Source code for thewalrus.symplectic
# 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.
"""
Symplectic Operations
=====================
**Module name:** :mod:`thewalrus.symplectic`
.. currentmodule:: thewalrus.symplectic
Contains some Gaussian operations and auxiliary functions.
Auxiliary functions
-------------------
.. autosummary::
expand
expand_vector
reduced_state
is_symplectic
sympmat
Gaussian states
---------------
.. autosummary::
vacuum_state
Gates and operations
--------------------
.. autosummary::
two_mode_squeezing
interferometer
loss
mean_photon_number
beam_splitter
rotation
Code details
^^^^^^^^^^^^
"""
import numpy as np
[docs]def expand(S, modes, N):
r"""Expands a Symplectic matrix S to act on the entire subsystem.
Args:
S (array): a :math:`2M\times 2M` Symplectic matrix
modes (Sequence[int]): the list of modes S acts on
N (int): full size of the subsystem
Returns:
array: the resulting :math:`2N\times 2N` Symplectic matrix
"""
M = len(S) // 2
S2 = np.identity(2 * N)
w = np.array(modes)
S2[w.reshape(-1, 1), w.reshape(1, -1)] = S[:M, :M].copy() # X
S2[(w + N).reshape(-1, 1), (w + N).reshape(1, -1)] = S[M:, M:].copy() # P
S2[w.reshape(-1, 1), (w + N).reshape(1, -1)] = S[:M, M:].copy() # XP
S2[(w + N).reshape(-1, 1), w.reshape(1, -1)] = S[M:, :M].copy() # PX
return S2
[docs]def expand_vector(alpha, mode, N, hbar=2.0):
"""Returns the phase-space displacement vector associated to a displacement.
Args:
alpha (complex): complex displacement
mode (int): mode index
N (int): number of modes
Returns:
array: phase-space displacement vector of size 2*N
"""
r = np.zeros(2 * N)
r[mode] = np.sqrt(2 * hbar) * alpha.real
r[N + mode] = np.sqrt(2 * hbar) * alpha.imag
return r
[docs]def reduced_state(mu, cov, modes):
r""" Returns the vector of means and the covariance matrix of the specified modes.
Args:
modes (int of Sequence[int]): indices of the requested modes
Returns:
tuple (means, cov): means is an array containing the vector of means,
and cov is a square array containing the covariance matrix. Both use the :math:`xp`-ordering.
"""
N = len(mu) // 2
if modes == list(range(N)):
# reduced state is full state
return mu, cov
# reduce state down to specified subsystems
if isinstance(modes, int):
modes = [modes]
if np.any(np.array(modes) > N):
raise ValueError("The specified modes cannot be larger than number of subsystems.")
ind = np.concatenate([np.array(modes), np.array(modes) + N])
rows = ind.reshape(-1, 1)
cols = ind.reshape(1, -1)
return mu.copy()[ind], cov.copy()[rows, cols]
[docs]def vacuum_state(modes, hbar=2.0):
r"""Returns the vacuum state.
Args:
modes (str): Returns the vector of means and the covariance matrix
hbar (float): (default 2) the value of :math:`\hbar` in the commutation
relation :math:`[\x,\p]=i\hbar`
Returns:
list[array]: the means vector and covariance matrix of the vacuum state
"""
means = np.zeros((2 * modes))
cov = np.identity(2 * modes) * hbar / 2
state = [means, cov]
return state
[docs]def squeezing(r, phi):
r"""Squeezing. In fock space this corresponds to \exp(\tfrac{1}{2}r e^{i \phi} (a^2 - a^{\dagger 2}) ).
Args:
r (float): squeezing magnitude
phi (float): rotation parameter
Returns:
array: symplectic transformation matrix
"""
# pylint: disable=assignment-from-no-return
cp = np.cos(phi)
sp = np.sin(phi)
ch = np.cosh(r)
sh = np.sinh(r)
S = np.array([[ch - cp * sh, -sp * sh], [-sp * sh, ch + cp * sh]])
return S
[docs]def two_mode_squeezing(r, phi):
"""Two-mode squeezing.
Args:
r (float): squeezing magnitude
phi (float): rotation parameter
Returns:
array: symplectic transformation matrix
"""
# pylint: disable=assignment-from-no-return
cp = np.cos(phi)
sp = np.sin(phi)
ch = np.cosh(r)
sh = np.sinh(r)
S = np.array(
[
[ch, cp * sh, 0, sp * sh],
[cp * sh, ch, sp * sh, 0],
[0, sp * sh, ch, -cp * sh],
[sp * sh, 0, -cp * sh, ch],
]
)
return S
[docs]def interferometer(U):
"""Interferometer.
Args:
U (array): unitary matrix
Returns:
array: symplectic transformation matrix
"""
X = U.real
Y = U.imag
S = np.vstack([np.hstack([X, -Y]), np.hstack([Y, X])])
return S
# pylint: disable=too-many-arguments
[docs]def loss(mu, cov, T, mode, nbar=0, hbar=2):
r"""Loss channel acting on a Gaussian state.
Args:
mu (array): means vector
cov (array): covariance matri
T (float): transmission; 1 corresponds to no
loss, 0 to full loss.
mode (int): mode to act on
nbar (float): thermal mean population (default 0)
hbar (float): (default 2) the value of :math:`\hbar` in the commutation
relation :math:`[\x,\p]=i\hbar`
Returns:
tuple[array]: the means vector and covariance matrix of the resulting state
"""
N = len(cov) // 2
cov_res = cov.copy()
mu_res = mu.copy()
for m in (mode, mode + N):
mu_res[m] *= np.sqrt(T)
cov_res[m, :] *= np.sqrt(T)
cov_res[:, m] *= np.sqrt(T)
cov_res[m, m] += (1 - T) * (2 * nbar + 1) * hbar / 2
return mu_res, cov_res
[docs]def mean_photon_number(mu, cov, hbar=2):
r"""Calculates the mean photon number for a given one-mode state.
Args:
mu (array): length-2 vector of means
cov (array): :math:`2\times 2` covariance matrix
hbar (float): (default 2) the value of :math:`\hbar` in the commutation
relation :math:`[\x,\p]=i\hbar`
Returns:
tuple: the photon number expectation and variance
"""
ex = (np.trace(cov) + mu.T @ mu) / (2 * hbar) - 1 / 2
var = (np.trace(cov @ cov) + 2 * mu.T @ cov @ mu) / (2 * hbar ** 2) - 1 / 4
return ex, var
[docs]def beam_splitter(theta, phi):
"""Beam-splitter.
Args:
theta (float): transmissivity parameter
phi (float): phase parameter
Returns:
array: symplectic-orthogonal transformation matrix of an interferometer with angles theta and phi
"""
U = np.array(
[
[np.cos(theta), -np.exp(-1j * phi) * np.sin(theta)],
[np.exp(1j * phi) * np.sin(theta), np.cos(theta)],
]
)
return interferometer(U)
[docs]def rotation(theta):
"""Rotation gate.
Args:
theta (float): rotation angle
Returns:
array: rotation matrix by angle theta
"""
V = np.identity(1) * np.exp(1j * theta)
return interferometer(V)
[docs]def sympmat(N):
r"""Returns the matrix :math:`\Omega_n = \begin{bmatrix}0 & I_n\\ -I_n & 0\end{bmatrix}`
Args:
N (int): positive integer
Returns:
array: :math:`2N\times 2N` array
"""
I = np.identity(N)
O = np.zeros_like(I)
S = np.block([[O, I], [-I, O]])
return S
[docs]def is_symplectic(S, rtol=1e-05, atol=1e-08):
r""" Checks if matrix S is a symplectic matrix
Args:
S (array): a square matrix
Returns:
(bool): whether the given matrix is symplectic
"""
n, m = S.shape
if n != m:
return False
if n % 2 != 0:
return False
nmodes = n // 2
Omega = sympmat(nmodes)
return np.allclose(S.T @ Omega @ S, Omega, rtol=rtol, atol=atol)
[docs]def autonne(A, rtol=1e-05, atol=1e-08, svd_order=True):
r"""Autonne-Takagi decomposition of a complex symmetric (not Hermitian!) matrix.
Args:
A (array): square, symmetric matrix
rtol (float): the relative tolerance parameter between ``A`` and ``A.T``
atol (float): the absolute tolerance parameter between ``A`` and ``A.T``
svd_order (boolean): whether to return result by ordering the singular values of ``A`` in descending (``True``) or asceding (``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")
if not np.allclose(A, A.T, rtol=rtol, atol=atol):
raise ValueError("The input matrix is not symmetric")
Areal = A.real
Aimag = A.imag
B = np.empty((2 * n, 2 * n))
B[:n, :n] = Areal
B[n : 2 * n, :n] = Aimag
B[:n, n : 2 * n] = Aimag
B[n : 2 * n, n : 2 * n] = -Areal
vals, vects = np.linalg.eigh(B)
U = vects[:n, n : 2 * n] + 1j * vects[n : 2 * n, n : 2 * n]
if svd_order:
return (vals[n : 2 * n])[::-1], U[:, ::-1]
return vals[n : 2 * n], U
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