
parameters to shift the local phases to the end are calculated such that
.. math::
M^{-1} D = D_{\mathrm{new}} M_{\mathrm{new}}
Args:
V (array): unitary matrix of size n_size
tol (int): the number of decimal places to use when determining
whether the matrix is unitary
Returns:
tuple[array]: returns a tuple of the form ``(tlist,np.diag(localV), None)``
where:
* ``tlist``: list containing ``[n, m, internal_phase, external_phase, n_size]`` of the T unitaries needed
* ``localV``: Diagonal unitary matrix to be applied at the end of circuit
* ``None``: the value ``None``, in order to make the return
signature identical to :func:`~.rectangular`
"""
tilist, diags, tlist = rectangular_MZ(V, tol)
new_tlist, new_diags = tilist.copy(), diags.copy()
# Push each beamsplitter through the diagonal unitary
for i in reversed(tlist):
em, en = int(i[0]), int(i[1])
alpha, beta = np.angle(new_diags[em]), np.angle(new_diags[en])
phi_i, phi_e = i[2], i[3]
# The new parameters required for D', MZ' st. MZ^(-1)D = D'MZ'
new_phi_e = (alpha - beta) % (2 * np.pi)
new_alpha = (beta - phi_e - phi_i + np.pi) % (2 * np.pi)
new_beta = (beta - phi_i + np.pi) % (2 * np.pi)
new_phi_i = phi_i % (2 * np.pi)
# repeat modulo operations , otherwise the input unitary
# numpy.identity(20) yields an external_phase of exactly 2 * pi
new_phi_i %= 2 * np.pi
new_phi_e %= 2 * np.pi
new_i = [i[0], i[1], new_phi_i, new_phi_e, i[4]]
new_diags[em], new_diags[en] = np.exp(1j * new_alpha), np.exp(1j * new_beta)
new_tlist = new_tlist + [new_i]
return new_tlist, new_diags, None
[docs]def triangular(V, tol=1e-11):
r"""Triangular decomposition of a unitary matrix due to Reck et al.
See :cite:`reck1994` for more details and :cite:`clements2016` for details on notation.
Args:
V (array[complex]): unitary matrix of size ``n_size``
tol (float): the tolerance used when checking if the matrix is unitary:
:math:`|VV^\dagger-I| \leq` tol
Returns:
tuple[array]: returns a tuple of the form ``(tlist,np.diag(localV), None)``
where:
* ``tlist``: list containing ``[n,m,theta,phi,n_size]`` of the T unitaries needed
* ``localV``: Diagonal unitary applied at the beginning of circuit
"""
localV = V
(nsize, _) = localV.shape
if not np.allclose(V @ V.conj().T, np.identity(nsize), atol=tol, rtol=0):
raise ValueError("The input matrix is not unitary")
tlist = []
for i in range(nsize - 2, -1, -1):
for j in range(i + 1):
tlist.append(nullT(nsize - j - 1, nsize - i - 2, localV))
localV = T(*tlist[-1]) @ localV
return list(reversed(tlist)), np.diag(localV), None
[docs]def M(n, sigma, delta, m):
r"""The symmetric Mach Zehnder interferometer matrix. (Eq 1 of the paper (arXiv:2104.0756).)
Args:
n (int): the starting mode of sMZI
sigma (complex): parameter of the sMZI :math:`\frac{(\theta_1+\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
delta (complex): parameter of the sMZI :math:`\frac{(\theta_1-\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
m (int): the length of the unitary matrix to be decomposed
Returns:
array[complex,complex]: the sMZI matrix between n-th and (n+1)-th mode
"""
mat = np.identity(m, dtype=np.complex128)
mat[n, n] = np.exp(1j * sigma) * np.sin(delta)
mat[n, n + 1] = np.exp(1j * sigma) * np.cos(delta)
mat[n + 1, n] = np.exp(1j * sigma) * np.cos(delta)
mat[n + 1, n + 1] = -np.exp(1j * sigma) * np.sin(delta)
return mat
[docs]def P(j, phi, m):
r"""The phase shifter matrix. (Eq 2 of the paper (arXiv:2104.0756).)
Args:
j (int): the starting mode of phase-shifter
phi (complex): parameter of the phase-shifter
m (int): the length of the unitary matrix to be decomposed
Returns:
array[complex,complex]: the phase-shifter matrix on the j-th mode
"""
mat = np.identity(m, dtype=np.complex128)
mat[j, j] = np.exp(1j * phi)
return mat
[docs]def triangular_compact(U, rtol=1e-12, atol=1e-12):
r"""Triangular decomposition of a unitary matrix with sMZIs and phase-shifters, as given in FIG. 2 and "The Reck Scheme" section of (arXiv:2104.0756).
Args:
U (array): unitary matrix
Returns:
dict: A dictionary containing the following items:
* ``m``: the length of the matrix
* ``phi_ins``: parameter of the phase-shifter at the beginning of the mode
* ``sigmas``: parameter of the sMZI :math:`\frac{(\theta_1+\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
* ``deltas``: parameter of the sMZI :math:`\frac{(\theta_1-\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
* ``zetas``: parameter of the phase-shifter at the end of the mode
"""
if not U.shape[0] == U.shape[1]:
raise ValueError("Matrix is not square")
if not np.allclose(U @ U.conj().T, np.eye(U.shape[0]), rtol=rtol, atol=atol):
raise ValueError("The input matrix is not unitary")
V = U.conj()
m = U.shape[0]
phases = {}
phases["m"] = m
phases["phi_ins"] = {} # mode : phi
phases["deltas"] = {} # (mode, layer) : delta
phases["sigmas"] = {} # (mode, layer) : sigma
phases["zetas"] = {} # mode : zeta
for j in range(m - 1):
x = m - 1
y = j
phi_j = -np.angle(V[x, y + 1]) + np.angle(V[x, y])
Pj = P(j + 1, phi_j, m)
phases["phi_ins"][j] = phi_j
V = V @ Pj
for k in range(j + 1):
n = j - k
if V[x, y] == 0:
delta = 0.5 * np.pi
else:
delta = np.arctan2(-abs(V[x, y + 1]), abs(V[x, y]))
V_temp = V @ M(n, 0, delta, m)
sigma = np.angle(V_temp[x - 1, y - 1]) - np.angle(V_temp[x - 1, y])
phases["deltas"][n, k] = delta
phases["sigmas"][n, k] = sigma
V = V @ M(n, sigma, delta, m)
x -= 1
y -= 1
# these next two lines are just to remove a global phase
zeta = -np.angle(V[0, 0])
phases["zetas"][0] = zeta
V = V @ P(0, zeta, m)
for j in range(1, m):
zeta = np.angle(V[0, 0]) - np.angle(V[j, j])
phases["zetas"][j] = zeta
V = V @ P(j, zeta, m)
assert np.allclose(V, np.eye(m), rtol=rtol, atol=atol), "decomposition failed"
return phases
def _rectangular_compact_init(
U, rtol=1e-12, atol=1e-12
): # pylint: disable=too-many-statements, too-many-branches
r"""Rectangular decomposition of a unitary with sMZIs and phase-shifters, as given in FIG. 3 and "The Clements Scheme" section of (arXiv:2104.0756).
Args:
U (array): unitary matrix
Returns:
dict: A dictionary containing the following items:
* ``m``: the length of the matrix
* ``phi_ins``: parameter of the phase-shifter at the beginning of the mode
* ``sigmas``: parameter of the sMZI :math:`\frac{(\theta_1+\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
* ``deltas``: parameter of the sMZI :math:`\frac{(\theta_1-\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
* ``zetas``: parameter of the phase-shifter at the middle of the mode
* ``phi_outs``: parameter of the phase-shifter at the end of the mode
"""
V = U.conj()
m = U.shape[0]
phases = {}
phases["m"] = m
phases["phi_ins"] = {} # mode : phi
phases["deltas"] = {} # (mode, layer) : delta
phases["sigmas"] = {} # (mode, layer) : sigma
phases["zetas"] = {} # mode : zeta
phases["phi_outs"] = {} # mode : phi
for j in range(m - 1):
if j % 2 == 0:
x = m - 1
y = j
phi_j = np.angle(V[x, y + 1]) - np.angle(V[x, y]) # reversed order from paper
V = V @ P(j, phi_j, m)
phases["phi_ins"][j] = phi_j
for k in range(j + 1):
if V[x, y] == 0:
delta = 0.5 * np.pi
else:
delta = np.arctan2(-abs(V[x, y + 1]), abs(V[x, y]))
n = j - k
V_temp = V @ M(n, 0, delta, m)
sigma = np.angle(V_temp[x - 1, y - 1]) - np.angle(V_temp[x - 1, y])
V = V @ M(n, sigma, delta, m)
phases["deltas"][n, k] = delta
phases["sigmas"][n, k] = sigma
x -= 1
y -= 1
else:
x = m - j - 1
y = 0
phi_j = np.angle(V[x - 1, y]) - np.angle(V[x, y])
V = P(x, phi_j, m) @ V
phases["phi_outs"][x] = phi_j
for k in range(j + 1):
if V[x, y] == 0.0:
delta = 0.5 * np.pi
else:
delta = np.arctan2(abs(V[x - 1, y]), abs(V[x, y]))
V_temp = M(x - 1, 0, delta, m) @ V
n = m + k - j - 2
if j != k:
sigma = np.angle(V_temp[x + 1, y + 1]) - np.angle(V_temp[x, y + 1])
else:
sigma = 0
phases["deltas"][n, m - k - 1] = delta
phases["sigmas"][n, m - k - 1] = sigma
V = M(n, sigma, delta, m) @ V
x += 1
y += 1
# these next two lines are just to remove a global phase
zeta = -np.angle(V[0, 0])
V = V @ P(0, zeta, m)
phases["zetas"][0] = zeta
for j in range(1, m):
zeta = np.angle(V[0, 0]) - np.angle(V[j, j])
V = V @ P(j, zeta, m)
phases["zetas"][j] = zeta
assert np.allclose(V, np.eye(m), rtol=rtol, atol=atol), "decomposition failed"
return phases
def _absorb_zeta(phases):
r"""Adjust rectangular decomposition to relocate residual phase-shifters of interferometer to edge-shifters, as given in FIG. 4 and "Relocating residual phase-shifts" section of (arXiv:2104.0756).
Args:
phases (dict): output of _rectangular_compact_init
Returns:
dict: A dictionary containing the following items:
* ``m``: the length of the matrix
* ``phi_ins``: parameter of the phase-shifter at the beginning of the mode
* ``sigmas``: parameter of the sMZI :math:`\frac{(\theta_1+\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
* ``deltas``: parameter of the sMZI :math:`\frac{(\theta_1-\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
* ``phi_edges``: parameters of the edge phase shifters
* ``phi_outs``: parameter of the phase-shifter at the end of the mode
"""
m = phases["m"]
new_phases = phases.copy()
del new_phases["zetas"]
new_phases["phi_edges"] = defaultdict(float) # (mode, layer) : phi
if m % 2 == 0:
new_phases["phi_outs"][0] = phases["zetas"][0]
for j in range(1, m):
zeta = phases["zetas"][j]
layer = m - j
for mode in range(j, m - 1, 2):
new_phases["sigmas"][mode, layer] += zeta
for mode in range(j + 1, m - 1, 2):
new_phases["sigmas"][mode, layer - 1] -= zeta
if layer % 2 == 1:
new_phases["phi_edges"][m - 1, layer] += zeta
else:
new_phases["phi_edges"][m - 1, layer - 1] -= zeta
else:
for j in range(m):
zeta = phases["zetas"][j]
layer = m - j - 1
for mode in range(j, m - 1, 2):
new_phases["sigmas"][mode, layer] += zeta
for mode in range(j + 1, m - 1, 2):
new_phases["sigmas"][mode, layer - 1] -= zeta
if layer % 2 == 0:
new_phases["phi_edges"][m - 1, layer] += zeta
else:
new_phases["phi_edges"][m - 1, layer - 1] -= zeta
return new_phases
[docs]def rectangular_compact(U, rtol=1e-12, atol=1e-12):
r"""Rectangular decomposition of a unitary with sMZIs and phase-shifters, as given in FIG. 3+4 and "The Clements Scheme" section of (arXiv:2104.0756).
Args:
U (array): unitary matrix
Returns:
dict: A dictionary containing the following items:
* ``m``: the length of the matrix
* ``phi_ins``: parameter of the phase-shifter at the beginning of the mode
* ``sigmas``: parameter of the sMZI :math:`\frac{(\theta_1+\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
* ``deltas``: parameter of the sMZI :math:`\frac{(\theta_1-\theta_2)}{2}`, where :math:`\theta_{1,2}` are the values of the two internal phase-shifts of sMZI
* ``phi_edges``: parameters of the edge phase shifters
* ``phi_outs``: parameter of the phase-shifter at the end of the mode
"""
if not U.shape[0] == U.shape[1]:
raise ValueError("Matrix is not square")
if not np.allclose(U @ U.conj().T, np.eye(U.shape[0]), rtol=rtol, atol=atol):
raise ValueError("The input matrix is not unitary")
phases_temp = _rectangular_compact_init(U, rtol=rtol, atol=atol)
return _absorb_zeta(phases_temp)
[docs]def williamson(V, tol=1e-11):
r"""Williamson decomposition of positive-definite (real) symmetric matrix.
See :ref:`williamson`.
Note that it is assumed that the symplectic form is
.. math:: \Omega = \begin{bmatrix}0&I\\-I&0\end{bmatrix}
where :math:`I` is the identity matrix and :math:`0` is the zero matrix.
See https://math.stackexchange.com/questions/1171842/finding-the-symplectic-matrix-in-williamsons-theorem/2682630#2682630
Args:
V (array[float]): positive definite symmetric (real) matrix
tol (float): the tolerance used when checking if the matrix is symmetric: :math:`|V-V^T| \leq` tol
Returns:
tuple[array,array]: ``(Db, S)`` where ``Db`` is a diagonal matrix
and ``S`` is a symplectic matrix such that :math:`V = S^T Db S`
"""
(n, m) = V.shape
if n != m:
raise ValueError("The input matrix is not square")
diffn = np.linalg.norm(V - np.transpose(V))
if diffn >= tol:
raise ValueError("The input matrix is not symmetric")
if n % 2 != 0:
raise ValueError("The input matrix must have an even number of rows/columns")
n = n // 2
omega = sympmat(n)
vals = np.linalg.eigvalsh(V)
for val in vals:
if val <= 0:
raise ValueError("Input matrix is not positive definite")
Mm12 = sqrtm(np.linalg.inv(V)).real
r1 = Mm12 @ omega @ Mm12
s1, K = schur(r1)
X = np.array([[0, 1], [1, 0]])
I = np.identity(2)
seq = []
# In what follows I construct a permutation matrix p so that the Schur matrix has
# only positive elements above the diagonal
# Also the Schur matrix uses the x_1,p_1, ..., x_n,p_n ordering thus I use rotmat to
# go to the ordering x_1, ..., x_n, p_1, ... , p_n
for i in range(n):
if s1[2 * i, 2 * i + 1] > 0:
seq.append(I)
else:
seq.append(X)
p = block_diag(*seq)
Kt = K @ p
s1t = p @ s1 @ p
dd = xpxp_to_xxpp(s1t)
perm_indices = xpxp_to_xxpp(np.arange(2 * n))
Ktt = Kt[:, perm_indices]
Db = np.diag([1 / dd[i, i + n] for i in range(n)] + [1 / dd[i, i + n] for i in range(n)])
S = Mm12 @ Ktt @ sqrtm(Db)
return Db, np.linalg.inv(S).T
[docs]def bloch_messiah(S, tol=1e-10, rounding=9):
r"""Bloch-Messiah decomposition of a symplectic matrix.
See :ref:`bloch_messiah`.
Decomposes a symplectic matrix into two symplectic unitaries and squeezing transformation.
It automatically sorts the squeezers so that they respect the canonical symplectic form.
Note that it is assumed that the symplectic form is
.. math:: \Omega = \begin{bmatrix}0&I\\-I&0\end{bmatrix}
where :math:`I` is the identity matrix and :math:`0` is the zero matrix.
As in the Takagi decomposition, the singular values of N are considered
equal if they are equal after np.round(values, rounding).
If S is a passive transformation, then return the S as the first passive
transformation, and set the the squeezing and second unitary matrices to
identity. This choice is not unique.
For more info see:
https://math.stackexchange.com/questions/1886038/finding-euler-decomposition-of-a-symplectic-matrix
Args:
S (array[float]): symplectic matrix
tol (float): the tolerance used when checking if the matrix is symplectic:
:math:`|S^T\Omega S-\Omega| \leq tol`
rounding (int): the number of decimal places to use when rounding the singular values
Returns:
tuple[array]: Returns the tuple ``(ut1, st1, vt1)``. ``ut1`` and ``vt1`` are symplectic orthogonal,
and ``st1`` is diagonal and of the form :math:`= \text{diag}(s1,\dots,s_n, 1/s_1,\dots,1/s_n)`
such that :math:`S = ut1 st1 v1`
"""
(n, m) = S.shape
if n != m:
raise ValueError("The input matrix is not square")
if n % 2 != 0:
raise ValueError("The input matrix must have an even number of rows/columns")
n = n // 2
omega = sympmat(n)
if np.linalg.norm(np.transpose(S) @ omega @ S - omega) >= tol:
raise ValueError("The input matrix is not symplectic")
if np.linalg.norm(np.transpose(S) @ S - np.eye(2 * n)) >= tol:
u, sigma = polar(S, side="left")
ss, uss = takagi(sigma, tol=tol, rounding=rounding)
# Apply a permutation matrix so that the squeezers appear in the order
# s_1,...,s_n, 1/s_1,...1/s_n
perm = np.array(list(range(0, n)) + list(reversed(range(n, 2 * n))))
pmat = np.identity(2 * n)[perm, :]
ut = uss @ pmat
# Apply a second permutation matrix to permute s
# (and their corresonding inverses) to get the canonical symplectic form
qomega = np.transpose(ut) @ (omega) @ ut
st = pmat @ np.diag(ss) @ pmat
# Identifying degenerate subspaces
result = []
for _k, g in groupby(np.round(np.diag(st), rounding)[:n]):
result.append(list(g))
stop_is = list(np.cumsum([len(res) for res in result]))
start_is = [0] + stop_is[:-1]
# Rotation matrices (not permutations) based on svd.
# See Appendix B2 of Serafini's book for more details.
u_list, v_list = [], []
for start_i, stop_i in zip(start_is, stop_is):
x = qomega[start_i:stop_i, n + start_i : n + stop_i].real
u_svd, _s_svd, v_svd = np.linalg.svd(x)
u_list = u_list + [u_svd]
v_list = v_list + [v_svd.T]
pmat1 = block_diag(*(u_list + v_list))
st1 = pmat1.T @ pmat @ np.diag(ss) @ pmat @ pmat1
ut1 = uss @ pmat @ pmat1
v1 = np.transpose(ut1) @ u
else:
ut1 = S
st1 = np.eye(2 * n)
v1 = np.eye(2 * n)
return ut1.real, st1.real, v1.real
[docs]def sun_compact(U, rtol=1e-12, atol=1e-12):
r"""Recursive factorization of unitary transfomations.
Decomposes elements of :math:`\mathrm{SU}(n)` as a sequence of :math:`\mathrm{SU}(2)`
transformations and entangling beamsplitters, see :cite:`deguise2018simple`.
This sequence of :math:`\mathrm{SU}(2)` transformations can then be mapped to an operation
on optical modes including two phase plates and one beam splitter.
This implementation is based on the authors' code at `github:glassnotes/Caspar
<https://github.com/glassnotes/Caspar>`_.
Args:
U (array): unitary matrix
rtol (float): relative tolerance used when checking if the matrix is unitary
atol (float): absolute tolerance used when checking if the matrix is unitary
Returns:
tuple[list[tuple,list], float]: Returns a list of operations with elements in
the form ``(i,i+1), [a, b, g]`` where the ``(i,i+1)`` indicates the modes of an
:math:`\mathrm{SU}(2)` transformation and ``[a, b, g]`` are the transformation parameters.
.. details::
Note that any unitary can be written in terms of an special unitary as
.. math:: U = e^{i \phi/n} S
where :math:`S \in \mathrm{SU}(n)` and :math:`e^{i\phi} = \mathrm{det}\,U`.
Here any :math:`S \in \mathrm{SU}(n)` is parametrized in terms of the Euler angles and written as
.. math::
S(\alpha, \beta, \gamma) =
\begin{pmatrix}
e^{i\alpha/2} & 0 \\
0 & e^{-i\alpha/2}
\end{pmatrix}
\begin{pmatrix}
\cos{\beta/2} & -\sin{\beta/2} \\
\sin{\beta/2} & \cos{\beta/2}
\end{pmatrix}
\begin{pmatrix}
e^{i\gamma/2} & 0 \\
0 & e^{-i\gamma/2}
\end{pmatrix}.
This factorization then determines the constructions of the :math:`\mathrm{SU}(2)` device
acting on the respective optical modes
.. math::
S(\alpha, \beta, \gamma) =
\left[ R(\alpha/2) \otimes R(-\alpha/2) \right] \,
BS(\beta/2) \,
\left[ R(\gamma/2) \otimes R(-\gamma/2) \right].
"""
n = U.shape[0]
parameters = []
global_phase = None
det = np.linalg.det(U)
if n < 3:
raise ValueError("Input matrix for decomposition must be at least 3x3.")
if not np.allclose(U @ U.conj().T, np.identity(n), rtol=rtol, atol=atol):
raise ValueError("The input matrix is not unitary.")
# if Unitary, factorize into phase times Special Unitary
SU = U.copy()
if not np.isclose(det, 1, rtol=rtol, atol=atol):
SU *= det ** (-1 / n)
global_phase = np.angle(det)
# Decompose the matrix
parameters_no_modes = _sun_parameters(SU, rtol, atol)
# Add the info about which modes each transformation is on
param_idx = 0
for md2 in range(2, n + 1):
for md1 in range(n - 1, md2 - 2, -1):
parameters.append([(md1 - 1, md1), parameters_no_modes[param_idx]])
param_idx += 1
return parameters, global_phase
def _sun_parameters(U, rtol=1e-12, atol=1e-12):
r"""Compute the set of parameters of the :math:`\mathrm{SU}(2)` transforms in the
factorization scheme.
Args:
U (array): unitary matrix
Returns:
list: a list of parameters ``[a, b, g]`` of an :math:`\mathrm{SU}(2)` operation
.. details::
This is a recursive process. The first step is to produce a
"staircase" of transformations on adjacent modes :math:`(d-1, d), (d-2, d-1), \dots`
so that what's left is an :math:`\mathrm{SU}(n-1)` transformation embedded in the lower
portion of the original system.
This is performed recursively down to the case of :math:`\mathrm{SU}(3)` where the
Rowe et al algorithm :cite:`rowe1999representations` is used to get the rest of the
transformation.
"""
if U.shape == (3, 3):
return _su3_parameters(U)
staircase_transformation, new_U = _build_staircase(U, rtol, atol)
Unm1 = new_U[1:, 1:]
return staircase_transformation + _sun_parameters(Unm1, rtol, atol)
# pylint: disable=too-many-branches
def _build_staircase(U, rtol=1e-12, atol=1e-12):
r"""Take a matrix in :math:`\mathrm{SU}(n)` and find the staircase of :math:`\mathrm{SU}(n)`
transformations which turns it into an :math:`\mathrm{SU}(n-1)` transformation on all but
the first mode.
Args:
U (array): unitary matrix
Returns:
list: Returns the list of parameters in the order in which they appear
graphically, e.g. for :math:`\mathrm{SU}(5)` will return parameters for a staircase
order as transformations on modes `(4,5)`, `(3,4)`, `(2,3)`, and finally `(1,2)`.
"""
n = U.shape[0]
# We need to do n - 1 transformations, starting from the bottom up.
transformations = []
running_prod = U
# There are a number of special cases to consider which occur when the
# left-most column contains all 0s except for one entry.
moduli = np.abs(U[:, 0])
if np.allclose(sorted(moduli), [0.0] * (n - 1) + [1], rtol, atol):
# In the special case where the top-most entry is a 1, or within some
# small tolerance of it, we basically already have an SU(n-1) transformation
# in there so just fill with empty parameters
if np.isclose(running_prod[0, 0], 1, rtol, atol):
transformations = [[0.0, 0.0, 0.0]] * (n - 1)
# Another special case is when the top left entry has modulus 1 (or close
# to it). Now we need to add a separate phase shift as well.
elif np.isclose(np.abs(running_prod[0, 0]), 1, rtol, atol):
# "Phase shift" by applying an SU(2) transformation to cancel out the
# top-most phase. Do nothing to everything else.
phase_su2 = np.array([[running_prod[0, 0].conjugate(), 0], [0, running_prod[0, 0]]])
transformations = [[0.0, 0.0, 0.0]] * (n - 2) + [_su2_parameters(phase_su2.conj().T)]
full_phase_su2 = np.identity(n, dtype=complex)
full_phase_su2[0:2, 0:2] = phase_su2
running_prod = full_phase_su2 @ running_prod
else:
# If the non-zero entry is lower down, permute until it
# reaches the top and then apply a phase transformation.
for rot_idx in range(n - 1, 0, -1):
if not np.isclose(running_prod[rot_idx, 0], 0, rtol, atol):
permmat = np.array([[0, -1], [1, 0]])
full_permmat = np.identity(n, dtype=complex)
full_permmat[rot_idx - 1 : rot_idx + 1, rot_idx - 1 : rot_idx + 1] = permmat
temp_product = full_permmat @ running_prod
if rot_idx == 1: # If we're at the top, add the phase too
phase_su2 = np.array(
[[temp_product[0, 0].conjugate(), 0], [0, temp_product[0, 0]]]
)
permmat = phase_su2 @ permmat
transformations.append(_su2_parameters(permmat.conj().T))
full_trans = np.identity(n, dtype=complex)
full_trans[rot_idx - 1 : rot_idx + 1, rot_idx - 1 : rot_idx + 1] = permmat
running_prod = full_trans @ running_prod
else: # Otherwise do nothing between these modes
transformations.append([0, 0, 0])
else:
for rot_idx in range(n - 1):
# Start at the bottom
i, j = n - rot_idx - 2, n - rot_idx - 1
# Initially we work with the inverses in order to "0 out" entries
# from the left; later we'll get the parameters from the "true" matrices.
Rij_inv = np.identity(2, dtype=complex)
full_Rij_inv = np.identity(n, dtype=complex)
if rot_idx != n - 2:
# The denominator of the transformation is the difference of
# absolute values of all columns *up* to this point.
sum_of_column = 0
for k in range(i):
sum_of_column += pow(np.absolute(running_prod[k, 0]), 2)
cf = np.sqrt(1 - sum_of_column)
y, z = running_prod[i, 0], running_prod[j, 0]
capY, capZ = y / cf, z / cf
# Build the SU(2) transformation and embed it into the larger matrix
Rij_inv = np.array([[np.conj(capY), np.conj(capZ)], [-capZ, capY]])
else:
# The last transformation, R12 is special and the rotation has
# a different form
x = U[0, 0]
cf = np.sqrt(1 - pow(np.absolute(x), 2))
Rij_inv = np.array([[np.conj(x), cf], [-cf, x]])
# Add the transformation to the sequence and update the product
Rij = Rij_inv.conj().T
transformations.append(_su2_parameters(Rij))
# Embed into larger space
full_Rij_inv[i : j + 1, i : j + 1] = Rij_inv
running_prod = full_Rij_inv @ running_prod
return transformations, running_prod
def _su2_parameters(U, tol=1e-10):
r"""Compute and return the parameters ``[a, b, g]`` of an :math:`\mathrm{SU}(2)` matrix.
Args:
U (array): unitary matrix of shape ``(2,2)`` with :math:`\det U = 1`
Returns:
list: a list of parameters ``[a, b, g]`` of the :math:`\mathrm{SU}(2)` matrix
.. details::
Given a matrix in :math:`\mathrm{SU}(2)`, parametrized as
.. math:
U(a, b, g) =
\begin{pmatrix}
e^{i(\alpha+\gamma)/2} \cos(\beta/2) & -e^{i(\alpha-\gamma)/2} \sin(\beta/2) \\
e^{-i(\alpha-\gamma)/2} \sin(\beta/2) & e^{-i(\alpha+\gamma)/2} \cos(\beta/2)
\end{pmatrix}
compute and return the parameters :math:`\alpha, \beta, \gamma`.
"""
if U.shape != (2, 2):
raise ValueError("Input matrix dimensions of _su2_parameters must be 2x2.")
if not np.isclose(np.linalg.det(U), 1, atol=tol, rtol=0):
raise ValueError(
"Input matrix must have determinant 1 to be decomposed into SU(2) parameters."
)
# Sometimes the absolute value of the matrix entry is very, very close to
# 1 and slightly above, when it should be 1 exactly. Isolate these cases
# to prevent us from getting NaN.
b = None
if np.isclose(np.absolute(U[0, 1]), 1, atol=tol, rtol=0):
b = 2 * np.arcsin(1)
else:
b = 2 * np.arcsin(np.absolute(U[0, 1]))
arg_pos = np.angle(U[0, 0]) # (a + g)/2
arg_neg = -np.angle(U[1, 0]) # (a - g)/2
a, g = arg_pos + arg_neg, arg_pos - arg_neg
return [a, b, g]
def _su3_parameters(U):
r"""Factorizes an :math:`\mathrm{SU}(3)` transformation into 3 :math:`\mathrm{SU}(2)`
transformations.
Args:
U (array): unitary matrix of shape ``(3,3)`` with :math:`\det U = 1`
Returns:
list[list]: a list containing three entries of the form ``[a, b, g]``, where every
entry has the parameters of an :math:`\mathrm{SU}(2)` matrix.
.. details::
Uses the factorization on :cite:`rowe1999representations` to factorize an
:math:`\mathrm{SU}(3)` transformation into 3 :math:`\mathrm{SU}(2)` transformations.
Parameters for each :math:`\mathrm{SU}(3)` transformation are returned as a list
:math:`[\alpha, \beta, \gamma]` (three-parameter transformation) or
:math:`[\alpha, \beta, \alpha]` (two-parameter transformation) where the matrices
are to be parametrized as
.. math::
SU_{ij}(\alpha, \beta, \gamma) =
\begin{pmatrix}
e^{i(\alpha+\gamma)/2} \cos(\beta/2) & -e^{i(\alpha-\gamma)/2} \sin(\beta/2) \\
e^{-i(\alpha-\gamma)/2} \sin(\beta/2) & e^{-i(\alpha+\gamma)/2} \cos(\beta/2)
\end{pmatrix}
The `ij` subscript indicates that the matrix should be embedded into modes `i` and `j` of the full
`n`-dimensional transformation.
The resultant matrix is expressed as
.. math::
U = SU_{23}(\alpha_1, \beta_1, \gamma_1)
SU_{12}(\alpha_2, \beta_2, \alpha_2)
SU_{23}(\alpha_3, \beta_3, \gamma_3).
"""
if U.shape != (3, 3):
raise ValueError("Input matrix dimensions of _su3_parameters must be 3x3.")
if not np.isclose(np.linalg.det(U), 1):
raise ValueError(
"Input matrix must have determinant 1 to be decomposed into SU(2) parameters."
)
# Grab the entries of the first row
x, y, z = U[0, 0], U[1, 0], U[2, 0]
# Special case: if the top left element is 1, then we essentially
# already have an SU(2) transformation embedded in an SU(3) transform,
# so all we need to do is get the parameters of that SU(2) transform.
if np.isclose(x, 1):
params = [[0.0, 0.0, 0.0], [0.0, 0.0, 0.0], _su2_parameters(U[1:, 1:])]
# Another special case: the modulus of the top left element is 1.
# Then we need to do a transformation on modes 1 and 2 to make the top
# entry 1, then an SU(2) transformation on modes 2 and 3 with what's left.
elif np.isclose(np.abs(x), 1):
# Compute the required phase matrix and embed into SU(3)
phase_su2 = np.array([[np.conj(x), 0], [0, x]])
full_phase_su2 = np.asarray(np.identity(3)) + 0j
full_phase_su2[0:2, 0:2] = phase_su2
# Compute what's left of the product, and the parameters
running_product = full_phase_su2 @ U
remainder_su2 = running_product[1:, 1:]
params = [
[0.0, 0.0, 0.0],
_su2_parameters(phase_su2.conj().T),
_su2_parameters(remainder_su2),
]
else:
# Typical case
cf = np.sqrt(1 - pow(np.absolute(x), 2))
capY, capZ = y / cf, z / cf
# Build the SU(2) transformation matrices
# SU_23(3) - three parameters
left = np.array([[1, 0, 0], [0, capY, -np.conj(capZ)], [0, capZ, np.conj(capY)]])
left_params = _su2_parameters(left[1:, 1:])
# SU_12(2) - only two parameters
middle = np.array([[x, -cf, 0], [cf, np.conj(x), 0], [0, 0, 1]])
middle_params = _su2_parameters(middle[0:2, 0:2])
# SU_23(3) - again three parameters
right = middle.conj().T @ left.conj().T @ U
right_params = _su2_parameters(right[1:, 1:])
params = [left_params, middle_params, right_params]
return params
[docs]def covmat_to_hamil(V, tol=1e-10): # pragma: no cover
r"""Converts a covariance matrix to a Hamiltonian.
Given a covariance matrix V of a Gaussian state :math:`\rho` in the xp ordering,
finds a positive matrix :math:`H` such that
.. math:: \rho = \exp(-Q^T H Q/2)/Z
where :math:`Q = (x_1,\dots,x_n,p_1,\dots,p_n)` are the canonical
operators, and Z is the partition function.
For more details, see https://arxiv.org/abs/1507.01941
Args:
V (array): Gaussian covariance matrix
tol (int): the number of decimal places to use when determining if the matrix is symmetric
Returns:
array: positive definite Hamiltonian matrix
"""
(n, m) = V.shape
if n != m:
raise ValueError("Input matrix must be square")
if np.linalg.norm(V - np.transpose(V)) >= tol:
raise ValueError("The input matrix is not symmetric")
n = n // 2
omega = sympmat(n)
vals = np.linalg.eigvalsh(V)
for val in vals:
if val <= 0:
raise ValueError("Input matrix is not positive definite")
W = 1j * V @ omega
l, v = np.linalg.eig(W)
H = (1j * omega @ (v @ np.diag(np.arctanh(1.0 / l.real)) @ np.linalg.inv(v))).real
return H
[docs]def hamil_to_covmat(H, tol=1e-10): # pragma: no cover
r"""Converts a Hamiltonian matrix to a covariance matrix.
Given a Hamiltonian matrix of a Gaussian state H, finds the equivalent covariance matrix
V in the xp ordering.
For more details, see https://arxiv.org/abs/1507.01941
Args:
H (array): positive definite Hamiltonian matrix
tol (int): the number of decimal places to use when determining if the Hamiltonian is symmetric
Returns:
array: Gaussian covariance matrix
"""
(n, m) = H.shape
if n != m:
raise ValueError("Input matrix must be square")
if np.linalg.norm(H - np.transpose(H)) >= tol:
raise ValueError("The input matrix is not symmetric")
vals = np.linalg.eigvalsh(H)
for val in vals:
if val <= 0:
raise ValueError("Input matrix is not positive definite")
n = n // 2
omega = sympmat(n)
Wi = 1j * omega @ H
l, v = np.linalg.eig(Wi)
V = (1j * (v @ np.diag(1.0 / np.tanh(l.real)) @ np.linalg.inv(v)) @ omega).real
return V
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