"""
An introduction to the bosonic backend
======================================
.. note::
This tutorial is first in a series on the ``bosonic`` backend. After getting acquainted
with the backend in this tutorial, go to :doc:`part 2 `
to learn about how the backend can be used to sample non-Gaussian states. After that,
check out :doc:`part 3 ` for a deep dive into using the ``bosonic``
backend to simulate qubits encoded in photonic modes. These tutorials accompany
our research paper [[#bourassa2021]_].
So far, Strawberry Fields allows its users to simulate photonic circuits using three
backends. Two of them, the ``fock`` and ``tf`` backends, represent the quantum state of the modes using the Fock or particle basis. These backends allow one to represent
arbitrary quantum states up to a photon cutoff, but this generality comes with an automatic
exponential increase in the space complexity as a function of the number of modes
with the base of the exponent scaling as the energy of the state of light.
At the exact opposite end of this trade-off is the ``gaussian`` backend, in which there is no
exponential scaling in the memory required, but at the same time there is only a subset of states
that can be represented.
In this tutorial, we introduce the new ``bosonic`` backend which borrows some of the best features
of all the previous backends. Specifically, the ``bosonic`` backend is tailored to simulate states
which can be represented as a linear combination of Gaussian functions in phase space.
It provides very succinct descriptions of Gaussian states, just
like the ``gaussian`` backend, but it can also provide descriptions of non-Gaussian states as well.
Moreover, like in the ``gaussian`` backend, the application of the most common active and passive linear
optical operations, like the displacement ``Dgate``, squeezing ``Sgate`` and beamsplitter ``BSgate`` gates,
is extremely efficient.
To motivate the ideas behind the backend, we will investigate how to represent the highly non-classical
and non-Gaussian cat state. Once we build our intuition we will introduce the ``bosonic`` backend and
investigate how it represents quantum states. Finally, we discuss why
it can provide advantages over other backends for many applications.
.. note::
We assume the reader is familiar with the basics of the phase-space picture of quantum mechanics,
including Wigner functions. For the uninitiated, a good place to start is the :doc:`CV quantum gate visualizations
tutorial `.
"""
######################################################################
# Of Cats and Kets
# ----------------
#
# Cat states are defined to be linear superpositions of two coherent states
#
# .. math::
# |k^\alpha \rangle_{\text{cat}} = \sqrt{\mathcal{N}}\left(|\alpha \rangle + e^{i \pi k} |-\alpha \rangle \right),
#
# where :math:`| \alpha \rangle` is a coherent state with amplitude
# :math:`\alpha`, :math:`k` is the phase parameter of the cat state and
#
# .. math:: \mathcal{N} = \frac{1}{2 (1+e^{-|\alpha|^2}\cos(\pi k) )}
#
# is a normalization constant.
#
# Given the universality of the Fock backend we can use it to simulate the preparation of this state
# and plot its Wigner function:
# Usual imports
import strawberryfields as sf
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
from matplotlib import cm
# Simulation and cat state parameters
nmodes = 1
cutoff = 30
q = 4.0
p = 0.0
hbar = 2
alpha = (q + 1j * p) / np.sqrt(2 * hbar)
k = 1
# SF program
prog_cat_fock = sf.Program(nmodes)
with prog_cat_fock.context as q:
sf.ops.Catstate(a=np.absolute(alpha), phi=np.angle(alpha), p=k) | q
eng = sf.Engine("fock", backend_options={"cutoff_dim": cutoff, "hbar": hbar})
state = eng.run(prog_cat_fock).state
# We now plot it
xvec = np.linspace(-15, 15, 401)
W = state.wigner(mode=0, xvec=xvec, pvec=xvec)
Wp = np.round(W.real, 4)
scale = np.max(Wp.real)
nrm = mpl.colors.Normalize(-scale, scale)
plt.axes().set_aspect("equal")
plt.contourf(xvec, xvec, Wp, 60, cmap=cm.RdBu, norm=nrm)
plt.show()
######################################################################
# Some parts of the Wigner function are easy to interpret: since the state
# is a coherent superposition of two widely separated coherent states,
# we expect to see two circular blobs centered around :math:`(q,p)=\pm\sqrt{2 \hbar}(\Re(\alpha), \Im(\alpha))`.
# Because a cat state is a *coherent* superposition of the two coherent states,
# we also see interference fringes in between the "classical" states of the cat.
# Mathematically, one can determine the nature of these interference fringes by
# looking at the density matrix of a cat state given by
#
# .. math::
# |k^\alpha \rangle \langle k^\alpha |_{\text{cat}} = \mathcal{N} \left( |\alpha \rangle \langle \alpha| + |-\alpha \rangle \langle -\alpha|+e^{i \pi k} |-\alpha \rangle \langle \alpha | + e^{-i \pi k} |\alpha \rangle \langle -\alpha |\right).
#
# The transformation from density matrix to Wigner function is linear, so the first two terms exactly correspond to the Wigner functions of the two coherent states
# :math:`|\pm \alpha \rangle` and are thus Gaussian functions centered around :math:`\boldsymbol{\mu} = \sqrt{2 \hbar}(\Re(\alpha), \Im(\alpha))`,
# with the usual vacuum noise giving them some finite width.
# The last two terms in the equation above are more complicated to interpret. We know that they are the Hermitian
# conjugates of each other :math:`(\left[ e^{i \pi k} |-\alpha \rangle \langle \alpha |\right]^\dagger = e^{-i \pi k} |\alpha \rangle \langle -\alpha |)`
# , and by elimination they must correspond to the interference features present close to the origin
# of phase space.
#
# In our recent preprint [[#bourassa2021]_], we show that the Wigner function of terms like :math:`|-\alpha \rangle \langle \alpha |`
# is also a Gaussian function, but with *complex* means :math:`\boldsymbol{\mu} = \sqrt{2 \hbar} i (\Im(\alpha),-\Re(\alpha))`.
# Therefore, we can express the Wigner function of the cat state as a weighted sum of four Gaussians in phase space ---
# as long as we allow the Gaussians to have complex means.
# Since a cat state can be written as a linear combination of Gaussian functions in phase space, we can simulate
# the same circuit presented above using the ``bosonic`` backend. Since this backend does not
# use Fock states, we don't need to pass a cutoff argument, thus we can write:
prog_cat_bosonic = sf.Program(nmodes)
with prog_cat_bosonic.context as q:
sf.ops.Catstate(a=np.absolute(alpha), phi=np.angle(alpha), p=k) | q
eng = sf.Engine("bosonic", backend_options={"hbar": hbar})
state = eng.run(prog_cat_bosonic).state
######################################################################
# We can now inspect the internal representation of the cat state inside
# the ``bosonic`` backend. To this end, we can print the attributes ``means``,
# ``covs`` and ``weights``.
# The ``means`` variable is a NumPy array containing the means of the
# four Gaussians needed to describe the state.
means = state.means()
print(means)
######################################################################
# The first axis of the array is the one labelling the different Gaussians.
# Similarly, ``covs`` contains the covariance matrices of the four Gaussians:
covs = state.covs()
print(covs)
######################################################################
# Finally, as noted earlier the Wigner function is a *weighted* sum of the
# four different Gaussians, the actual ``weights`` or coefficients are given by
weights = state.weights()
print(weights)
######################################################################
# Note that both the ``weights`` and ``means`` are complex.
# With the information provided from the backend, we are ready to verify
# that we got the correct cat. To this end, we will first create a simple
# wrapper function to generate a Gaussian with mean ``mu`` and covariance
# matrix ``V`` and a convenience function to evaluate it on a grid of points:
def gaussian_func_gen(mu, V):
"""Generates a function that when evaluated returns
the value of a normalized Gaussian specified in terms
of a vector of means and a covariance matrix.
Args:
mu (array): vector of means
V (array): covariance matrix
Returns:
(callable): a normalized Gaussian function
"""
Vi = np.linalg.inv(V)
norm = 1.0 / np.sqrt(np.linalg.det(2 * np.pi * V))
fun = lambda x: norm * np.exp(-0.5 * (x - mu) @ Vi @ (x - mu))
return fun
def evaluate_fun(fun, xvec, yvec):
"""Evaluate a function a 2D in a grid of points.
Args:
fun (callable): function to evaluate
xvec (array): values of the first variable of the function
yvec (array): values of the second variable of the function
Returns:
(array): value of the function in the grid
"""
return np.array([[fun(np.array([x, y])) for x in xvec] for y in xvec])
funs = [gaussian_func_gen(means[i], covs[i]) for i in range(len(means))]
Wps = [weights[i] * evaluate_fun(funs[i], xvec, xvec) for i in range(len(weights))]
######################################################################
# The list ``Wps`` contains the values of the different Gaussians making
# the Wigner function of the cat state. We can easily verify that when we
# sum the different components for every point in phase space we obtain
# a zero imaginary part:
print(np.allclose(sum(Wps).imag, 0))
######################################################################
# In fact, we can verify that the last two components are the complex
# conjugate of each other (thus their sum is real),
# and that the first two components are strictly real:
print(np.allclose(Wps[2], Wps[3].conj()))
print(np.allclose(Wps[0].imag, 0))
print(np.allclose(Wps[1].imag, 0))
######################################################################
# We can look individually at each of the Gaussians as well:
fig, axs = plt.subplots(1, 4, figsize=(10, 2.2))
for i in range(4):
Wp = np.round(Wps[i].real, 4)
axs[i].contourf(xvec, xvec, Wp, 60, cmap=cm.RdBu, norm=nrm)
if i != 0:
axs[i].set_yticks([])
plt.show()
######################################################################
# And also put them together to obtain exactly the same Wigner function
# from before:
Wcat = sum(Wps)
plt.axes().set_aspect("equal")
plt.contourf(xvec, xvec, Wcat.real, 60, cmap=cm.RdBu, norm=nrm)
plt.show()
######################################################################
# Why is the bosonic backend useful?
# ----------------------------------
# Now that we understand the internal workings of the new ``bosonic``
# backend, let's illustrate its utility. To this end, let us consider
# a slightly more complicated circuit where we displace a cat state.
# This is how it looks in the Fock backend:
x = 6
y = 0
beta = (x + 1j * y) / np.sqrt(2 * hbar)
prog_cat_fock_displaced = sf.Program(nmodes)
with prog_cat_fock_displaced.context as q:
sf.ops.Catstate(a=np.absolute(alpha), phi=np.angle(alpha), p=k) | q
sf.ops.Dgate(beta) | q
eng = sf.Engine("fock", backend_options={"cutoff_dim": cutoff, "hbar": hbar})
state = eng.run(prog_cat_fock_displaced).state
######################################################################
# Plotting the Wigner function of the returned state:
W = state.wigner(mode=0, xvec=xvec, pvec=xvec)
plt.axes().set_aspect("equal")
plt.contourf(xvec, xvec, W, 60, cmap=cm.RdBu, norm=nrm)
plt.show()
######################################################################
# This plot does not meet our expectation that a ``Dgate`` gate simply
# displaces the Wigner function of a state. Indeed, we see that significant distortion occurs in the
# right-hand-side .
# This happened because the ``cutoff`` we choose is not sufficient to faithfully
# represent the *displaced* cat. A simple solution to this problem is to increase
# the ``cutoff`` in the simulation, but that will lead to an increase in memory and
# a slower simulation.
# The behaviour of the ``fock`` backend can be contrasted
# with the one of the ``bosonic`` backend where we obtain
prog_cat_bosonic_displaced = sf.Program(nmodes)
with prog_cat_bosonic_displaced.context as q:
sf.ops.Catstate(a=np.absolute(alpha), phi=np.angle(alpha), p=k) | q
sf.ops.Dgate(beta) | q
eng = sf.Engine("bosonic", backend_options={"hbar": hbar})
state = eng.run(prog_cat_bosonic_displaced).state
######################################################################
# Just like in the case of the ``fock`` backend,
# we can also use the ``wigner`` method to generate Wigner functions:
Wps = state.wigner(mode=0, xvec=xvec, pvec=xvec)
plt.axes().set_aspect("equal")
plt.contourf(xvec, xvec, Wps, 60, cmap=cm.RdBu, norm=nrm)
plt.show()
######################################################################
# We can easily verify that in the internal representation in the backend
# we have only modified the displacement relative to the first program
# we investigated:
print(state.means())
######################################################################
# Conclusions and Outlook
# -----------------------
# In this tutorial, we have introduced the new ``bosonic`` backend of Strawberry Fields,
# explained the basic idea of how it represents quantum states and
# showcased some of the advantages it has with respect to other backends.
# We observed that when the energy of the cat state being considered was
# increased by displacement, the Fock backend gave unexpected results,
# and it actually requires higher cutoff and more memory for accurate results.
# On the other hand, the bosonic backend can quickly and easily deal with
# situations such as this on by merely changing the means and covariance
# matrices of the represented state.
# For a more advanced feature, check out :doc:`part 2 `
# of this series to see how the bosonic backend can be used for sampling.
#
######################################################################
# References
# ----------
#
# .. [#bourassa2021]
#
# J. Eli Bourassa, Nicolás Quesada, Ilan Tzitrin, Antal Száva, Theodor Isacsson,
# Josh Izaac, Krishna Kumar Sabapathy, Guillaume Dauphinais, and Ish Dhand.
# Fast simulation of bosonic qubits via Gaussian functions in phase space.
# `arXiv:2103.05530 `_, 2021.