# Source code for thewalrus.decompositions

# Copyright 2019 Xanadu Quantum Technologies Inc.

# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
"""
Decompositions
==============

**Module name:** :mod:thewalrus.decompositions

.. currentmodule:: thewalrus.decompositions

This module implements common shared matrix decompositions that are
used to perform gate decompositions.

Summary
-------

.. autosummary::
williamson
symplectic_eigenvals
blochmessiah
takagi

Code details
------------
"""
import numpy as np

from scipy.linalg import sqrtm, schur, polar
from thewalrus.symplectic import sympmat
from thewalrus.quantum.gaussian_checks import is_symplectic

[docs]
def williamson(V, rtol=1e-05, atol=1e-08):
r"""Williamson decomposition of positive-definite (real) symmetric matrix.

See this thread <https://math.stackexchange.com/questions/1171842/finding-the-symplectic-matrix-in-williamsons-theorem/2682630#2682630>_
and the Williamson decomposition documentation <https://strawberryfields.ai/photonics/conventions/decompositions.html#williamson-decomposition>_

Args:
V (array[float]): positive definite symmetric (real) matrix
rtol (float): the relative tolerance parameter used in np.allclose
atol (float): the absolute tolerance parameter used in np.allclose

Returns:
tuple[array,array]: (Db, S) where Db is a diagonal matrix
and S is a symplectic matrix such that :math:V = S Db S^T
"""
(n, m) = V.shape

if n != m:
raise ValueError("The input matrix is not square")

if not np.allclose(V, V.T, rtol=rtol, atol=atol):
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)

if not np.all(vals > 0):
raise ValueError("Input matrix is not positive definite")

M12 = np.real_if_close(sqrtm(V))
Mm12 = np.linalg.inv(M12)
Gamma = Mm12 @ omega @ Mm12
a, O = schur(Gamma)
# In what follows a permutation matrix perm is constructed 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 the permutation perm is updated
# to go to the ordering x_1, ..., x_n, p_1, ... , p_n
perm = np.arange(2 * n)
for i in range(n):
if a[2 * i, 2 * i + 1] <= 0:
(perm[2 * i], perm[2 * i + 1]) = (perm[2 * i + 1], perm[2 * i])

perm = np.array([perm[2 * i] for i in range(n)] + [perm[2 * i + 1] for i in range(n)])

O = O[:, perm]
phi = np.abs(np.diag(a, k=1)[::2])
dd = np.concatenate([1 / phi, 1 / phi])
ddsqrt = 1 / np.sqrt(dd)
S = M12 @ O * ddsqrt
return np.diag(dd), S

[docs]
def symplectic_eigenvals(cov):
r"""Returns the symplectic eigenvalues of a covariance matrix.

Args:
cov (array): a covariance matrix

Returns:
(array): symplectic eigenvalues
"""
M = len(cov) // 2
Omega = sympmat(M)
return np.real_if_close(-1j * np.linalg.eigvals(Omega @ cov))[::2]

[docs]
def blochmessiah(S):
"""Returns the Bloch-Messiah decomposition of a symplectic matrix S = O @ D @ Q
where O and Q are orthogonal symplectic matrices and D is a positive-definite diagonal matrix
of the form diag(d1,d2,...,dn,d1^-1, d2^-1,...,dn^-1),

Args:
S (array[float]): 2N x 2N real symplectic matrix

Returns:
tuple(array[float],  : orthogonal symplectic matrix O
array[float],  : diagonal matrix D
array[float])  : orthogonal symplectic matrix Q
"""

N, _ = S.shape

if not is_symplectic(S):
raise ValueError("Input matrix is not symplectic.")
N = N // 2
V, P = polar(S, side="left")
A = P[:N, :N]
B = P[:N, N:]
C = P[N:, N:]
M = A - C + 1j * (B + B.T)
Lam, W = takagi(M)
Lam = 0.5 * Lam
O = np.block([[W.real, -W.imag], [W.imag, W.real]])
Q = O.T @ V
sqrt1pLam2 = np.sqrt(1 + Lam**2)
D = np.diag(np.concatenate([sqrt1pLam2 + Lam, sqrt1pLam2 - Lam]))
return O, D, Q

[docs]
def takagi(A, svd_order=True):
r"""Autonne-Takagi decomposition of a complex symmetric (not Hermitian!) matrix.
Note that the input matrix is internally symmetrized by taking its upper triangular part.
If the input matrix is indeed symmetric this leaves it unchanged.
See Carl Caves note. <http://info.phys.unm.edu/~caves/courses/qinfo-s17/lectures/polarsingularAutonne.pdf>_

Args:
A (array): square, symmetric matrix
svd_order (boolean): whether to return result by ordering the singular values of A in descending (True) or ascending (False) order.

Returns:
tuple[array, array]: (r, U), where r are the singular values,
and U is the Autonne-Takagi unitary, such that :math:A = U \diag(r) U^T.
"""

n, m = A.shape
if n != m:
raise ValueError("The input matrix is not square")
# Here we build a Symmetric matrix from the top right triangular part
A = np.triu(A) + np.triu(A, k=1).T

A = np.real_if_close(A)

if np.allclose(A, 0):
return np.zeros(n), np.eye(n)

if np.isrealobj(A):
# If the matrix A is real one can be more clever and use its eigendecomposition
ls, U = np.linalg.eigh(A)
vals = np.abs(ls)  # These are the Takagi eigenvalues
signs = (-1) ** (1 + np.heaviside(ls, 1))
phases = np.sqrt(np.complex128(signs))
Uc = U * phases  # One needs to readjust the phases
# Find the permutation to sort in decreasing order
perm = np.argsort(vals)
# if svd_order reverse it
if svd_order:
perm = perm[::-1]
return vals[perm], Uc[:, perm]

# Find the element with the largest absolute value
pos = np.unravel_index(np.argmax(np.abs(A)), (n, n))
# Use it to find whether the input is a global phase times a real matrix
phi = np.angle(A[pos])
Amr = np.real_if_close(np.exp(-1j * phi) * A)
if np.isrealobj(Amr):
vals, U = takagi(Amr, svd_order=svd_order)
return vals, U * np.exp(1j * phi / 2)

u, d, v = np.linalg.svd(A)
U = u @ sqrtm((v @ np.conjugate(u)).T)
if svd_order is False:
return d[::-1], U[:, ::-1]
return d, U