r"""
.. role:: html(raw)
:format: html
.. _gate_teleport:
Gate teleportation
==================
"Entanglement-assisted communication becomes entanglement-assisted computation"
--- Furusawa [[1]_]
In the quantum state teleportation algorithm, the quantum state is transferred from the sender
to the receiver exactly. However, quantum teleportation can be used in a much more powerful
manner, by simultaneously processing and manipulating the teleported state; this is known as
**gate teleportation**.
But the biggest departure from its namesake is the method in which the gate to be 'teleported'
is applied; rather than applying a quantum unitary directly to the first qumode in the system,
the unitary is applied via the projective measurement of the first qumode onto a particular basis.
This measurement-based approach provides significant advantages over applying unitary gates
directly, for example by reducing resources, and in the application of experimentally
hard-to-implement gates [[1]_]. In fact, gate teleportation forms a universal
quantum computing primitive, and is a precursor to cluster state models of quantum computation
[[2]_][[3]_].
Implementation
--------------
First described by Gottesman and Chuang [[2]_] in the case of qubits, gate
teleportation was generalized for the CV case by Bartlett and Munro in 2003 [[4]_].
In an analogous process to the discrete-variable case, you begin with the algorithm for
**local state teleportation**:
.. image:: /tutorials/images/gate_teleport1.svg
:align: center
:width: 50%
:target: javascript:void(0);
:html:`
`
Note that:
* Unlike the spatially-separated quantum state teleportation we considered in the previous section,
**local teleportation** can transport the state using only two qumodes; the state we are
teleporting is entangled directly with the squeezed vacuum state in the momentum space through
the use of a controlled-phase gate.
* The state is then teleported to qumode :math:`q_1` via a homodyne measurement in the computational
basis (the position quadrature).
* Like in the previous section, to recover the teleported state exactly, we must perform Weyl-Heisenberg
corrections to :math:`q_1`; here, that would be :math:`F^\dagger X(m)^\dagger`. However, for convenience and
simplicity, we write the circuit without the corrections applied explicitly.
Rather than simply teleporting the state as-is, we can introduce an arbitrary unitary :math:`U` that
acts upon :math:`\ket{\psi}`, as follows:
.. image:: /tutorials/images/gate_teleport2.svg
:align: center
:width: 50%
:target: javascript:void(0);
:html:`
`
Now, the action of the unitary :math:`U` is similarly teleported along with the initial state
--- this is a trivial extension of the local teleportation circuit. In order to view this in
as a measurement-based universal quantum computing primitive, we make a couple of important changes:
* The inverse Fourier gate is absorbed into the measurement, making it a homodyne detector in
the momentum quadrature
* The unitary gate :math:`U`, if diagonal in the computational basis (i.e., it is of the form
:math:`U=e^{i f(\hat{x}^i)}`), commutes with the controlled-phase gate
(:math:`CZ(s)=e^{i s ~\hat{x_1}\otimes\hat{x_2}/\hbar}`), and can be moved to the right of it.
It is then also absorbed into the projective measurement.
.. image:: /tutorials/images/gate_teleport3.svg
:align: center
:width: 50%
:target: javascript:void(0);
:html:`
`
Additional gates can now be added simply by introducing additional qumodes with the appropriate
projective measurements, all 'stacked vertically' (i.e., coupled to the each consecutive qumode
via a controlled-phase gate). From this primitive, the model of cluster state quantum computation
can be derived [[3]_].
.. note::
What happens if the unitary is *not* diagonal in the computational basis? In this case,
**feedforward** is required; additional qumodes and projective measurements are introduced,
with successive measurements dependent on the previous result [[5]_].
Code
----
Consider the following gate teleportation circuit,
.. image:: /tutorials/images/gate_teleport_ex.svg
:align: center
:width: 70%
:target: javascript:void(0);
:html:`
`
Here, the state :math:`\ket{\psi}`, a squeezed state with :math:`r=0.1`, is teleported to the
final qumode, with the quadratic phase gate (:class:`~strawberryfields.ops.Pgate`)
:math:`P(s)=e^{is\hat{x}^2/2\hbar}` teleported to act on it - with the quadratic phase gate
chosen as it is diagonal in the :math:`\x` quadrature. This can be easily implemented
using Strawberry Fields:
"""
import numpy as np
# set the random seed
np.random.seed(42)
import strawberryfields as sf
from strawberryfields.ops import *
gate_teleportation = sf.Program(3)
with gate_teleportation.context as q:
# create initial states
Squeezed(0.1) | q[0]
Squeezed(-2) | q[1]
Squeezed(-2) | q[2]
# apply the gate to be teleported
Pgate(0.5) | q[1]
# conditional phase entanglement
CZgate(1) | (q[0], q[1])
CZgate(1) | (q[1], q[2])
# projective measurement onto
# the position quadrature
Fourier.H | q[0]
MeasureX | q[0]
Fourier.H | q[1]
MeasureX | q[1]
######################################################################
# Some important notes:
#
# * As with the :doc:`state teleportation circuit `, perfectly squeezed vacuum states
# are not physically realizable; preparing the states with a squeezing factor of :math:`|r|=2`
# (:math:`\sim 18\text{dB}`) is a reasonable approximation.
#
# ..
#
# * The Blackbird notation ``Operator.H`` denotes the Hermitian conjugate of the corresponding
# operator.
#
# ..
#
# * Here, we do not make the corrections to the final state; this is left as an exercise to the reader.
# For additional details, see the gate teleportation commutation relations derived by van Loock
# [[5]_].
#
# Since all operations in the above gate teleportation circuit are Gaussian, we can use the
# ``"gaussian"`` backend:
eng = sf.Engine(backend="gaussian")
######################################################################
# Running the engine, and printing out the reduced state of modes 2
# and 3:
results = eng.run(gate_teleportation)
_, cov = results.state.reduced_gaussian([2])
print("Covariance matrix:\n", cov)
######################################################################
# We can also extract the measurement results of modes 0 and 1:
m0, m1 = results.samples[0]
print(m0, m1)
######################################################################
# To easily check that the output of the circuit is as expected, we can make sure that it agrees
# with the (uncorrected) state
#
# .. math:: X({q_1})FP(0.5)X(q_0)F \ket{z}.
check = sf.Program(1)
with check.context as q:
# compare against the expected output
# X(q1/sqrt(2)).F.P(0.5).X(q0/sqrt(0.5)).F.|z>
# not including the corrections
Squeezed(0.1) | q[0]
Fourier | q[0]
Xgate(m0) | q[0]
Pgate(0.5) | q[0]
Fourier | q[0]
Xgate(m1) | q[0]
eng = sf.Engine(backend="gaussian")
results = eng.run(check)
expected_cov = results.state.cov()
print("Expected covariance matrix:\n", expected_cov)
######################################################################
# These two states should be identical:
print(np.allclose(cov, expected_cov, atol=0.05, rtol=0))
######################################################################
# References
# ----------
#
# .. [1] A. Furusawa and P. van Loock. Quantum Teleportation and Entanglement:
# A Hybrid Approach to Optical Quantum Information Processing. Wiley, 2011.
# ISBN 9783527635290. URL: https://books.google.ca/books?id=eKxHZ0UHEU4C.
#
# .. [2] D. Gottesman and I. L. Chuang. Demonstrating the viability of universal quantum
# computation using teleportation and single-qubit operations. Nature,
# 402:390–393, Nov 1999. arXiv:quant-ph/9908010, doi:10.1038/46503.
#
# .. [3] Mile Gu, Christian Weedbrook, Nicolas C. Menicucci, Timothy C. Ralph, and
# Peter van Loock. Quantum computing with continuous-variable clusters.
# Physical Review A, 79:062318, Jun 2009. doi:10.1103/PhysRevA.79.062318.
#
# .. [4] Stephen D. Bartlett and William J. Munro. Quantum teleportation of
# optical quantum gates. Physical Review Letters, 90:117901, Mar 2003.
# doi:10.1103/PhysRevLett.90.117901.
#
# .. [5] Peter van Loock. Examples of gaussian cluster computation. Journal of the
# Optical Society of America B, 24(2):340–346, Feb 2007. doi:10.1364/JOSAB.24.000340.