# Building a large-scale quantum computer with continuous-variable optical
technologies

Kosuke Fukui & Shuntaro Takeda  Department of Applied Physics, School of
Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656,
Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Realizing a large-scale quantum computer requires hardware platforms that can
simultaneously achieve universality, scalability, and fault tolerance. As a
viable pathway to meeting these requirements, quantum computation based on
continuous-variable optical systems has recently gained more attention due to
its unique advantages and approaches. This review introduces several topics of
recent experimental and theoretical progress in the optical continuous-
variable quantum computation that we believe are promising. In particular, we
focus on scaling-up technologies enabled by time multiplexing, bandwidth
broadening, and integrated optics, as well as hardware-efficient and robust
bosonic quantum error correction schemes.

Keywords: quantum optics, continuous-variable quantum computation, bosonic
quantum error correction, one-way quantum computation, time multiplexing

##  1 Introduction

Over the past decades, much effort has been dedicated to developing quantum
computers with the promise of performing previously impossible computing
tasks. Various quantum computing (QC) hardware platforms have emerged, such as
superconducting circuits [1, 2], trapped ions [3, 4], semiconductor quantum
dots [5, 6], and photonic circuits [7, 8]. One of the leading platforms is the
superconducting circuits, as highlighted by a recent demonstration of “quantum
supremacy” with a 53-qubit superconducting quantum computer [1]. This device
successfully performed a specific computing task that was classically hard to
simulate even with the state-of-the-art supercomputer. The current quantum
computers are, however, still too small and noisy to perform practical quantum
algorithms. Recently, near-term applications with such noisy intermediate-
scale quantum computers have been pursued with several approaches, such as the
development of quantum-classical hybrid algorithms and error mitigation
techniques [9, 10]. However, the applicability of these approaches is
currently limited and still unclear. Scaling up quantum computers is an
inevitable direction to take advantage of the full potential of quantum
computers.

What are the requirements imposed on hardware platforms to realize practical,
large-scale quantum computers? Here we will focus on three requirements. The
first requirement is “universality”: the ability to perform a set of
operations necessary to construct arbitrary quantum computation (QC). This
requirement is essential to realize general-purpose quantum computers which
can run any kinds of quantum algorithms. The second is “scalability”: the
possibility to increase the number of qubits and operational steps efficiently
without constraints. The difficulty of this requirement lies in the fact that
quantum processors for more and more qubits and operations often suffer from
increasing noise channels and more severe restrictions originating from their
environmental conditions. The third is “fault tolerance”: the functionality to
correct errors during computation and derive reliable results. Various quantum
error-correction (QEC) schemes are already known [11], but they not only
require a low error rate below 1% but also incur tremendous resource overhead
that makes their implementation a daunting challenge.

As a viable pathway to meeting these severe requirements, QC based on
continuous variables (CVs) has recently gained more and more attention. The
mainstream of QC utilizes qubits as fundamental quantum information units and
encodes them into two-level physical systems. In contrast, QC based on CVs is
a complementary approach expressing quantum information by quantum states in
an infinite-dimensional Hilbert space [12]. This infinite-dimensional space
offers computational models unique to CVs, such as Gaussian boson sampling and
its related algorithms [13, 14, 15, 16], quantum machine learning based on CVs
[17, 18, 19], and other CV quantum algorithms and simulations [20, 21, 22].
Even more powerful tools in CVs are bosonic quantum error-correcting codes
(QECCs) [23, 24]. The bosonic codes can encode an error-correctable logical
qubit into a single-mode quantum state by exploiting its infinite-dimensional
space as a resource to introduce redundancy. These codes are more efficient
than the standard error-correcting codes requiring additional two-level
physical systems to increase the dimensionality and introduce redundancy.
Therefore, the bosonic codes potentially reduce resource overhead and thus
provide hardware-efficient implementations of QEC.

Several quantum systems that can potentially perform CVQC are known, such as
microwave modes in superconducting circuits [25, 26], vibrational modes of
trapped ions [27, 28] or mechanical oscillators [29], atomic spin ensembles
[30, 31], and optical modes [32, 33, 34]. For microwave and optical modes,
several architectures for fault-tolerant CVQC have been recently proposed [24,
35, 36, 37, 38, 39]. Here in this review, we shed light on “optical” CVQC,
expecting its potential to satisfy the universality, scalability, and fault
tolerance simultaneously. The optical CV approach has recently made remarkable
progress in realizing scalability, as exemplified by the deterministic
generation of large-scale entangled states [40, 41, 42] and their application
to many steps of QC [43, 44]. This scalability comes from deterministic
quantum resources available in CV schemes and optical multiplexing techniques
to increase optical modes efficiently. In addition, optical platforms suffer
from fewer limitations to scalability since they do not require severe
environments such as very low temperature or vacuum. In terms of fault
tolerance, the optical CV approach has unique advantages. Optical quantum
systems are inherently robust because light has almost no interaction with
environments. The dominant cause of errors is optical losses, but the bosonic
QECCs can deal with such loss errors in the hardware-efficient ways. The last
requirement, universality, has long been one of the major difficulties in the
optical QC. This is because the strong optical nonlinearity required for
universal quantum operations has been hard to achieve deterministically.
However, recent progress in the CV approach has found several feasible
methodologies to realize deterministic nonlinear optical quantum operations
[45, 46, 47]. For these reasons, various types of optical architectures for
scalable, fault-tolerant, and universal CVQC have recently been proposed [37,
48, 38].

This review aims to introduce several topics of recent experimental and
theoretical progress in the optical CVQC that we believe opens a promising
path to realizing universality, scalability, and fault tolerance at the same
time. First, Section II focuses on the universality. We define basic
operations in the CVQC and show how to perform the universal QC in theory. We
also outline the one-way CVQC model, which is the leading approach in this
field. Then, Section III focuses on the scalability. After briefly summarizing
elementary technologies developed thus far, we mainly concentrate on
experimental progress in three topics related to scalability, including
optical multiplexing, bandwidth broadening, and integrated photonic chips.
Section IV focuses on the fault tolerance. We introduce bosonic QECCs,
focusing on Gottesman-Kitaev-Preskill (GKP) qubits [23] in particular. We also
discuss the methodologies for the fault-tolerant optical CVQC with GKP qubits.
Finally, Sec. V summarizes and concludes this review.

##  2 Universal quantum computation with CVs

###  2.1 Continuous variables

In the beginning, we review the operators and the measurement of quantum
states for CVQC. The Hamiltonian of a quantized electromagnetic mode is
described by that of a quantum harmonic oscillator, where quadratures of a
single mode $k$ correspond to the harmonic oscillator’s position and momentum
operators $\hat{q}_{k}$ and $\hat{p}_{k}$, respectively. The Hamiltonian of
the quantum harmonic oscillator can be written by these operators as

$\hat{H}_{k}=\frac{1}{2}(\hat{p}^{2}_{k}+\omega^{2}_{k}\hat{q}^{2}_{k})=\hbar\omega_{k}\left(\hat{a}^{{\dagger}}_{k}\hat{a}_{k}+\frac{1}{2}\right),$
(1)

where $\hat{a}_{k}$ and $\hat{a}^{{\dagger}}_{k}$ are annihilation and
creation operators, $\omega_{k}$ is the center frequency $\omega$ of the mode
$k$, and $\hbar$ is the Planck constant divided by $2\pi$. The position and
momentum operators are described by creation and annihilation operators as

$\hat{q}_{k}=\sqrt{\frac{\hbar}{2\omega}_{k}}(\hat{a}_{k}+\hat{a}^{{\dagger}}_{k}),\hskip
10.0pt\hat{p}_{k}=-i\sqrt{\frac{\hbar{\omega}_{k}}{2}}(\hat{a}_{k}-\hat{a}^{{\dagger}}_{k}),$
(2)

which satisfy the commutation relation,
$[\hat{q}_{k},\hat{p}_{k^{\prime}}]=i\hbar\delta_{kk^{\prime}}$,
$[\hat{a}_{k},\hat{a}^{{\dagger}}_{k^{\prime}}]=\delta_{kk^{\prime}}$, and
$[\hat{a}_{k},\hat{a}_{k^{\prime}}]=0$ ($\delta_{kk^{\prime}}$ is the
Kronecker delta). The position and momentum quadratures then represent a
conjugate pair, and the well-known uncertainty relation becomes

$\langle(\Delta\hat{q}_{k})^{2}\rangle\langle(\Delta\hat{p}_{k})^{2}\rangle\geq\frac{1}{4}|\langle\left[\hat{q}_{k},\hat{p}_{k}\right]\rangle|^{2}=\frac{\hbar^{2}}{4}.$
(3)

In the following, we will use the position and momentum quadratures with
${\omega}_{k}=1$ and $\hbar=1$, and may omit the label $k$.

Table 1: Comparison between qubits and qumodes in quantum information processing. |  Discrete variables (qubits) |  Continuous variables (qumodes)
---|---|---
Computational basis |  $\\{\ket{0}_{\rm L},\ket{1}_{\rm L}\\}$ |  $\\{\ket{s}_{q}\\}_{s\in\mathbb{R}}$
Conjugate basis |  $\\{\ket{\pm}_{\rm L}=\frac{1}{\sqrt{2}}(\ket{0}_{\rm L}\pm\ket{1}_{\rm L})\\}$ |  $\\{\ket{t}_{p}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}ds\hskip 2.0pte^{ist}\ket{s}_{q}\\}_{t\in\mathbb{R}}$
Encoding |  $\ket{\psi}=\alpha\ket{0}_{\rm L}+\beta\ket{1}_{\rm L}$ $(|\alpha|^{2}+|\beta|^{2}=1)$ |  $\ket{\psi}=\int_{-\infty}^{\infty}ds\psi(s)\ket{s}_{q}$ $(\int_{-\infty}^{\infty}ds|\psi(s)|^{2}=1)$
Detector |  Photon detector |  Homodyne detector
|  Bit-flip gate: $\hat{X}$ |  Displacement in $\hat{q}$ basis: $\hat{X}(v)=e^{-iv\hat{p}}$ $(v\in\mathbb{R})$
|  $\hat{X}\ket{0}_{\rm L}=\ket{1}_{\rm L},\hskip 2.0pt\hat{X}\ket{1}_{\rm L}=\ket{0}_{\rm L}$ |  $\hat{X}(v)\ket{s}_{q}=\ket{s+v}_{q}$
|  Phase-flip gate: $\hat{Z}$ |  Displacement in $\hat{p}$ basis :$\hat{Z}(u)=e^{iu\hat{q}}$ $(u\in\mathbb{R})$
|  $\hat{Z}\ket{0}_{\rm L}=\ket{0}_{\rm L},\hskip 2.0pt\hat{Z}\ket{1}_{\rm L}=-\ket{1}_{\rm L}$ |  $\hat{Z}(u)\ket{t}_{p}=\ket{t+u}_{p}$
Quantum gate |  Hadamard gate: $\hat{H}$ |  Fourier gate: $\hat{R}{(\frac{\pi}{2})}=e^{i\frac{\pi}{4}(\hat{q}^{2}+\hat{p}^{2})}$
|  $\hat{H}\ket{0}_{\rm L}=\ket{+}_{\rm L},\hskip 2.0pt\hat{H}\ket{1}_{\rm L}=\ket{-}_{\rm L}$ |  $\hat{R}{(\frac{\pi}{2})}\ket{s}_{q}=\ket{s}_{p}$, $\hat{R}{(\frac{\pi}{2})}\ket{t}_{p}=\ket{-t}_{q}$
|  Controlled-NOT (CX) gate: $\hat{CX}$ |  CX gate: $\hat{CX}=e^{-i\hat{q}_{1}\hat{p}_{2}}$
|  $\hat{CX}\ket{0}_{\rm L}\ket{0(1)}_{\rm L}=\ket{0}_{\rm L}\ket{0(1)}_{\rm L}$ |  $\hat{CX}\ket{s_{1}}_{q_{1}}\ket{s_{2}}_{q_{2}}=\hat{CX}\ket{s_{1}}_{q_{1}}\ket{s_{2}+s_{1}}_{q_{2}}$
|  $\hat{CX}\ket{1}_{\rm L}\ket{0(1)}_{\rm L}=\ket{1}_{\rm L}\ket{1(0)}_{\rm L}$ |  $\hat{CX}\ket{t_{1}}_{p_{1}}\ket{t_{2}}_{p_{2}}=\hat{CX}\ket{t_{1}-t_{2}}_{p_{1}}\ket{t_{2}}_{p_{2}}$
Carrier |  Degrees of freedom of a photon |  Quadratures of a light field

The measurement in the quadrature eigenbasis is implemented by the homodyne
measurement, which is an essential part of implementing CVQC. In the
(balanced) homodyne measurement [49], the signal mode to be measured and the
local oscillator (LO) mode are coupled by a beam splitter with a
transmissivity 50%, and these two modes are measured by photodetectors. The
difference of the output photocurrents is given by
$\delta\hat{i}=c(\hat{a}_{\rm LO}^{{\dagger}}\hat{a}_{\rm sig}+\hat{a}_{\rm
LO}\hat{a}^{{\dagger}}_{\rm sig})$ with constant $c$, where $\hat{a}_{\rm
sig}(\hat{a}_{\rm LO})$ is the annihilation operator for the signal (LO) mode.
Assuming a large amplitude of the LO mode, i.e., $\hat{a}_{\rm
LO}\to\alpha_{\rm LO}=|\alpha_{\rm LO}|e^{i\Phi}$ with the relative phase
between signal and LO modes $\Phi$, we can describe $\delta\hat{i}$ as
$\delta\hat{i}\propto|\alpha_{\rm LO}|(\hat{q}_{\rm sig}{\rm
cos}\Phi+\hat{p}_{\rm sig}{\rm sin}\Phi)$. By changing $\Phi$ we can measure
the quadratures of the signal mode. The cases for $\Phi=0$ and $\pi/2$
correspond to the measurement in the $q$ and $p$ quadratures, respectively.

###  2.2 Quantum computation with CVs

####  2.2.1 Continuous and discrete variables

For quantum information processing based on the qubit, the eigenbasis
$\left\\{\ket{0}_{\rm L},\ket{1}_{\rm L}\right\\}$ is used for the
computational basis. For CVs, the eigenbasis of $\hat{q}$,
$\\{\ket{s}_{q}\\}_{s\in\mathbb{R}}$, is conventionally used as the
computational basis, and it spans an infinite-dimensional Hilbert space.
Comparison between qubits and qumodes in quantum information processing is
summarized in Table 1, where qumodes are CV equivalent of the qubits. Reviews
for quantum information processing with CVs were provided in Refs. [50, 51,
33, 34].

QC has a great deal of potential to efficiently solve some hard problems for
conventional computers. On the other hand, the Gottesman-Knill theorem for
qubits [52] shows that a class of QC with qubits which employs only Clifford
gates as well as projective measurements in the computational basis can be
efficiently simulated by a classical computer. Thus, non-Clifford gates play
an important role in providing a speedup over a classical computer, and allow
us to realize universality defined by the condition that any unitary
transformation can be implemented with arbitrary accuracy. It is known that
universality can be achieved by a finite set of Clifford and non-Clifford
gates, which is called a universal gate set.

CVQC was proposed by Lloyd and Braunstein [12], showing the way to use CVs for
QC and the universal gate set for CVQC. For universal CVQC, non-Gaussian gates
corresponding to the non-Clifford gates are required. The CV analogue of the
Gottesman–Knill theorem for qubits [52] was formulated by Bartlett et al.
[53], showing that CVQC cannot overcome a classical computer if initial
states, gates, and measurements are all Gaussian. We note that Gaussian (non-
Gaussian) gates linearly (nonlinearly) transform quadratures by Hamiltonians
with the second or lower order (the third or higher order) of quadrature
operators, requiring only second-order (third or higher-order) optical
nonlinearity. In the following Secs. 2.2.2 and 2.2.3, we see the universal
gate set consisting of a finite set of Gaussian and non-Gaussian gates.

####  2.2.2 Gaussian quantum gates

A typical Gaussian quantum gate set for universality consists of
{$\hat{Z}(t)$, $\hat{P}(\eta)$, $\hat{R}{(\theta)}$, $\hat{CZ}$}, where
$\hat{Z}(t)$ is $p$ quadrature displacement, $\hat{P}(\eta)$ is shearing (also
referred to as the phase gate), $\hat{R}{(\theta)}$ is rotation (also referred
to as the phase shift), and $\hat{CZ}$ is the controlled-phase (CZ) gate. The
above Gaussian gates can be partly substituted by $q$ quadrature displacement
$\hat{X}(t)$, squeezing $\hat{S}(r)$, the controlled-NOT (CX) gate, and a
beam-splitter coupling $\hat{BS}{(\theta)}$. In terms of an experimental
realization in an optical setup, the Gaussian quantum gate set for
universality, {$\hat{S}(r)$, $\hat{Z}(t)$ (or $\hat{X}(t)$),
$\hat{R}{(\theta)}$, $\hat{BS}{(\theta)}$}, is also employed since these
Gaussian gates are more feasible than others.

We see the Gaussian quantum gates in more detail. Displacement operators in
position and momentum quadratures are defined as $\hat{X}(v)=e^{-iv\hat{p}}$
and $\hat{Z}(u)=e^{iu\hat{q}}$ with $v,u\in\mathbb{R}$, respectively. These
operators act on the quadrature eigenbasis as

$\hat{X}(v)\ket{s}_{q}=\ket{s+v}_{q},\hat{Z}(u)\ket{t}_{p}=\ket{t+u}_{p}.$ (4)

In the Heisenberg picture, $\hat{X}(v)$ and $\hat{Z}(u)$ transform the
position and momentum quadratures as $\hat{q}\to\hat{q}+v$ and
$\hat{p}\to\hat{p}+u$, respectively. $\hat{X}(v)$ and $\hat{Z}(u)$ are
analogous to bit- and phase-flip gates for the qubit, called Pauli $X$ and $Z$
gates, which act on the computational basis as $\hat{X}\ket{0}_{\rm
L}(\hat{X}\ket{1}_{\rm L})=\ket{1}_{\rm L}(\ket{0}_{\rm L})$ and
$\hat{Z}\ket{0}_{\rm L}(\hat{Z}\ket{1}_{\rm L})=\ket{0}_{\rm L}(-\ket{1}_{\rm
L})$, respectively. Additionally, $\hat{X}(v)$ and $\hat{Z}(u)$ are
generalized to the Weyl-Heisenberg group of the displacement operator
$\hat{D}(\alpha)$ which is any linear combination of $\hat{X}(v)$ and
$\hat{Z}(u)$ and defined as $\hat{D}(\alpha)={\rm
exp}(\alpha\hat{a}^{{\dagger}}-\alpha^{\ast}\hat{a})$ with the complex
amplitude $\alpha=(v+iu)/\sqrt{2}$.

The squeezing, phase, and rotation gates are defied as
$\hat{S}(r)=e^{i\frac{r}{2}(\hat{q}\hat{p}+\hat{p}\hat{q})}$,
$\hat{P}(\eta)=e^{i\frac{\eta}{2}\hat{q}^{2}}$, and
$\hat{R}{(\theta)}=e^{i\frac{\theta}{2}(\hat{q}^{2}+\hat{p}^{2})}$,
respectively. In the Heisenberg picture, they transform quadratures as

$\displaystyle\hat{S}(r)$ $\displaystyle:\hat{q}\to e^{-r}\hat{q},\hskip
10.0pt\hat{p}\to e^{r}\hat{p},$ (5) $\displaystyle\hat{P}(\eta)$
$\displaystyle:\hat{q}\to\hat{q},\hskip 10.0pt\hat{p}\to\hat{p}+\eta\hat{q},$
(6) $\displaystyle\hat{R}{(\theta)}$
$\displaystyle:\hat{q}\to\hat{q}\mathrm{cos}\theta-\hat{p}\mathrm{sin}\theta,\hskip
10.0pt\hat{p}\to\hat{q}\mathrm{sin}\theta+\hat{p}\mathrm{cos}\theta.$ (7)

In addition to the single-mode Gaussian gates described above, two-mode
Gaussian gates are also required for CVQC. The typical example is the CZ gate
$\hat{CZ}=e^{i\hat{q}_{1}\hat{q}_{2}}$, which transforms the quadratures for
the modes 1 and 2 as

$\hat{q}_{1}\to\hat{q}_{1},\hskip
10.0pt\hat{p}_{1}\to\hat{p}_{1}+\hat{q}_{2},\hskip
10.0pt\hat{q}_{2}\to\hat{q}_{2},\hskip
10.0pt\hat{p}_{2}\to\hat{p}_{2}+\hat{q}_{1}.$ (8)

In addition, the CX gate is defined as
$\hat{CX}=e^{-i\hat{q}_{1}\hat{p}_{2}}$, which transforms the quadratures as

$\hat{q}_{1}\to\hat{q}_{1},\hskip
10.0pt\hat{p}_{1}\to\hat{p}_{1}-\hat{p}_{2},\hskip
10.0pt\hat{q}_{2}\to\hat{q}_{2}+\hat{q}_{1},\hskip
10.0pt\hat{p}_{2}\to\hat{p}_{2}.$ (9)

The CX gate for CVs is also referred to as the quantum-nondemolition (QND)
gate or the sum gate. We note that, more generally, CZ and CX gates for CVs
can be defined as $\hat{CZ}(g)=e^{ig\hat{q}_{1}\hat{q}_{2}}$ and
$\hat{CX}(g)=e^{-ig\hat{q}_{1}\hat{p}_{2}}$, respectively, with interaction
strength $g$, but $g$=1 is used throughout this review.

Experimentally, the CZ and CX gates can be implemented by linear optics and
ancillary squeezed vacuum states, $\hat{S}(r)\ket{0}$ [54, 55, 56]. However,
the ancilla state introduces additional errors due to finite squeezing.
Instead, a beam splitter operation can be generally used for a two-mode gate
due to the experimental feasibility in an optical setup. The beam-splitter
operation $\hat{BS}{(\theta)}$ with the parameter $\theta$ is defined as
$\hat{BS}{(\theta)}=e^{\frac{\theta}{2}(\hat{a}_{1}^{{\dagger}}\hat{a}_{2}-\hat{a}_{1}\hat{a}_{2}^{{\dagger}})}=e^{\frac{i\theta}{2}(\hat{q}_{1}\hat{p}_{2}+\hat{p}_{1}\hat{q}_{2})}$,
where $\hat{a}_{1(2)}$ is the annihilation operator for the mode 1(2) and
$\theta$ determines the transmissivity of the beam splitter $T={\rm
cos}^{2}\frac{\theta}{2}\in[0,1]$. $\hat{BS}{(\theta)}$ transforms the
quadratures for two modes 1 and 2 as

$\displaystyle\hat{q}_{1}$ $\displaystyle\to$
$\displaystyle\sqrt{T}\hat{q}_{1}+\sqrt{R}\hat{q}_{2},\hskip
12.0pt\hat{p}_{1}\to\sqrt{T}\hat{p}_{1}+\sqrt{R}\hat{p}_{2},$ (10)
$\displaystyle\hat{q}_{2}$ $\displaystyle\to$
$\displaystyle-\sqrt{R}\hat{q}_{1}+\sqrt{T}\hat{q}_{2},\hskip
5.0pt\hat{p}_{2}\to-\sqrt{R}\hat{p}_{1}+\sqrt{T}\hat{p}_{2},$ (11)

where $R=1-T$ is the reflectivity of the beam splitter.

Note that a combination of the above Gaussian gates can generate any Gaussian
gate which transforms the quadratures linearly in the Heisenberg picture. On
the other hand, these gates provide only Gaussian operations, and are thus not
sufficient to generate an arbitrary unitary gate.

####  2.2.3 Non-Gaussian quantum gates and universality

Next, we see the non-Gaussian quantum gates for universality. A typical non-
Gaussian gate is a unitary $\hat{U}_{n}(t)=e^{i\frac{t}{n}\hat{q}^{n}}$ with a
natural number $n\geq 3$ and the nonlinear interaction strength $t$. The
unitary $\hat{U}_{3}(t)$ is referred to as the cubic phase gate. We can see
$\hat{U}_{1}(t)$ and $\hat{U}_{2}(t)$ correspond to $\hat{Z}(t)$ and
$\hat{P}(t)$, respectively. In the Heisenberg picture,
$e^{i\frac{t}{n}\hat{q}^{n}}$ with $n\geq 3$ nonlinearly transforms
quadratures as

$\hat{q}\to\hat{q},\hskip 10.0pt\hat{p}\to\hat{p}+t\hat{q}^{n-1}.$ (12)

It is known that an arbitrary unitary operator can be constructed from
Gaussian gates and $\hat{U}_{3}(t)$. Thus, the universal gate set for an
optical setup is given by a finite set of gates, e.g.,{$\hat{S}(r)$,
$\hat{Z}(t)$ (or $\hat{X}(t)$), $\hat{R}{(\theta)}$,
$\hat{BS}{(\theta)}$,$\hat{U}_{3}(t)$}.

The way to construct approximately an arbitrary unitary operator was
introduced by Lloyd et al. [12] and formulated by Sefi et al. [57, 58]. Also,
there have been many studies on the decomposition of an arbitrary or specific
unitary operator exactly or approximately [59, 60, 61, 62, 63]. An arbitrary
unitary $\hat{U}$ is described as $\hat{U}={\rm exp}(i\hat{H})$, and $\hat{H}$
is the Hermitian operator consisting of a sum of products of quadrature
operators. Universality is realized when $\hat{H}$ can be approximated with
arbitrary precision by using Hamiltonians composing the universal gate set. To
decompose and approximate a given Hamiltonian, the following approximations
for Hamiltonian operators ${\hat{A}}$ and ${\hat{B}}$ are typically used:

$\displaystyle e^{it(\hat{A}+\hat{B})}$ $\displaystyle\approx$ $\displaystyle
e^{i\hat{A}\frac{t}{2}}e^{i\hat{B}t}e^{i\hat{A}\frac{t}{2}}+O(t^{3}),$ (13)
$\displaystyle e^{t^{2}[\hat{A},\hat{B}]}$ $\displaystyle\approx$
$\displaystyle
e^{i\hat{B}t}e^{i\hat{A}t}e^{-i\hat{B}t}e^{-i\hat{A}t}+O(t^{3}),$ (14)

where $t$ is a time to apply each of the Hamiltonians [64, 57]. When we apply
these operators repeatedly by replacing $t$ with $t/N$, Eqs. (13) and (14) are
transformed as

$\displaystyle e^{it(\hat{A}+\hat{B})}$ $\displaystyle\approx$
$\displaystyle(e^{i\hat{A}\frac{t}{2N}}e^{i\hat{B}\frac{t}{N}}e^{i\hat{A}\frac{t}{2N}})^{N}+O\left(\frac{t^{3}}{N^{2}}\right),$
(15) $\displaystyle e^{t^{2}[\hat{A},\hat{B}]}$ $\displaystyle\approx$
$\displaystyle(e^{i\hat{B}\frac{t}{N}}e^{i\hat{A}\frac{t}{N}}e^{-i\hat{B}\frac{t}{N}}e^{-i\hat{A}\frac{t}{N}})^{N^{2}}+O\left(\frac{t^{3}}{N}\right),$
(16)

where $N$ is the number of repetitions of sequential operations consisting of
operators $\hat{A}$ and $\hat{B}$. Thus, we can realize $(\hat{A}+\hat{B})$
and $[\hat{A},\hat{B}]$ from $\hat{A}$ and $\hat{B}$ with arbitrary precision
in principle, assuming sufficiently large $N$ so that
$O\left({t^{3}}/{N^{2}}\right),O\left({t^{3}}/{N}\right)\mapsto 0$. This fact
enables us to construct higher-order Hamiltonians of $\hat{q}$ and $\hat{p}$
from lower-order ones. Additionally, the several exact gate decompositions
without any approximations have been shown [57, 63], e.g.,

$\displaystyle\hat{q}^{m+1}$ $\displaystyle=$
$\displaystyle-\frac{2}{3m}[\hat{q}^{m},[\hat{q}^{3},\hat{p}^{2}]],$ (17)
$\displaystyle\hat{q}^{m}p^{n}+\hat{p}^{n}\hat{q}^{m}$ $\displaystyle=$
$\displaystyle-\frac{4i}{(n+1)(m+1)}[\hat{q}^{m+1},\hat{p}^{n+1}]$ (18)
$\displaystyle-$
$\displaystyle\frac{1}{n+1}\sum_{k=1}^{n-1}[\hat{p}^{n-k},[\hat{q}^{m},\hat{p}^{k}]].$

The above results allow us to implement an arbitrary Hamiltonian and thus an
arbitrary unitary from a universal gate set composed of a finite number of
gates.

Figure 1: Cubic phase gate. (a)The cubic phase gate by the measurement-induced
scheme using the cubic phase state. (b) The preparation of the cubic phase
state [65, 66].

####  2.2.4 Non-Gaussian gates in an optical setup

As already mentioned in the previous section, universal QC can be implemented
by a finite set of Gaussian gates and a single non-Gaussian gate. Optical non-
Gaussian gates require third or higher-order optical nonlinear effects.
However, such nonlinearity is extremely weak in general and difficult to
achieve for optical quantum states at the single-photon level. In fact, this
lack of sufficient nonlinearity has been a major problem in realizing two-
qubit gates in the optical discrete variable approach. One possible way to
circumvent this problem is to introduce ancillary photons and projective
measurement to probabilistically induce effective nonlinearity, as proposed by
Knill, Laflamme, and Milburn [67]. However, this scheme requires a substantial
resource overhead to make the two-qubit gate deterministic. Another way is to
enhance the nonlinearity by using strong interaction between Rydberg atoms
[68], or by coupling two-level systems with photons in cavities or waveguides
[69, 70, 71, 72, 73].

In the case of the CV approach, a simple measurement-based scheme to
deterministically perform a non-Gaussian gate was proposed by Gottesman,
Kitaev, and Preskill. Figure 1(a) shows their scheme to perform the cubic
phase gate $\hat{V}(\gamma)=\hat{U}_{3}(\gamma)={\rm
exp}(i\frac{\gamma}{3}\hat{q}^{3})$ with $\gamma\in\mathbb{R}$. In their
scheme, the cubic phase state is used as a resource to implement the cubic
phase gate, where the ideal cubic phase state is defined as
$\ket{\gamma}=\hat{V}(\gamma)\ket{0}_{p}\propto\int
dse^{i\frac{\gamma}{3}s^{3}}\ket{s}_{q}$. Figure 1(a) shows the implementation
of the cubic phase gate on the input state $\ket{\psi}$. In Fig. 1(a), the CX
gate between $\ket{\gamma}$ and $\ket{\psi}$ is implemented, and the upper
mode is measured in the $q$ quadrature. After the measurement, the $p$
quadrature for the input state is transformed as
$\hat{p}\to\hat{p}+\gamma\hat{q}^{2}+2m\gamma\hat{q}+m^{2}\gamma$ in the
Heisenberg picture, where $m$ is the homodyne measurement outcome. Finally,
$\hat{V}(\gamma)\ket{\psi}$ is obtained after canceling the terms
$2m\gamma\hat{q}$ and $m^{2}\gamma$ via feedforward operations
$\hat{P}(-2m\gamma)$ and $\hat{Z}(-m^{2}\gamma)$ depending on $m$.

The state $\ket{\gamma}$ can be prepared approximately by using squeezed
vacuum states, a beam-splitter coupling, a displacement operation, and an
adaptive squeezing operation depending on the result of a photon counting
measurement [66], as shown in Fig. 1(b). This scheme, however, requires a high
squeezing level for the preparation of the cubic phase gate. For the
experimental feasibility, thus, a superposition of Fock states can be
alternatively used to approximate the cubic phase state. For example, the
state $\ket{\phi}_{\rm cps}\propto\ket{0}+c\ket{3}$ with
$c={i\gamma\sqrt{3}}/{2}$ has the position wave function $\phi(q)\propto
1+i\gamma(q^{3}-3q/2)$. Since the wave function of $\ket{\gamma}$ can be
described as $1+i\gamma q^{3}+O(\gamma^{2})$, the state $\ket{\phi}_{\rm cps}$
effectively approximates $\ket{\gamma}$ with $\gamma\ll 1$ except for the term
$3\gamma q/2$ whose effect can be canceled out later. For the implementation
of a cubic phase gate including the preparation of the cubic phase state,
there are many theoretical efforts [46, 74, 66, 47, 75] and experimental
efforts described in Sec. 3.1.1.

###  2.3 One-way CVQC

####  2.3.1 One-way QC

A model of QC analogous to classical computation is called a quantum circuit
model. In the circuit model, unitary quantum gates are performed on the input
quantum states in a quantum register. After the unitary time evolution
required for the algorithm, the quantum states are measured in the register.
To implement a practical quantum algorithm, this model requires a number of
deterministic unitary quantum gates with coherent control of individual qubits
[76].

Instead of the quantum circuit model, Raussendorf and Briegel formulated one-
way QC [77], also called measurement-based QC. Qubit-based one-way QC provides
the ability to perform universal QC using only measurements of the qubits
composing the multipartite entangled state, called the cluster state;
accordingly, an arbitrary quantum algorithm can be performed via a sequence of
single-qubit measurements on qubit-based cluster states. In an optical setup,
the measurement of qubits is usually easier than the quantum gate based on the
quantum circuit model, which requires coherent control of qubits. Thus, one-
way QC has been a promising model to implement large-scale QC in optical
systems if one can prepare a large-scale cluster state. Later, one-way CVQC
was established by Menicucci et al. [78] and Gu et al. [65]. The CV version of
the cluster state can implement any computational algorithm based on CVQC.

The cluster state is typically described by a graph $G=(V,E)$ with the sets of
vertices $V$ and edges $E$, as illustrated in Fig. 2(a), where the elements of
$V$ and $E$ correspond to qubits (qumodes) and CZ gates, respectively. The
cluster state for qubits is generated by implementing CZ gates between
neighboring qubits which are initially prepared to $\ket{+}_{\rm
L}=(\ket{0}_{\rm L}+\ket{1}_{\rm L})/\sqrt{2}$, as illustrated in Fig. 2(b).
The CV cluster state is generated by performing the CZ gate between
neighboring qumodes which are initially prepared to momentum-squeezed vacuum
states, as illustrated in Fig. 2(c). The cluster state is efficiently
described by the stabilizer formalism. For CVs, the operator $\hat{X}(v)$ is a
stabilizer operator for the state $\ket{0}_{p}$ because of
$\hat{X}(v)\ket{0}_{p}=\ket{0}_{p}$. For the CV cluster state, the stabilizers
for $n$ modes can be defined as the set of
$\hat{\mathfrak{S}}_{i}(v)=\hat{X}_{i}(v)\prod_{j\in N(i)}\hat{Z}_{j}(v)$ for
$i=1,\cdots,n$ and for all $v\in\mathbb{R}$, where $N(i)$ means the set of
vertices in the neighborhood of the vertice $i$. Here
$\hat{\mathfrak{S}}_{i}(v)$ is described by using nullifiers $\hat{H}_{i}$ as
$\hat{\mathfrak{S}}_{i}(v)=e^{-iv{\hat{H}}_{i}}$ for all $v\in\mathbb{R}$.
Considering the expression of $\hat{\mathfrak{S}}_{i}(v)$ for the CV cluster
state, $\hat{H}_{i}$ can be written as $\hat{p}_{i}-\sum_{j\in
N(i)}\hat{q}_{j}$. For a simple example, the nullifiers of the cluster state
in Fig. 2(d) are given by $\hat{p}_{1}-\hat{q}_{2},$
$\hat{p}_{2}-\hat{q}_{1}-\hat{q}_{3},$ and $\hat{p}_{3}-\hat{q}_{2}$. The
stabilizer generators are written as $\hat{X}_{1}(v)\hat{Z}_{2}(v)$,
$\hat{Z}_{1}(v)\hat{X}_{2}(v)\hat{Z}_{3}(v)$, and
$\hat{Z}_{2}(v)\hat{X}_{3}(v)$ for all $v$. The +1 eigenstate of the
stabilizer generators, however, requires infinite squeezing and thus infinite
energy. Since such a state is not realistic, a common way to approximate the
CV cluster state is to replace each eigenstate $\ket{0}_{p}$ with a squeezed
vacuum state whose variance in the $p$ quadrature is equal to
$\langle(\Delta\hat{p})^{2}\rangle=\frac{1}{2}e^{-2r}$ with a finite squeezing
parameter $r$.

Universal one-way CVQC requires the preparation of two-dimensional cluster
states where qumodes are connected in a two-dimensional array. The preparation
of three-dimensional cluster states has also been considered to introduce
fault tolerance into one-way QC [79, 80]. To prepare the cluster state in an
optical setup, a beam-splitter operation is generally used for a two-mode
gate, instead of the CZ gate which requires a complicated setup and introduces
additional noise from finitely-squeezed ancilla states [54, 55, 56]. Thus, the
CZ gates are replaced by beam-splitter networks with appropriate configuration
and parameters. The way to generate the cluster state using beam-splitter
networks has been developed over the past two decades [81, 82, 83, 84, 85, 41,
42, 86, 48], and related experimental progress is explained in Sec. 3.2.

Figure 2: Cluster state. (a) Graph $G=(V,E)$ (b) The cluster state for qubits.
(c) The CV cluster state. (d) The three mode cluster state.

####  2.3.2 Quantum gates in one-way CVQC

One-way CVQC is performed on the CV cluster state described above. As a simple
example, we consider the Fourier gate using the entangled pair prepared by the
CZ gate. Figures 3(a) and (b) show the Fourier gate in one-way QC using the
quantum circuit representation and the cluster state representation,
respectively. Before the CZ gate, the $q$ and $p$ quadratures for the mode
1(2) are described as $(\hat{q}_{1(2)},\hat{p}_{1(2)})$, respectively. The CZ
gate transforms the quadratures as
$(\hat{q}_{1(2)},\hat{p}_{1(2)}+\hat{q}_{2(1)})$. When the input mode 1 is
measured in the $p$ quadrature, the quadratures for mode 2 are transformed to
$(\hat{p}_{1},\hat{p}_{2}+\hat{q}_{1})$ after the feed-forward operation in
the $q$ quadrature for the mode 2. In the case of infinite squeezing, i.e.,
$\langle(\Delta\hat{p}_{2})^{2}\rangle\to 0$, the quadratures
$(-\hat{p}_{1},\hat{p}_{2}+\hat{q}_{1})$ become $(-\hat{p}_{1},\hat{q}_{1})$.
Thus, the above circuit with the infinite squeezing corresponds to the Fourier
gate $\hat{R}(\pi/2)$ described in Eq. (7). In a practical implementation, due
to a finite squeezing effect in a practical implementation, the state after
this circuit suffers from a noise corresponding to
$\langle(\Delta\hat{p}_{2})^{2}\rangle=\frac{1}{2}e^{-2r}$. For the
implementation of the other quantum gates, it is known that an arbitrary
single-mode Gaussian gate can be achieved by using a four-mode linear cluster
state via changing the measurement basis [87]. For example, given the
measurement on the $i$-th mode in the quadrature $\hat{p}+m_{i}\hat{q}$, the
phase gate is implemented by four measurements with
$(m_{1},m_{2},m_{3},m_{4})=(1,0,0,0)$ [87, 88]. In Sec. 3.1.1, the
experimental realization of the single- and two-mode gates in one-way CVQC is
described. To achieve universality, the cubic phase gate should be implemented
within the framework of one-way QC. Figure 3(c) shows the quantum circuit
representation for the cubic phase gate based on the measurement-induced
scheme using the cubic phase state $\ket{\gamma}$. Figure 3(d) shows the
cluster state representation of Fig. 3(c) [78, 65].

Figure 3: Quantum gates in one-way QC. (a) Quantum circuit for the Fourier
gate. (b) Cluster state representation for the Fourier gate. (c) Quantum
circuit for the cubic phase gate. (d) Cluster state representation for the
cubic phase gate [65].

##  3 Scalable quantum computation with CVs

###  3.1 Scalability of photonic QC

####  3.1.1 Building blocks for photonic QC

Recent experimental progress has found unique strategies that make optical
CVQC highly scalable. Before discussing such strategies, let us first briefly
review the history of developing basic building blocks for optical CVQC over
the past several decades [89, 32, 34, 33].

One of the most essential building blocks for optical QC is quantum light
sources. The most commonly used quantum light source in CV systems is the
sources for squeezed vacuum states $\hat{S}(r)\ket{0}$ because these states
not only well approximate the quadrature eigenstate $\ket{0}_{q}$ in the high
squeezing limit ($r\to\infty$) but also play an important role in generating
various CV entangled states [90]. The first generation of squeezed light in
1985 used four-wave mixing in an atomic vapor of sodium atoms [91]. However,
it was later found that high-level squeezed light can be generated more
conveniently by an optical parametric oscillator (OPO), which is an optical
cavity with a second-order nonlinear crystal inside and pumped by a
continuous-wave beam [92]. In this configuration, more than 10 dB (up to 15dB)
of squeezing has been observed [93, 94, 95]. Thus far we have only focused on
single-mode squeezing produced from OPOs operated below threshold. However,
depending on the configuration and operating condition, OPOs can also produce
multimode squeezing and entanglement, such as standard two-mode squeezing
[96], entanglement between spatial or temporal modes [97, 98, 99, 100],
frequency modes [40, 101, 102], and different colors [103, 104]. The OPOs are
also used to generate non-Gaussian states with high purity, such as single
photon states [105], Fock-state superposition states [106], and Schrödinger’s
cat states [107, 108, 109], by sending part of the squeezed light to photon
detectors and adopting specific detection events.

Figure 4: Optical implementation of a universal CV gate set. $\chi^{(2)}$ and
$\chi^{(3)}$ represent second- and third-order nonlinearity, respectively.
Measurement-induced implementations are based on Refs. [54, 46]. EOM, electro-
optic modulator; Disp., displacement operation.

The central challenge in optical QC has been the realization of a universal
set of CV quantum gates, the combination of which allows us to perform an
arbitrary unitary transformation on optical quantum states (Fig. 4). From the
experimental point of view, here we consider a universal set composed of four
Gaussian gates, including phase shift $\hat{R}(\theta)$, beam splitter
$\hat{BS}(\theta)$, displacement $\hat{D}(\alpha)$, and squeezing operations
$\hat{S}(r)$, as well as at least one non-Gaussian gate such as a cubic phase
gate $\hat{V}(\gamma)$ (their definitions are given in Sec. 2.2).
Deterministic implementation of all these Gaussian gates has already been
well-established. Phase shift and beam splitter operations are easy to
implement with only passive linear optics (Figs. 4(a) and (b)). Displacement
operations are also easily implemented by partly mixing a modulated ancillary
beam to the target beam (Fig. 4(c)). In contrast, implementing squeezing
operations requires second-order nonlinearity and thus is nontrivial. A
straightforward way to perform squeezing is to directly couple the target beam
to a nonlinear medium, but this method often suffers from unwanted coupling
loss that degrades the operational fidelity. This problem is avoided by a
measurement-induced squeezing gate [54], where an ancillary squeezed vacuum
state is consumed to indirectly apply a squeezing operation to the target
state (Fig. 4(d)). Such a measurement-induced squeezing gate has been
demonstrated [110] and further extended to implement a two-mode CX gate for
CVs [55]. In the early experiments, these gates were performed on Gaussian
input states defined in optical narrow frequency sidebands. They were later
updated to cover much broader bandwidth so that they can be applied to non-
Gaussian input states that are usually created in optical wave packets with
broad frequency spectra [111, 56].

Instead of constructing these quantum gates one by one in optical circuits,
one-way QC based on CV cluster states has also been pursued as an alternative
approach. In the early experiments, small-scale CV cluster states with up-to
eight modes have been generated by preparing multiple squeezed light sources
in parallel and mixing the generated squeezed vacua with multiple beam
splitters [112, 113, 114]. Gaussian quantum gates, including single-mode
[115], two-mode [116], and also sequential gates [117], were demonstrated by
performing homodyne measurements on these cluster states with appropriate
measurement bases.

Thus far, all demonstrated CV gates were limited to Gaussian gates. Non-
Gaussian gates have yet to be achieved, and they have been one of the most
challenging building blocks in optical CVQC. However, there has been
continuous theoretical and experimental progress towards the realization of
non-Gaussian gates based on measurement-induced schemes. As already mentioned
in Sec. 2.3.2, Ref. [23] proposed an original measurement-induced scheme for
deterministic cubic phase gates $\hat{V}(\gamma)$, requiring only Gaussian
operations except for the preparation for an ancillary cubic phase state
$\hat{V}(\gamma)\ket{0}_{p}$. The original proposal was later rearranged to a
simpler implementation method which is experimentally more feasible [45, 46]
(Fig. 4(e)). Moreover, it was also found that higher-order phase gates can be
implemented similarly with additional non-Gaussian ancillary states [47]. A
probabilistic source of approximated cubic phase states [106] and the
feedforward system for the cubic phase gate [118] have been reported already.
Toward deterministic cubic phase gates, an all-optical quantum memory to store
the cubic phase states has also been investigated [119]. These technologies
are expected to lead to the demonstration of a cubic phase gate shortly.

####  3.1.2 Integrating building blocks to scale up QC

Once all the necessary basic building blocks for photonic QC are realized, it
is in principle possible to scale up photonic QC by building large optical
circuits incorporating all these ingredients. However, this kind of in-
principle scalability does not mean that the implementation is experimentally
efficient. All the early proof-of-principle demonstrations introduced in Sec.
3.1.1 used bulky optical components in free space and beam-path encoding where
one beam path represents one quantum mode. Such implementations are flexible
and suitable for building simple optical circuits designed for specific
purposes. However, the number of optical components and spaces linearly
increases with the number of optical modes and operational steps. A vast
number of gates and modes are estimated to be required for quantum computers
to outperform current supercomputers in solving practical and meaningful
quantum algorithms [120, 121]. In Ref. [121], the number of the physical gates
and modes for factoring 2048 bit integers are estimated as $\sim 10^{9}$ and
$2\times 10^{7}$, respectively. Building such a powerful quantum computer by
the straightforward extension of the previous implementations is almost
impractical.

However, recent progress has revealed that the property of light enables us to
overcome the problem and efficiently scale up photonic QC. Here, we focus on
three important research directions to scale up photonic QC: one is optical
multiplexing techniques to use the rich degrees of freedom of light; second is
the increase in operational bandwidth to utilize the large information
capacity of light; third is photonic chips to miniaturize and integrate
optical components. In the following Secs. 3.2 to 3.4, we introduce the basic
idea and recent experimental progress for each direction.

###  3.2 Scaling up with multiplexing

####  3.2.1 Types of multiplexing

The basic implementation of optical quantum information processing uses path
encoding. However, there exist many other degrees of freedom of light for
encoding, such as frequency, time, and spatial modes. These degrees of freedom
enable us to potentially pack a large amount of quantum information in a
single optical path and perform large-scale QC more efficiently. This approach
is called multiplexing and one of the unique advantages in optical systems. We
introduce three different options for multiplexing below, each of which has
advantages and disadvantages. Note that these options can be combined and
simultaneously exploited.

The first option is spatial-mode multiplexing using different spatial
distributions of light. In principle, one can define arbitrarily many
orthogonal spatial modes in a single optical beam. For example, the experiment
in Ref. [122] generated multimode CV entangled states in a single optical beam
by defining modes to be combinations of different spatial regions of one beam.
In another approach, several spatial modes with different orbital angular
momentum (OAM) were used to generate entangled states [123, 124] or realize
quantum teleportation of several modes in parallel [125]. These approaches may
be suitable for enhancing the data-carrying capacity in quantum communication
protocols by combining multiple channels into a single optical transmission
channel. However, it is difficult to individually access or separate each
spatial mode in a single beam, and thus the efficient scheme to implement QC
in this strategy is unclear.

The second option is frequency multiplexing using different frequency bins.
One prominent approach in this direction is to use equally spaced resonant
frequency modes (optical frequency combs) of an OPO [126, 127, 33]. By pumping
a single OPO with multi-frequency continuous-wave pump beams, up to 60
frequency modes were demonstrated to be entangled [40]. In contrast, another
approach repeatedly pumps an OPO by synchronized pulsed pump beams, succeeding
in generating entangled states of $\sim 10$ independent frequency modes [101,
102] and non-Gaussian states in selectable frequency modes [128]. The
challenges in these approaches include the limitation of maximum frequency
modes (for example, the phase-matching bandwidth limits the modes in the
frequency-comb approach on the order of $10^{4}$ [33]) as well as difficulties
in frequency-sensitive measurement and efficient separation of individual
frequency modes.

The third option is time multiplexing using different time bins [34]. By
dividing an optical beam into non-overlapping time bins, an unlimited number
of wave-packet modes can be defined. These modes are individually accessible,
and the same optical components can be repeatedly used at different times for
operations and measurements of the modes. Time multiplexing is currently one
of the leading approaches in optical CVQC, as highlighted by recent
demonstrations of ultra-large-scale cluster states and one-way QC with these
states [84, 85, 41, 42, 43, 44]. One disadvantage of this approach is the
lossy long optical delay lines required for introducing interaction between
optical pulses at different times and increasing the number of processable
modes. In the following Secs. 3.2.2 and 3.2.3, we focus on two specific
approaches in this direction and explain more details of technical progress
and challenges.

####  3.2.2 Time-multiplexed one-way QC

Figure 5: Experimental schematic for generating time-multiplexed cluster
states. (a) One-dimensional cluster states in Refs. [84, 85]. (b) Two-
dimensional cluster states in Ref. [41]. (c) Two-dimensional cluster states in
Ref. [42]. OPO, optical parametric oscillator; BS, beam splitter; HD, homodyne
detector; LO, local oscillator.

Here, we review the recent impressive experimental progress in one-way CVQC
with a time-multiplexing approach. As mentioned in Sec. 3.1.1, the early
demonstrations of one-way QC relied only on path encoding [112, 113, 114].
However, this encoding requires one squeezed light source for one quantum
mode, making it infeasible to generate large-scale cluster states. In around
2010, novel ideas to efficiently generate large-scale cluster states with time
multiplexing were proposed [129, 83]. Based on these ideas, the generation of
more than 10,000-mode cluster states was verified in 2013 [84]. Later the
number of modes was increased up to one million by technical improvement of
the experimental system [85]. In fact, this is the largest number of modes
confirmed to be entangled in any physical system. The schematic of these
experiments is shown in Fig. 5(a). In this setup, two OPOs continuously
generated squeezed beams, each of which was divided into time bins to define a
train of pulsed squeezed states. These squeezed states were then combined at
the first beam splitter to produce an Einstein-Podolsky-Rosen (EPR) entangled
state. The optical delay line followed by the second beam splitter combined
one part of the EPR state with one part of another EPR state. This interaction
finally generated a large-scale entangled state, where quantum modes were
connected in a one-dimensional chain fashion. The generated states were proven
to be equivalent to one-dimensional CV cluster states, a resource for one-
input one-output QC.

Figure 6: Graphical representation of the two-dimensional CV cluster states.
Reproduced from [41].

In 2019, two research groups further extended this approach to generate two-
dimensional cluster states, a universal resource for multi-input multi-output
one-way QC [41, 42]. Although experimental setups were slightly different
(Figs. 5(b) and (c)), these two demonstrations share the common idea that two
optical delay lines with different lengths allow for entangling optical modes
in two dimensions. The shorter delay line was for connecting neighboring modes
to produce one-dimensional cluster states and the longer one was for
connecting these cluster states in another dimension to produce two-
dimensional cluster states. The graph of the resultant cluster state can be
depicted as quantum modes arranged on a continuous cylindrical structure in
both experiments, as shown in Fig. 6. The number of modes along the cylinder
circumference was determined by the length of the longer delay line and limits
the number of input modes for QC. On the other hand, the number of modes along
the cylindrical axis was infinite in principle, enabling an unlimited number
of quantum operations.

Recently, these groups have demonstrated one-way QC using these time-
multiplexed cluster states. CV one-way QC can be performed by homodyne
measurement of each mode of cluster states with appropriate measurement basis
and the following displacement operation on specific modes. When using time-
multiplexed cluster states, it is necessary to change the measurement basis
pulse-by-pulse in time. In fact, rapidly changing the homodyne measurement
basis is easily achieved by phase-modulating the LO beams with electro-optic
modulators. The phase modulation pattern can be defined by programmable
electric signals, meaning that the gate sequence can be easily programmed and
changed according to computational purposes. It is an advantage over the
standard circuit-model QC whose gate sequence is determined by the arrangement
of optical components and thus not easily variable. Thus far, one-input
100-step quantum gates with one-dimensional cluster states [43] and three-
input 12-step quantum gates with two-dimensional cluster states [44] have been
experimentally demonstrated. In these experiments, optical displacement
operations were replaced by classical signal processing after the measurement.
This replacement is valid when the purpose is to know only classical
calculation results at the final step.

Despite these impressive experiments demonstrating scalability, there still
exist a lot of challenges to be overcome in time-multiplexed one-way QC.
First, the current implementations rely on optical delay lines to increase the
number of processable modes. An optimal choice for long delay lines is optical
fibers having the minimum loss of 0.2 dB/km at telecommunication wavelength.
For example, transmitting a 1-km optical fiber introduces 5% loss, which is
below the fault-tolerant threshold for some QEC schemes [130]. The minimum
time interval of wave packets in the recent experiments is 40 ns (12 m in
length) [43], and thus the 1-km fiber enables QC with the order of 100
processable modes. Further scale-up requires a shorter time interval or
modified architectures (one methodology to shorten the time interval is given
in Sec. 3.3). Another issue is that all the previous implementations have been
limited to the Gaussian regime, which is efficiently simulatable by classical
computers [53]. Possible options for introducing non-Gaussian elements include
photon-number resolving measurement on cluster states [78, 65] or injection of
non-Gaussian states as ancillary states [131]. Finally, QEC needs to be
incorporated into the one-way QC. For this purpose, the extension from two- to
three-dimensional cluster states and the introduction of bosonic QECCs are
being considered [48, 86, 37, 38].

####  3.2.3 Loop-based architecture for time-multiplexed QC

Another emergent approach for scalable time-multiplexed optical QC is to use
loop-based architectures, where optical pulses circulating in loops transmit
the same optical components repeatedly. The repeated use of the same optical
components leads to a dramatic reduction of the circuit size, making photonic
QC more scalable. In 2014, Ref. [132] proposed a dual-loop architecture that
realizes an arbitrary multi-mode linear interferometer in a compact setup.
This architecture was proven to have applications in boson sampling and
universal QC based on single photons [132, 133], and several related
experiments have been reported already [134, 135].

Figure 7: Loop-based architecture for CVQC. (a) A universal multi-mode quantum
processor proposed in Ref. [136]. (b) A single-mode quantum processor
demonstrated in Ref. [137]. EOM, electro-optic modulator; VPS, variable phase
shifter; VBS, variable beam splitter.

This idea was later extended to a dual-loop architecture for universal CVQC in
Fig. 7(a) [136]. The essential working principle of this architecture is
simple. The outer loop is a quantum memory to store a train of optical pulses
for input states and ancillary states for QC. The inner one is a quantum
processor to sequentially perform gates to the pulses by dynamically
controlling several parameters, such as beam splitter transmissivity $(T)$,
switch polarity, phase shift $(\theta)$, measurement basis $(\phi)$, and
feedforward gain $(g)$. This configuration enables us to perform a universal
set of gates either directly or by measurement-induced schemes [54, 46], once
necessary ancillary states are provided. This architecture can deal with
arbitrarily many modes and operational steps without increasing the number of
optical components. Moreover, the gate sequence can be easily modified by
changing the control signal of the parameters, meaning that the QC is
programmable.

One advantage of the above loop-based scheme compared to the one-way QC scheme
in Sec 3.2.2 is the decrease in the required resources. One-way QC first
prepares universal entanglement resources for arbitrary QC. When performing a
specific computational task, unnecessary parts of the entanglement have to be
removed by measurement and feedforward operations [131, 138]. This process
increases operational steps and often introduces additional noise and
imperfection to the computation. In contrast, the loop-based scheme requires
fewer steps because it creates entanglement only when necessary. However, the
challenge is the requirement for fast and precise control of many variable
parameters, which is unnecessary for the one-way QC.

Recent efforts have tackled the technical challenge and partly demonstrated
the advantageous functionalities of the dual-loop architecture. Based on the
original proposal, Ref. [139] dynamically controlled a single-loop circuit
with a variable beam splitter and phase shifter to demonstrate the
programmable and scalable generation of various entangled states from two-mode
EPR states to 1,000-mode cluster states. A later experiment updated this
circuit to a single-mode photonic quantum processor in Fig. 7(b),
demonstrating programmable and multi-step Gaussian operations [137]. This
demonstration included optical feedforward that was omitted in the time-
multiplexed one-way QC experiments [41, 42]. This inclusion of feedforward
means that this processor can finalize quantum operations and export the
output optical quantum states for further use, thereby working as a versatile
photonic quantum processor with potential applications to quantum
communication and sensing. The next challenge will be the demonstration of
non-Gaussian gates and multi-mode gates.

The idea of using loop structures has been introduced in several proposals for
optical CV quantum information processing. In contrast to the dual-loop
architecture in Ref. [136], the chain-loop architecture was introduced in Ref.
[140] to implement universal QC with lower loss. The loop structure is also
effective for increasing the success probability of photon subtraction [47]
and efficient implementation and verification of Gaussian boson sampling
[141].

###  3.3 Scaling up with broadening bandwidth

####  3.3.1 Why broadband?

The carrier frequency of a light field is a few hundred THz, and the available
broad frequency space is a rich resource to encode large-capacity quantum
information. This resource will ultimately enable us to process unlimitedly
many independent frequency modes simultaneously in frequency multiplexing or
ultra-short optical pulse trains at an ultra-high clock frequency in time
multiplexing. However, the operational bandwidth in actual optical QC is
inherently limited to the electronically accessible MHz-to-GHz range, posing a
dramatic gap between potentially usable bandwidth and accessible bandwidth.
For example, the maximum bandwidth in recent experiments on CV quantum
operations is $\sim 100$ MHz [56], limiting the operational clock frequency in
time-multiplexed processing to 25 MHz at most [43]. This limitation came from
the bandwidth of squeezed light sources and electronics to measure and
manipulate optical states. Broadening the bandwidth of these elements is an
important research direction to exploit the optical bandwidth maximally.
Below, we refer to recent advancements in this direction and finally explain,
as an ultimate dream, an all-optical QC scheme that is not limited by
electronic bandwidth anymore.

####  3.3.2 Ultra-broadband squeezed light source

One of the most fundamental light sources in CVs is squeezed light sources,
which are used for producing various Gaussian and non-Gaussian states. The
bandwidth of the squeezed light sources usually limits the bandwidth of the
generated optical states and their generation rates. As already mentioned in
Sec. 3.1.1, the most successful squeezed light source to date is an OPO, where
second-order nonlinearity is enhanced in optical cavities. Instead, these
cavities limit the bandwidth of the generated squeezed states. For example,
typical ring-cavity OPOs used in state-of-the-art time-multiplexed experiments
[41, 43, 139, 137] have only 65 MHz bandwidth [142], which mainly limited the
computational speed of these experiments. The bandwidth can be broader by
fabricating smaller cavities but has been limited to around 2 GHz even with a
monolithically integrated OPO [143].

Figure 8: Design of the periodically poled LiNbO3 waveguide for OPA.
Reproduced from [144].

This bandwidth limit can be broken by removing cavities and using a single-
pass optical parametric amplifier (OPA). The OPA’s bandwidth is no longer
limited by the cavities and reaches THz, limited only by dispersion or phase-
matching conditions of the nonlinear crystal itself. In general, high-level
squeezing cannot be produced by simply injecting a continuous-wave pump beam
into a standard nonlinear crystal without cavity enhancement. Instead, pulsed
pump beams with intense peak power can induce sufficient nonlinearity in this
single-pass OPA [145]. However, the pulsed configuration introduces another
technical difficulty in optimizing spatiotemporal mode matching between pulsed
beams and thus often degrades the quality of the squeezed light [146]. Another
option is to use waveguide nonlinear crystals, enhancing nonlinearity by
transverse field confinement of pump beams in small cross-sections over long
interaction length. Several experiments have reported squeezed light
generation from waveguide OPAs with continuous-wave pump beams [147, 148], but
until recently, the squeezing level was limited to around 2 dB [149, 150]. In
2020, 4.0-dB squeezing was reported by a fiber-coupled OPA module with a
periodically poled LiNbO3 (PPLN) waveguide [151]. Without the fiber coupling,
the same waveguide produced 6.3-dB squeezing and was confirmed to have the
squeezing bandwidth of 2.5 THz [144] (Fig. 8). In these experiments, a higher
level of squeezing is expected by improving the fabrication process of the
waveguides to reduce the optical propagation loss due to structural
imperfections. Such broadband squeezed light sources will further strengthen
the capacity of frequency- and time-multiplexing. For example, their THz-order
bandwidth enables us to use micrometer wave-packet modes for time
multiplexing, much shorter than the tens-of-meter modes in recent experiments
[41, 42, 43, 44, 139]. It dramatically reduces the length of required delay
lines and downsizes the optical circuits, ultimately leading to large-scale
time-multiplexed QC on compact photonic chips.

####  3.3.3 Ultra-broadband quadrature measurement

Figure 9: Standard homodyne detection (a) and OPA-based quadrature measurement
(b). Reproduced from [152].

In addition to the broadband light sources, broadband light detectors are
required for broadening the operational bandwidth of photonic QC. In CVs, the
most commonly used detector is homodyne detectors to measure quadratures of
light fields. Among the electronics in CVQC, the bandwidth of the homodyne
detector most severely limits the overall operational bandwidth (other signal
processing, such as amplification and modulation, can easily reach GHz-
bandwidth operation). Standard homodyne detection lets an input optical signal
interfere with a LO beam, converts the optical signal into photocurrent by
photodiodes, and then amplifies the signal by transimpedance amplifiers. The
response of photodiodes and amplifiers usually limit the speed performance of
the detector. Conventional high-quantum-efficiency homodyne detectors for
quantum optics have typically had bandwidths of $\sim 100$ MHz or below. [153,
154]. Recently, high-speed homodyne detectors with up to 1.1-GHz bandwidth
have been reported with free-space photodiodes [155]. Another experiment has
developed a 1.7-GHz bandwidth homodyne detector with on-chip photodiodes and
measured squeezed light with up to 9 GHz [156]. These GHz-bandwidth homodyne
detectors will lift the bandwidth of QC from the current MHz range to the GHz
range. However, as far as electronics are concerned, a further dramatic
increase in bandwidth cannot be expected.

A completely different approach to overcome this limitation is to use high-
gain OPAs for quadrature measurement, as shown in Fig. 9 [157, 152, 158]. The
parametric amplification process selects one of the quadratures of an optical
field and amplifies it without adding noise while attenuating the other
quadrature. This process is equivalent to the standard homodyne detection,
where the LO beam selects a quadrature of interest and amplifies it to an
electrical signal level sufficiently above the electric noise of the detection
system. In the high-gain OPA, a high-power pump beam plays the role of the LO
beam, amplifying the optical signal to the level sufficiently above the vacuum
noise level of light. There are several advantages to this measurement scheme.
First, the same nonlinear process can be used both for squeezed light
generation and quadrature measurement, thus guaranteeing that the measurement
system can cover the entire bandwidth of the squeezed light. Second, the high-
gain OPA amplifies the originally quantum signal of quadratures to a classical
optical signal which is insensitive to optical losses, including propagation
losses, mode mismatching, and detector inefficiencies [159, 160]. Therefore,
the quadrature can be measured with high quantum efficiency even by low-
quantum-efficiency detectors.

In 2018, a high-gain OPA was realized by third-order nonlinearity of a
photonic crystal fiber to achieve broadband quadrature measurement of squeezed
light [157]. In this experiment, a pulsed laser pumped the fiber to generate
pulsed squeezed light through the nonlinearity. Its quadrature was then
amplified by using the same nonlinear process again and finally detected by a
spectrometer. This configuration realized the observation of squeezing over 55
THz and also showed its robustness to detection inefficiencies. In
applications to CVQC, a more suitable option for high-gain OPAs is to use
second-order nonlinearity of waveguide crystals with continuous-wave pump
beams. The difficulty of this approach lies in the fact that the crystal for
high-gain OPAs is required to have high optical conversion efficiency as well
as high durability to a high-power pump beam. In 2019, an over-30-dB-gain OPA
was reported by a highly durable PPLN waveguide with a continuous pump beam
[161]. In 2020, the PPLN waveguides were used both for squeezed light
generation and quadrature measurement [152]. As a result, 3-dB squeezing was
observed over 3 THz by detecting the amplified optical signal with an optical
spectrum analyzer. The emergence of these broadband quadrature measurement
schemes with high-gain OPAs will open a promising pathway for ultrahigh
capacity quantum communication and computation.

####  3.3.4 Towards ultra-broadband QC by all-optical means

As an ultimate form of ultra-broadband optical QC, there is a potential to
realize “all-optical” QC. Its essential idea is to replace all electronics
with corresponding optical components and break the electrical bandwidth
limit. The resultant all-optical quantum computer may be able to exploit the
entire THz bandwidth of light.

This idea dates back to the original proposal of “all-optical” CV quantum
teleportation in 1999 [162] (Fig. 10). CV quantum teleportation [163, 164] is
the fundamental building block for one-way QC and measurement-induced quantum
gates in CVs. The original protocol of optical CV quantum teleportation
proceeds in the following four steps: (i) generation of an EPR-entangled state
shared by a sender and a receiver, (ii) sender’s joint homodyne measurement on
an input state and his/her part of the entangled state, (iii) transmission of
the measured electric signals to the receiver through classical channels, and
(iv) receiver’s feedforward operation to his/her part of the entangled state
through electro-optic modulation. The all-optical teleportation scheme (Fig.
10) replaces the homodyne detectors in step (ii) with high-gain OPAs,
amplifying the optical signals without converting them to electric signals.
The amplified optical signals can be sent through lossy classical optical
channels to the receiver because they are well above the vacuum noise level
and tolerant to losses. These optical signals are then directly used for
feedforward operation by weakly injecting the beam to the target beam via a
beam splitter. In this way, the teleportation protocol can be performed all-
optically without any electronics, leading to potentially ultra-broadband
operation.

Figure 10: All-optical quantum teleportation proposed in Ref. [162] and
demonstrated in Ref. [125].

Recently, all-optical quantum teleportation was experimentally demonstrated as
a means to simultaneously teleport nine channels multiplexed in the OAM of a
single beam [125]. In this experiment, OPAs based on four-wave mixing were
used to amplify the optical signals in different OAM modes simultaneously. The
amplified optical beam was used to perform feedforward operation directly for
all nine channels in parallel. This demonstration showed that the all-optical
method is a promising path to simultaneous processing of a large amount of
quantum information encoded in degrees of freedom of a single light beam. In
terms of exploiting its frequency degree of freedom, the recent development of
ultra-broadband squeezed light sources and measurement based on waveguide OPAs
[151, 144, 152] will be an enabler for ultra-broadband all-optical quantum
teleportation circuits. Such all-optical teleportation circuits can be further
extended to more advanced QC protocols, such as one-way QC and measurement-
induced gate operations, opening the path for the all-optical QC with over-THz
bandwidth.

Figure 11: Integrated photonic chips for CV quantum information processing.
(a) The silica-based photonic chip for generating CV entanglement in Ref.
[165]. (b) The LN-based photonic chip for generating, mixing, and measuring
squeezed beams. Reprinted from [166]. © The Authors, some rights reserved;
exclusive licensee AAAS. Distributed under a CC BY-NC 4.0 license
http://creativecommons.org/licenses/by-nc/4.0/.

###  3.4 Scaling up with integrated optics

Another important research direction for scaling up optical QC is to
miniaturize optical circuits on small chips [167, 168]. Traditional quantum
optics experiments were based on bulky optical components separately sitting
on a tabletop. In this case, the optical circuit is unavoidably large and
susceptible to the instability of the surrounding environment, thus lacking
scalability. In the same way as the evolution of on-chip integration of
electronic components, there has been a tremendous effort in integrating and
miniaturizing optical components on photonic chips. Photonics chips offer a
promising path to packaging many quantum light sources, interferometers, and
detectors compactly. There are many advantages for the on-chip integration,
not to mention the miniaturization of large-scale optical circuits. First, on-
chip integration improves the stability of the optical circuits compared to
the bulky setups that are susceptible to vibration and temperature change.
Second, it enhances the precision of quantum operations by avoiding spatial
mode mismatching between propagating beams and reducing propagation losses
with an appropriate choice of materials. Third, optical circuits become
reconfigurable and versatile because on-chip beam splitter transmissivity and
phase shifts can be externally controllable. Finally, electronic circuits for
measurement and feedforward operations can also be integrated into the same
chip, enabling very fast and low latency operations. There are several
material options for photonic chips, each of which has advantages for
realizing a practical integrated quantum photonics platform. Quantum light
sources for single-photon or squeezed states can be integrated on the chips if
the material has sufficient second-order nonlinearity, converting one pump
photon into two photons, or third-order nonlinearity, converting two pump
photons into another two photons.

One of the most typical materials for the integration is fused silica (SiO2)
due to its low propagation loss and good coupling efficiency to optical
fibers. Silica-based photonic chips integrating beam splitters and phase
shifters have been used for the early demonstrations in single-photon-based
quantum information processing, such as high-fidelity two-photon interference
and two-qubit gates [169], Shor’s factoring algorithm [170], multi-particle
quantum walks [171], and boson sampling [172]. The silica-based chips were
later applied to CV quantum information processing, such as CV entanglement
generation [165] (Fig. 11(a)) and measurement-induced squeezing operations
[173]. However, the silica-based circuits are based on low-refractive-index-
contrast waveguides whose low transverse confinement leads to relatively large
chip size. Even simple circuits require a chip size of $\sim 10$ cm.
Therefore, the silica-based platform is not suitable for the fabrication of
large complex photonic circuits. In addition, silica has no second-order
nonlinearity and also much lower third-order nonlinearity compared to silicon.
Thus, silica is not suitable for integrating high-level squeezed light
sources, although single-photon sources on silica chips have been reported
[174].

On the other hand, silicon-based platforms such as silicon (Si) and silicon
nitride (Si3N4) offer a very high-refractive-index contrast as well as
compatibility with existing foundary processes [175, 176]. Its high-index
contrast leads to not only high component density by reducing the circuit size
but also high transverse confinement that is suitable for nonlinear processes.
These materials do not possess second-order nonlinearity but have sufficient
third-order nonlinearity, enabling us to fabricate and integrate on-chip
squeezed light sources. There have been several CV experiments with silicon-
based platforms. For example, Ref. [177] used a silicon chip to produce
squeezed states by four-wave mixing and perform Gaussian boson sampling in a
fixed configuration. Silicon nitride ring resonators have also been
demonstrated to generate non-degenerate two-mode squeezed states [178] and
degenerate single-mode squeezed states [179, 180, 181] by four-wave mixing or
self-phase modulation. One of such sources were later integrated into a
programmable photonic chip based on silicon nitride, demonstrating Gaussian
boson sampling and its related algorithms [8].

Another reasonable choice for the material is lithium niobate (LiNbO3, LN)
since it has second-order nonlinearity for more efficient squeezed light
generation [182, 183]. In addition, its large electro-optic coefficient
enables fast amplitude and phase modulation, which is suitable for switching
and feedforward operations in CVQC. LN waveguides themselves have been
investigated as single-pass squeezed light sources [151, 144, 147, 184].
Several experiments reported more advanced integrated photonic chips based on
LN waveguide, integrating periodically poled waveguide segments for squeezed
light generation and other segments for couplers and phase shifters [185, 166,
149] (Fig. 11(b)). These experiments used proton-exchanged LN waveguides
having low index contrast. In contrast, high index contrast LN waveguides can
be fabricated by a lithium-niobate-on-insulator (LNOI) platform, which offers
much better scaling properties similar to silicon-based platforms. Fabrication
of LNOI waveguides for quantum state generation and manipulation has also been
investigated [186, 183, 187].

Photonic QC on integrated chips has often been troubled with optical losses
caused by waveguide roughness, material absorption, and coupling efficiency at
input and output ports. These effects are expected to be reduced by suitable
material choice and improved fabrication processes. Ultimately, it is
favorable to integrate all elements, including quantum light sources,
interferometers, and detectors, on the same chip to avoid coupling losses. In
addition to integration of sources and interferometers described above, there
has been much effort towards integration of detectors, such as homodyne
detectors [188, 189, 156], superconducting nanowire single-photon detectors
[190, 191], and transition-edge sensors for photon number resolving detection
[192, 193]. Integrated chips for time-multiplexed CVQC may use off-chip fiber
delay lines, so the development of structures for near-unity chip-fiber
coupling efficiency is also desired. On-chip integration of suitable CVQC
architectures with broad operational bandwidth would be the ultimate form of
photonic quantum computers, leading to scalable and universal QC at an ultra-
high clock rate.

##  4 Fault-tolerant quantum computation with CVs

###  4.1 Quantum error correction with CVs

In the previous section, we have seen the progress to scale up photonic QC by
several approaches, which enable us to increase the number of processable
modes and gate steps. Considering the errors during quantum gates, however,
the errors are accumulated and propagated via quantum gate operations, which
leads to the destruction of quantum information even if each of these errors
is small. Thus, we must remove the errors to implement large-scale QC. To
solve the problem of errors during QC, the QEC for qubits [194] has been
widely developed. By using the QECC such as Knill’s ${\rm C}_{4}/{\rm C}_{6}$
code [195], Steane’s seven qubit code [196, 197], or surface codes [198, 199]
in a fault-tolerant manner, the errors can be substantially reduced if the
error per quantum operation is below a constant threshold value [200, 201,
202]. This is called a threshold theorem.

The QEC for CVs was introduced by Braunstein [203], Lloyd and Slotine [204],
where the standard qubit-based QECC such as Shor’s nine qubit code [205] was
applied to the quadrature eigenbasis $\\{\ket{s}_{q}\\}_{s\in\mathbb{R}}$. The
QECC by Braunstein has been demonstrated experimentally in Ref. [206], where
Shor’s nine qubit code was encoded and decoded by using squeezed light and
linear optics. Although it is worthwhile demonstrating the principle of the
QECC for CVs, unfortunately, error models of these protocols do not correspond
to the realistic errors, for example, photon loss during QC. Bosonic codes
overcome this problem, as described in Sec. 4.2.

Table 2: A family of bosonic codes.

|  Superposition of (squeezed) coherent states |  Fock state superposition |  Hybrid approach
---|---|---|---

Example |  Cat code [207] |  GKP qubit [23] |  Binomial code [208] |  Cat-photon qubit [209]
---|---|---|---|---
${\ket{0}}_{\rm L}$ state |  $\ket{\alpha}$
|  $\sum\limits_{m=-\infty}^{\infty}\ket{2m\sqrt{\pi}}_{q}$
|  $\frac{1}{\sqrt{2^{N}}}\sum\limits_{p\in{\rm
even}}^{[0,N+1]}\sqrt{\binom{N+1}{p}}{\small\ket{pM}}$
|  $\ket{+}\ket{\alpha}$

${\ket{1}}_{\rm L}$ state |  $\ket{-\alpha}$
|  $\sum\limits_{m=-\infty}^{\infty}\ket{(2m+1)\sqrt{\pi}}_{q}$
|  $\frac{1}{\sqrt{2^{N}}}\sum\limits_{p\in{\rm
odd}}^{[0,N+1]}\sqrt{\binom{N+1}{p}}{\small\ket{pM}}$
|  $\ket{-}\ket{-\alpha}$

Experiment (examples) |  Optical setup [210], Superconducting circuits [211] |  Ion trap system [212], Superconducting circuits [213]. |  Superconducting circuits
[214] |  Optical setup [215]

Other code examples |  Multicomponent cat code [216], Rotation symmetric cat code [217], Pair cat code [218] |  NOON code [219, 220], Loss tolerant code [221, 222, 223] |  N/A

---|---|---|---

###  4.2 Bosonic codes

####  4.2.1 A family of bosonic codes

Bosonic codes encode discrete quantum information into bosonic modes. The
bosonic code is a key ingredient to protect quantum information against errors
from an environment such as photon loss. By concatenating bosonic codes with
standard qubit-based QECCs such as Steane’s seven qubit code [196, 197], one
can implement large-scale QC thanks to the threshold theorem. There are a
variety of bosonic codes for encoding quantum information in a CV system
[224], where they could be classified into three categories: a superposition
of coherent states (or squeezed coherent states), a superposition of a finite
number of Fock states, and a hybrid approach of them. In this review, we
classify the GKP qubit [23] into the first category. We will explain more
details of the GKP qubit in subsequent subsections. Table 2 summarizes
examples of the bosonic codes.

The first category includes the cat code, which is a superposition of a few
coherent states [225, 218, 226]. The logical 0 and 1 states for one of the
simplest cat codes are defined as $\ket{0}_{\rm L}=\ket{\alpha}$ and
$\ket{1}_{\rm L}=\ket{-\alpha}$, respectively, where $\ket{\alpha}$ is a
coherent state with a real amplitude $\alpha$. Since the overlap
$\braket{-\alpha}{\alpha}=e^{-2\alpha^{2}}$ is not zero, there is the
probability of misidentifying the logical state, i.e., the bit-flip error. By
using large $\alpha$, this probability becomes effectively zero [227, 228,
229]. For a photon loss channel with the efficiency $\eta$, the probabilities
of the bit- and phase-flip errors after photon loss are given by
$e^{-2\eta\alpha^{2}}$ and $\frac{1}{2}(1-e^{-2(1-\eta)\alpha^{2}})$,
respectively [229].

The second category includes the binomial code composed of a superposition of
Fock states with coefficients which obey a binomial distribution [208]. The
binomial code can correct amplitude damping, displacement, and dephasing
errors by detecting the number parity of photons. The logical 0 and 1 states
for the binomial code, which can correct $L$ photon losses, $G$ photon gain
errors, and $D$ dephasing events, are given by $\ket{0(1)}_{\rm
L}=\frac{1}{\sqrt{2^{N}}}\sum_{p\in{\rm
even(odd)}}^{[0,N+1]}\sqrt{\binom{N+1}{p}}{\small\ket{pM}}$, where $M=L+G+1$
and $N={\rm max}\\{L,G,2D\\}$.

For the third category, there are several proposals for a hybrid approach of
cat and photonic qubits [209, 215, 32, 230, 231, 232, 233]. The logical 0 and
1 states for a typical cat-photon qubit are given by $\ket{0}_{\rm
L}=\ket{+}\ket{\alpha}$ and $\ket{1}_{\rm L}=\ket{-}\ket{-\alpha}$ with
$\ket{\pm}=(\ket{H}\pm\ket{V})\sqrt{2}$, respectively, where $\ket{H}$ and
$\ket{V}$ are horizontal and vertical polarization states of a photon,
respectively. The cat-photon qubit allows us to implement near-deterministic
quantum teleportation [209]. For large-scale QC with the hybrid approach, the
architecture for fault-tolerant QC with the hybrid of photon and cat qubits
has been proposed [234, 235].

####  4.2.2 GKP qubit

Among bosonic codes, the GKP qubit proposed by Gottesman, Kitaev, and Preskill
[23] is the most promising alternative to photon-based qubits, which provides
both universality and fault tolerance for CVQC. A GKP qubit encodes a qubit
into the oscillator’s position and momentum quadratures. Indeed, the GKP qubit
can correct small random displacement errors for both $q$ and $p$ quadratures.
Furthermore, the GKP qubit has an excellent error tolerance against noises
that frequently occur in an optical system, e.g., photon loss, compared with
cat and binomial codes [224]. GKP qubits inherit the advantage of squeezed
vacuum states on optical implementation; they can be entangled by only beam-
splitter coupling. By virtue of this feature, the large-scale CV cluster state
combined with the GKP qubits allows large-scale QC if these states have a
sufficient squeezing level [88]. Experiments involving trapped ions [212] and
superconducting circuits [213] have demonstrated the generation of the GKP
qubit. The squeezing level of the GKP qubit generated in Ref. [213] is
sufficient for fault tolerance [130, 236]. In recent years, the GKP qubit has
gathered a lot of interest for various applications. In addition to the
application for QC [237, 238, 239], the GKP qubit performs well for quantum
communication [240, 241, 242] thanks to the robustness against photon loss
[224]. Indeed, recent results show that using GKP qubits may greatly enhance
the distance of quantum communication [243, 244].

Figure 12: The codeword for the GKP qubit. (a) Ideal GKP qubit with infinite
squeezing. (b) The GKP qubit with finite squeezing.

The ideal code states of the GKP code are Dirac combs in $q$ and $p$
quadratures and are described as

$\displaystyle\ket{{0}}_{\rm
GKP}=\sum_{m=-\infty}^{\infty}\ket{2m\sqrt{\pi}}_{q},$ (19)
$\displaystyle\ket{{1}}_{\rm
GKP}=\sum_{m=-\infty}^{\infty}\ket{(2m+1)\sqrt{\pi}}_{q},$ (20)

respectively, as shown in Fig. 12(a). Since the ideal GKP code is not a
normalizable state, physical states for the GKP code are finitely squeezed
approximations to these and are often modeled as a comb of Gaussian peaks of
variance $\delta^{2}$ with separation $2\sqrt{\pi}$ damped by a larger
Gaussian envelope of variance$1/\kappa^{2}$. In the position basis, the
logical 0 and 1 states of approximate code states, $\ket{\widetilde{0}}_{\rm
GKP}$ and $\ket{\widetilde{1}}_{\rm GKP}$, are given by

$\displaystyle\ket{\widetilde{0}}_{\rm GKP}$ $\displaystyle\propto$
$\displaystyle\sum_{m=-\infty}^{\infty}\int
e^{-\frac{[(2m)\sqrt{\pi}]^{2}}{2(1/\kappa^{2})}}e^{-\frac{(s-2m\sqrt{\pi})^{2}}{2\delta^{2}}}\ket{s}_{q}ds,$
(21) $\displaystyle\ket{\widetilde{1}}_{\rm GKP}$ $\displaystyle\propto$
$\displaystyle\sum_{m=-\infty}^{\infty}\int
e^{-\frac{[(2m+1)\sqrt{\pi}]^{2}}{2(1/\kappa^{2})}}e^{-\frac{[s-(2m+1)\sqrt{\pi}]^{2}}{2\delta^{2}}}\ket{s}_{q}ds,$
(22)

respectively. In the case of infinite squeezing ($\kappa\rightarrow 0$,
$\delta\rightarrow 0$) the states become ideal GKP qubits. In this review, we
choose $\kappa^{2}$ and $\delta^{2}$ so that the variance of each peak in the
position and momentum observables is equal to $\sigma^{2}_{\rm gkp}$, i.e.,
$\kappa^{2}=\delta^{2}=2\sigma^{2}_{\rm gkp}$. These approximate states are
not orthogonal, and there is a probability of misidentifying
$\ket{\widetilde{0}}_{\rm GKP}$ as $\ket{\widetilde{1}}_{\rm GKP}$ (and vice
versa) in the computational basis measurement. The computational basis
measurement is implemented by the measurement in the $q$ quadrature and
binning to the integer multiple of $\sqrt{\pi}$. A qubit-level measurement
error occurs when the measured outcome is more than $\sqrt{\pi}/2$ away from
the correct outcome. The probability of misidentifying the bit value of a non-
ideal GKP qubit with the variance $\sigma_{\rm gkp}^{2}$, $P_{\rm
fail}(\sigma_{\rm gkp}^{2})$, is approximately given by

$P_{\rm fail}({\sigma_{\rm
gkp}}^{2})=1-\int_{\frac{-\sqrt{\pi}}{2}}^{\frac{\sqrt{\pi}}{2}}dx\frac{1}{\sqrt{2\pi{\sigma_{\rm
gkp}}^{2}}}{\rm exp}\left(-\frac{x^{2}}{2{\sigma_{\rm gkp}}^{2}}\right).$ (23)

####  4.2.3 Representation of the GKP qubit

We review further studies for the mathematical formalism of the GKP qubit,
which is an important research direction to understand CVQC with the GKP
qubits. Here we will see three formalisms for the GKP qubits.

The first formalism is the normalization of the GKP qubit. In addition to the
state in Eqs. (21) and (22), there are various ways to approximate the ideal
states in Eqs. (19) and (20) as physical states, i.e., normalizable states. In
Ref. [245], Tzitrin et al. summarized various ways to describe the GKP qubit
as a physical state such as the following ways: (1) The states in Eqs. (21)
and (22). (2) The state suffered from a Fock damping operator
$\hat{E}(\epsilon)=e^{-\epsilon\hat{n}}$ with $\epsilon>0$ and
$\hat{n}=\hat{a}^{{\dagger}}\hat{a}$, described by
$\hat{E}(\epsilon)\ket{{0/1}}_{\rm GKP}$ [88]. (3) The state suffered from an
operator of coherent random shifts $\hat{G}=\sqrt{2/\pi\Delta^{2}}\int
d^{2}\alpha e^{-|\alpha|^{2}/\Delta^{2}}\hat{D}(\alpha)$, defined by
$\hat{G}\ket{{0/1}}_{\rm GKP}$. The equivalence of the above ways (1)-(3) was
analytically shown in Ref. [246].

The second formalism is the modular subsystem decomposition developed by
Pantaleoni et al. [247], where arbitrary CV state can be decomposed into
discrete variable and CV parts. For a particular case of this formalism, the
ideal GKP qubit can be described as $\ket{\phi}_{\rm
GKP}=\ket{\phi}_{\mathcal{L}}\otimes\ket{+_{I}}_{\mathcal{G}}$, where
$\ket{\phi}_{\mathcal{L}}=c_{0}\ket{{0}}_{{\rm
GKP},{\mathcal{L}}}+c_{1}\ket{{1}}_{{\rm GKP},{\mathcal{L}}}$
$(|c_{0}|^{2}+|c_{1}|^{2}=1)$ represents the two-dimensional qubit subspace
for the logical mode, and
$\ket{+_{I}}_{\mathcal{G}}=\sum_{m=-\infty}^{\infty}\ket{m\sqrt{\pi}}_{q,{\mathcal{G}}}$
is represented as a state in the other infinite-dimensional space for the
gauge mode [247]. Then, the normalizable GKP qubit can be described as
$\hat{E}(\epsilon)\ket{\phi}_{\rm
GKP}=\hat{E}(\epsilon)(\ket{\phi}_{\mathcal{L}}\otimes\ket{+_{I}}_{\mathcal{G}})$
by using the Fock damping operator $\hat{E}(\epsilon)$. One of the
applications of this technique is to define the Bloch sphere for the GKP
qubit. Specifically, the density operator of the GKP qubit is obtained by
tracing out the gauge mode as $\hat{\rho}(\epsilon)={\rm
Tr}_{\mathcal{G}}[\hat{E}(\epsilon)\ket{\phi}_{\rm
GKP}\bra{\phi}\hat{E}(\epsilon)]$. Then, $\hat{\rho}(\epsilon)$ provides
information about the Bloch sphere for the qubit from
$\hat{\rho}(\epsilon)=\frac{1}{2}\sum_{i=0}^{3}s_{i}\hat{\sigma}_{i}$, where
$\hat{\sigma}_{i}$ and $s_{i}$ are the Pauli/identity matrix and the Stokes
parameter [245]. Additionally, many studies have recently been conducted on
the mathematical formalism using the modular subsystem decomposition [248,
249, 250, 245, 251, 252].

The third formalism is the Wigner representation of the GKP qubit, which is
also useful to characterize the GKP qubit. As mentioned in the original GKP’s
paper [23], the Wigner function of the ideal GKP qubit, $\ket{{0/1}}_{\rm
GKP}$, can be described by the sum of delta functions. In Ref. [253], it has
been shown that the Wigner function of the normalizable GKP qubit,
$\hat{E}(\epsilon)\ket{{0/1}}_{\rm GKP}$, can be also described by the sum of
Gaussian functions. This enables us to efficiently simulate the unitary time
evolution of the GKP qubit. Also, other non-Gaussian states such as the cat
code and Fock states can be represented by this formalism. Consequently, this
formalism allows us to efficiently perform the numerical simulation of quantum
information processing where several bosonic codes evolve simultaneously
[254].

####  4.2.4 Universality for the GKP qubit

To realize non-Gaussian gates and universality for fault-tolerant QC with GKP
qubits, the cubic phase gate is expected to be a typical resource [23]. The
cubic phase gate, however, may not be appropriate for the GKP qubit in terms
of the gate fidelity [255]. To achieve universality with the GKP qubit, the
GKP magic state teleportation will be preferable compared with the cubic phase
gate if we can prepare the GKP magic state with sufficient fidelity for the
magic state distillation [256, 257, 258]. Ref. [259] has investigated the way
to implement the GKP magic state teleportation in an optical setup. For the
preparation of the magic state of the GKP qubit, Baragiola et al. proposed an
impressive scheme to prepare a noisy magic state from a logical GKP qubit and
a vacuum state by only Gaussian operations and measurement. Then, a high-
fidelity magic state is prepared from many noisy magic states by using the
magic state distillation [256]. This implies that the preparation of the GKP
logical basis and Gaussian operations allow us to implement universal and
fault-tolerant QC. In contrast, Ref. [260] proposed a more efficient scheme to
achieve universal and fault-tolerant QC without the magic state distillation,
where high-fidelity GKP magic states are directly prepared and combined with
Gaussian operations.

####  4.2.5 Error model for the GKP qubit

Here we will review an error model for the GKP qubit to lay the groundwork for
the description of the QEC with the GKP qubit. We describe the additive
Gaussian noise (AGN) [261, 23, 262], also known as a Gaussian random
displacement channel, as a convenient error model. The AGN is a kind of the
Gaussian quantum channel (GQC) and is a common type of noise in bosonic
systems. The AGN randomly displaces the state in phase space according to a
Gaussian distribution, which is described by the superoperator
$\mathcal{G}_{\xi}$ acting on density operator $\hat{\rho}$ as

$\displaystyle\hat{\rho}\to\mathcal{G}_{\xi}(\hat{\rho})$
$\displaystyle=\frac{1}{\pi{\xi}^{2}}\int
d^{2}\alpha\,\mathrm{e}^{-|\alpha|^{2}/{{\xi}^{2}}}\hat{D}(\alpha)\hat{\rho}\hat{D}^{\dagger}(\alpha),$
(24)

where $\hat{D}(\alpha)=e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}$ is the
phase-space displacement operator. With $\alpha=(v+iu)/\sqrt{2}$, the position
$q$ and momentum $p$ are displaced independently as $\hat{q}\to\hat{q}+v$,
$\hat{p}\to\hat{p}+u$, where $v$ and $u$ are real Gaussian random variables
with mean zero and variance $\xi^{2}$. Therefore, the AGN maintains the
locations of the Gaussian peaks in the probability for the measurement
outcome, but it increases the variance of each peak by $\xi^{2}$ in both
quadratures. Notice that the outputs of this channel are mixed states, even
when $\hat{\rho}$ is pure.

The dominant noise channel in an optical setup is photon loss. We will see
that photon loss combined with the amplification techniques can be modeled by
the increase of a variance, likewise the AGN. The equivalence between the
photon loss combined with amplifiers and the AGN can provide the way to apply
the numerical results such as the threshold for fault-tolerant QC with the AGN
to that with the photon-loss channel.

The photon-loss channel can be modeled as an unwanted beam-splitter coupling
with an environmental vacuum state. In the Heisenberg picture, the loss
channel transforms the $\hat{q}$ and $\hat{p}$ quadratures as

$\hat{q}\to\sqrt{\eta}\hat{q}+\sqrt{1-\eta}\hat{q}_{\rm vac},\hskip
10.0pt\hat{p}\to\sqrt{\eta}\hat{p}+\sqrt{1-\eta}\hat{p}_{\rm vac},$ (25)

where $\eta$ is the efficiency, and $\hat{q}_{\rm vac}(\hat{p}_{\rm vac})$ is
the position (momentum) quadrature of the vacuum state. After the loss
channel, the variances of the GKP states in the $\hat{q}$ and $\hat{p}$
quadratures are transformed as

${\sigma^{2}}_{\rm gkp}\to{\eta}\sigma^{2}_{{\rm gkp}}+\frac{1-\eta}{2}.$ (26)

In Refs. [224, 263], the equivalence between the AGN and the photon-loss
channel combined with the amplification has been studied. At the cost of
additional Gaussian noise, photon loss can be converted into a random
displacement error channel, i.e., AGN by applying phase-insensitive
amplifiers. The phase-insensitive amplifiers may be implemented either after
or before the photon-loss channel, and these two methods are called post- and
preamplification, respectively. For the postamplification, the quadrature
variances of the GKP qubits in both $\hat{q}$ and $\hat{p}$ are transformed as
$\sigma^{2}_{\rm gkp}\to\sigma^{2}_{\rm gkp}+\frac{1-\eta}{\eta}$. For the
preamplification, the variances are transformed as $\sigma^{2}_{\rm
gkp}\to\sigma^{2}_{\rm gkp}+1-\eta.$ Note that the latter introduces less
noise than the former. In addition to the phase-insensitive amplifiers,
rescaling the outcome of the homodyne measurement is another method for the
amplification. Unlike the phase-insensitive amplifiers, the rescaling is
implemented after the measurement of the target quantum state. In the
rescaling, the measurement outcome $m$ is rescaled on a classical computer as
$m/\sqrt{\eta}$. The variances are transformed as $\sigma^{2}_{\rm
gkp}\to\sigma^{2}_{\rm gkp}+(1-\eta)/2\eta$. The performance of the above
three techniques has been compared in the context of the quantum repeater
protocol in Ref. [243]. Additionally, the rescaling technique has been applied
to fault-tolerant QC with the GKP qubits [236].

####  4.2.6 QEC with the GKP qubit

We review the way to correct small displacement errors in $q$ and $p$
quadratures [23], which is referred to as the single-qubit level QEC (SQEC) in
the following. We consider errors derived from the variances of the input
qubit and auxiliary states.

In the SQEC for the $q$ quadrature, a single ancilla qubit is entangled with
the data qubit by the CX gate, where the ancilla qubit is the target qubit, as
illustrated in Fig. 13. The ancilla qubit is prepared in the state
$\ket{\widetilde{+}}_{\rm GKP}$ to prevent us from identifying the bit value
of the data qubit in the subsequent syndrome measurement. The CX gate, which
corresponds to the operator $e^{-i\hat{q}_{\rm D}\hat{p}_{\rm A}}$ for CVs,
transforms the quadratures as

$\displaystyle\hat{q}_{\rm D}$ $\displaystyle\to$ $\displaystyle\hat{q}_{\rm
D},\hskip 10.0pt\hat{p}_{\rm D}\to\hat{p}_{\rm D}-\hat{p}_{\rm A},$ (27)
$\displaystyle\hat{q}_{\rm A}$ $\displaystyle\to$ $\displaystyle\hat{q}_{\rm
A}+\hat{q}_{\rm D},\hskip 10.0pt\hat{p}_{\rm A}\to\hat{p}_{\rm A},$ (28)

where $\hat{q}_{\rm D}(\hat{p}_{\rm D})$ and $\hat{q}_{\rm A}(\hat{p}_{\rm
A})$ are the quadrature operators of the data and ancilla qubits in the
$q$($p$) quadrature, respectively. Regarding the displacement errors in the
$q$ and $p$ quadratures, the CX gate transforms the displacement errors as

$\displaystyle\overline{\Delta}_{\rm{\it q},{\rm D}}$ $\displaystyle\to$
$\displaystyle\overline{\Delta}_{\rm{\it q},{\rm D}},\hskip
10.0pt\overline{\Delta}_{\rm{\it p},{\rm D}}\to\overline{\Delta}_{\rm{\it
p},{\rm D}}-\overline{\Delta}_{\rm{\it p},A},$ (29)
$\displaystyle\overline{\Delta}_{\rm{\it q},A}$ $\displaystyle\to$
$\displaystyle\overline{\Delta}_{\rm{\it q},A}+\overline{\Delta}_{\rm{\it
q},{\rm D}},\hskip 10.0pt\overline{\Delta}_{\rm{\it
p},A}\to\overline{\Delta}_{\rm{\it p},A},$ (30)

where $\overline{\Delta}_{\rm{\it q},D}(\overline{\Delta}_{\rm{\it p},D})$ and
$\overline{\Delta}_{\rm{\it q},{\rm A}}(\overline{\Delta}_{\rm{\it p},{\rm
A}})$ are the true values for the displacements of the data and ancilla qubits
in $q$($p$), respectively. We assume that the true values for the data
(ancilla) qubit in the $q$ and $p$ quadratures obey the Gaussian distribution
with the variance $\sigma^{2}_{{\rm D(A)},q}$ and $\sigma^{2}_{{\rm D(A)},p}$,
respectively. After the CX gate, we measure the ancilla qubit in the $q$
quadrature. When $|\overline{\Delta}_{\rm{\it q},A}+\overline{\Delta}_{\rm{\it
q},D}|$ < $\sqrt{\pi}/2$, we obtain the measured displacement error of the
ancilla qubit as ${\Delta}_{\rm m{\it q},A}=\overline{\Delta}_{\rm{\it
q},A}+\overline{\Delta}_{\rm{\it q},D}$. Then, we perform the displacement
operation on the $q$ quadrature of the data qubit by $-{\Delta}_{\rm m{\it
q},A}$, and the true value of the displacement becomes
$\overline{\Delta}_{\rm{\it q},D}-{\Delta}_{\rm m{\it q},A}=-{\Delta}_{\rm{\it
q},A}$. On other hand, when $|\overline{\Delta}_{\rm{\it
q},A}+\overline{\Delta}_{\rm{\it q},D}|$ > $\sqrt{\pi}/2$, we obtain the
outcome of the displacement ${\Delta}_{\rm m{\it
q},A}=\overline{\Delta}_{\rm{\it q},A}+\overline{\Delta}_{\rm{\it
q},D}-\sqrt{\pi}$, assuming $\overline{\Delta}_{\rm{\it
q},A},\overline{\Delta}_{\rm{\it q},D}>0$.

Figure 13: A quantum circuit for the single-qubit level QEC (SQEC) with the
GKP qubit.

After the displacement operation by $-{\Delta}_{\rm m{\it q},A}$, the true
value for the data qubit in the $q$ quadrature becomes
$\overline{\Delta}_{\rm{\it q},D}-{\Delta}_{\rm m{\it
q},A}=\overline{\Delta}_{\rm{\it q},A}+\sqrt{\pi}$. Since the displacement by
$|\sqrt{\pi}|$ in the $q$ quadrature corresponds to the bit-flip error, the
qubit-level error occurs on the data qubit in the $q$ quadrature if
$|\overline{\Delta}_{\rm{\it q},A}+\overline{\Delta}_{\rm{\it q},D}|$ >
$\sqrt{\pi}/2$. Assuming that the true value $\overline{\Delta}_{\rm{\it
q},D(A)}$ obeys the Gaussian distribution with the variance $\sigma^{2}_{{\rm
D(A)},q}$, the probability of the bit-flip error is given by $P_{\rm
fail}(\sigma^{2}_{{\rm D},q}+{\sigma^{2}_{{\rm A},q}})$ using Eq. (23).
Consequently, the SQEC in the $q$ quadrature can reduce the variance of the
data qubit in the $q$ quadrature from $\sigma^{2}_{{\rm D},q}$ to
${\sigma^{2}_{{\rm A},q}}$ when $\sigma^{2}_{{\rm D},q}>{\sigma^{2}_{{\rm
A},q}}$. For the variance of the data qubit in the $p$ quadrature, the initial
variance $\sigma^{2}_{{\rm D},p}$ becomes $\sigma^{2}_{{\rm
D},p}+{\sigma^{2}_{{\rm A},p}}$ after the SQEC in the $q$ quadrature.

The SQEC in the $p$ quadrature can then be performed using the second ancilla
qubit with the CX gate, where the ancilla is prepared in
$\ket{\widetilde{0}}_{\rm GKP}$ and the data qubit is assumed to be the target
qubit. In a similar manner to the SQEC in $q$, but with the measurement of the
ancilla in the $p$ quadrature, the SQEC for the data qubit in the $p$
quadrature can reduce the variance of the data qubit in the $p$ quadrature
from $\sigma^{2}_{{\rm D},p}+{\sigma^{2}_{{\rm A},p}}$ to ${\sigma^{2}_{{\rm
A2},p}}$, where ${\sigma^{2}_{{\rm A2},p(q)}}$ is the variance in the $p(q)$
quadrature for the second ancilla qubit. In the SQEC in $p$, there is the
phase-flip error whose probability is given by $P_{\rm fail}(\sigma^{2}_{{\rm
D},p}+{\sigma^{2}_{{\rm A},p}}+{\sigma^{2}_{{\rm A2},p}})$ using Eq. (23).

As a result of the sequential SQECs, the variances of the data qubit are
replaced by those of the ancilla qubits, as $(\sigma^{2}_{{\rm
D},q},\sigma^{2}_{{\rm D},p})\to({\sigma^{2}_{{\rm A},q}}+{\sigma^{2}_{{\rm
A2},q}},{\sigma^{2}_{{\rm A2},p}})$. This kind of variance reduction with the
SQEC is helpful for the QEC against the AGN which increases the variance, if
we can prepare ancilla qubits whose variance is sufficiently small. We note
that the SQEC can be applied to a photon-loss channel by transforming the
channel into an AGN with the amplification technique as described in Sec.
4.2.5. While we have used the CX gate for the SQEC, the SQEC can be also
performed by using a beam-splitter coupling and a squeezing operation instead
of the CX gate. The error analysis using such a setup was studied in Refs.
[264, 265].

###  4.3 Fault-tolerant QC with the GKP qubits

####  4.3.1 Higher-level encoding for GKP qubits

Although the SQEC works well for the small displacement error, we need to use
the logical-qubit level QEC to correct the displacement error greater than
$\sqrt{\pi}/2$. In general, the error probability for each logical-qubit level
operation should be below $10^{-12}-10^{-15}$ [120] to employ useful quantum
algorithms. Therefore, when we use GKP qubits for such algorithms, we must
sufficiently reduce the logical qubit level errors by concatenating the GKP
qubits with the higher-level encoding, i.e., the qubit-based standard QECCs
such as Steane’s seven qubit code [196, 197], Knill’s $C_{4}/C_{6}$ code
[195], the surface code [198, 199], and so on. Then, the logical-qubit level
error probability of the QECC can be suppressed to an arbitrary value if the
physical-qubit level error is smaller than the threshold value. In the case of
the GKP qubits, the threshold value corresponds to the required initial
squeezing level of the resource state, which determines the physical-qubit
level error probability of the GKP qubits.

Figure 14: One-way QC based on the CV cluster state with the GKP qubit. (a) A
quantum circuit for the SEQC in the $q$ and $p$ quadratures, which corresponds
to Fig. 13 (up to Fourier gates). (b) Cluster state representation
corresponding to (a). (c) A quantum circuit for one-way QC to implement an
arbitrary single-mode quantum gate with the SQECs in the $q$ and $p$
quadratures. (d) Cluster state representation of the cluster state to
implement the CZ gate with the SQECs in the $q$ and $p$ quadratures. (e)
Cluster state representation after the CZ gate between the data qubits,
$\ket{\widetilde{\psi}}_{\rm GKP}$ and $\ket{\widetilde{\phi}}_{\rm GKP}$, via
the measurements of the two ancilla qubits in the $p$ quadrature [88].

####  4.3.2 Threshold for fault-tolerant QC

A squeezing level is commonly used to represent the threshold for fault-
tolerant CVQC, where a squeezing level $s$ is given by $s=-10{\rm
log}_{10}2\sigma^{2}_{\rm gkp}$. Now, we move on to the threshold value for
fault-tolerant CVQC with the GKP qubits. The first work for the threshold of
the squeezing level was provided by Menicucci in 2014 [88]. Also, Ref. [88] is
the first work that showed fault-tolerant QC is possible within the framework
of one-way QC using large-scale CV cluster states. In the case of one-way
CVQC, besides errors from an environment such as photon loss, errors derived
from the CV cluster state itself accumulate during QC [88]. The GKP qubit
overcomes this problem by performing the SQEC with CV cluster states, and
qubit-level Clifford gates on the GKP qubits are implemented simultaneously by
homodyne measurements on CV cluster states. In the following, we see the
implementation of the SQEC on the CV cluster state and the procedure to obtain
the threshold of the squeezing level.

First, we see the way to perform the SQEC during one-way CVQC. Figures 14(a)
and (b) show a quantum circuit and the cluster state representation for the
SQECs, respectively, where Fig. 14(a) corresponds to the quantum circuit in
Fig. 13 up to Fourier gates. Figure 14(c) shows the cluster state to perform a
single-mode quantum gate and the SQECs in the $q$ and $p$ quadratures. For the
implementation of a single-mode quantum gate, the first four modes in the
transverse direction are used (we recall that an arbitrary single-mode quantum
gate can be implemented by the four-mode linear cluster state). The SQECs in
$q$ and $p$ quadratures are implemented by using ancillary GKP qubits
entangled with the last two modes in the transverse direction, as can be
understood from the correspondence between Figs. 14(b) and (c). Therefore, the
cluster state in Fig. 14(c) enables us to perform the SQECs in $q$ and $p$
quadratures during single-mode one-way QC. We note that there is the failure
probability of SQECs, and the error probability for the single mode gate is
given by the probability that at least one of the two SQECs fails.

In general, the CZ or CX gate sets the threshold for qubit-based fault-
tolerant QC since the two-qubit gate is the noisiest gate among the universal
gate set for qubits. Figures 14(d) and (e) show the CZ gate between the data
GKP qubits, $\ket{\widetilde{\psi}}_{\rm GKP}$ and
$\ket{\widetilde{\phi}}_{\rm GKP}$, by the measurement of the squeezed vacuum
states between the data qubits in the $p$ quadrature. In this model [88], the
CZ gate is followed by four SQECs, and thus the error probability for the CZ
gate is given by the probability that at least one of the four SQECs fails. We
assume that the initial variance $\sigma^{2}_{\rm gkp}$ for the GKP qubit is
the same as that of the initial momentum-squeezed vacuum states used to make
the CV cluster state with a finite squeezing parameter $r$, $\sigma^{2}_{\rm
in}=\frac{1}{2}e^{-2r}$. Then, the Gaussian-distributed shift errors in two of
the four SQECs (including noise of ancillary squeezed vacuum states after
CVQC) have variance $7\sigma^{2}_{\rm in}$, and two others have variance
$5\sigma^{2}_{\rm in}$ [88]. Therefore, the probability that at least one of
those corrections fails is $p_{\rm err}(\sigma^{2}_{\rm in})=1-[1-P_{\rm
fail}(7\sigma^{2}_{\rm in})]^{2}[1-P_{\rm fail}(5\sigma^{2}_{\rm in})]^{2}$.
Then, it can be expected that fault-tolerant QC is possible if $p_{\rm err}$
is smaller than the threshold value $p_{\rm FT}$ of the qubit-based QECCs
[194, 196, 197, 195]. The threshold value $p_{\rm FT}$ depends on QECCs for
qubits, e.g. $10^{-6}$–$10^{-2}$ in Refs. [200, 201, 198, 199, 195, 266]. The
threshold of the squeezing level can be obtained from the condition $p_{\rm
FT}=p_{\rm err}(\sigma^{2}_{\rm in})$. For example, when we use $p_{\rm
FT}=10^{-6}$ for Steane’s seven qubit code [197], fault-tolerant QC can be
realized if $\sigma^{2}_{\rm in}$ < 4.44 $\times$ $10^{-3}$. In this case, we
obtain the threshold of the squeezing level as $-10{\rm
log}_{10}2\sigma^{2}_{\rm in}\sim 20.5$ dB. The GKP qubit enables us to
implement large-scale QC, but it is still challenging to experimentally
generate the GKP qubit with the squeezing level 15.6–20.5 dB for qubit-based
QECCs [88]. In Secs. 4.3.3, 4.3.4, and 4.3.5, we will see several efforts to
reduce the threshold of the squeezing level.

Figure 15: Analog quantum error correction. Reproduced from [267].

####  4.3.3 Analog QEC with GKP qubits

The effect of noise on CV states is observed as the displacement error in an
analog measurement outcome, which includes beneficial information for the QEC.
Thus, the analog measurement outcome is expected to improve the QEC
performance compared to the QEC based on only binary information. Analog QEC
[267] provides such an improvement of the QEC performance by using the real-
valued syndrome of a GKP qubit. In the measurement of the GKP qubit, we obtain
the measurement outcome $(2t+k)\sqrt{\pi}+\Delta_{\rm m}$, as shown in Fig.
15(a), where $t$, $k$, and $\Delta_{\rm m}$ are integer, the bit value, and
the measured displacement error. As described in Sec. 4.2.6, there are two
possible events: one is no error, i.e., the true value for the displacement
error $\overline{\Delta}$ is equal to $\Delta_{\rm m}$, as shown in Fig.
15(b). The other is the bit(phase)-flip error in the $q(p)$ quadrature, i.e.,
$|\overline{\Delta}|$ is equal to $\sqrt{\pi}-|\Delta_{\rm m}|$, as shown in
Fig. 15(c). Then, we obtain two likelihoods for the two events as
$f(\overline{\Delta})=1/\sqrt{2\pi\sigma^{2}}e^{-\overline{\Delta}^{2}/(2\sigma^{2})}$
since $\overline{\Delta}$ obeys the Gaussian distribution
$f(\overline{\Delta})$. The analog QEC applies these likelihoods to higher-
level encoding, namely the qubit-based standard QECC. Then, the most likely
error pattern is selected to improve the performance of the decoding in the
QEC. For example, the likelihood is used for procedures of the decoding, such
as a belief propagation for the concatenated code [268, 269] and a minimum-
weight perfect-matching algorithm for the surface code [198, 199, 270].

One may wonder how the analog QEC improves the QEC performance. We focus on
the attainable rate against the AGN with the code capacity model. We note that
the attainable rate does not correspond to the threshold for fault-tolerant
QC, while we see it as a simple example of the analog QEC. In the code
capacity model, the noise is parametrized by a single variance $\sigma^{2}$ of
the AGN, where the GKP qubits with zero variance (i.e., infinite squeezing)
become those with the variance $\sigma^{2}$ after the AGN. The attainable rate
for the GQC without the analog QEC can be obtained by the quantum capacities
of the binary channel [194, 271], assuming that bit- and phase-flip errors
occur with the probability $p_{\rm e}$ independently. Using the binary
Calderbank-Shor-Steane code [194, 196], the attainable rate $R$ is defined by
$R>1-2H_{2}(p_{\rm e})=1+2p_{\rm e}{\rm log}_{2}p_{\rm e}+2(1-p_{\rm e}){\rm
log}_{2}(1-p_{\rm e})$, i.e., the rate is nonzero when $p_{\rm e}<0.11$, where
$H_{2}(p_{\rm e})$ is the binary entropy function. For the case of the GKP
qubit, the error probability $p_{\rm e}$ is equal to $P_{\rm
fail}(\sigma^{2})$ described in Eq. (23). Since $P_{\rm fail}(\sigma^{2})$
with $\sigma~{}\sim 0.555$ corresponds to $p_{\rm e}\sim 0.11$, the
nonvanishing quantum capacity for the AGN is defined when $\sigma<0.555$. In
Refs. [267, 130], it has been shown the analog QEC with Knill’s ${\rm
C}_{4}/{\rm C}_{6}$ code [195] and the surface code [198, 199] can achieve
$\sigma~{}\sim 0.607$, which is larger than $\sim 0.555$. This value reaches
the hashing bound of the GQC, also known as a lower bound on the quantum
capacity of the GQC [23, 262]. Thus, the analog QEC considerably improves the
error tolerance, indicating the potential to improve the threshold for fault-
tolerant QC.

####  4.3.4 The highly reliable measurement

Another technique to improve the threshold was proposed in Ref. [130], where
the postselected measurement, referred to as the highly reliable measurement
(HRM), is used to prevent the squeezing level from decreasing during the
cluster state preparation. Figures 16(a) and (b) show the schematic of the
HRM. Here we see one of the peaks for the logical 0 state, which lies in
position $2k\sqrt{\pi}$ with integer $k$ in the $q$ quadrature. In Fig. 16(a),
the plain blue region and the red region with vertical lines represent the
different code words of the logical 0 and 1 states, respectively. In the
conventional measurement, the red regions marked with vertical lines
correspond to the probability of incorrect decision of the bit value. In the
HRM, we set the dotted lines which represent an upper limit $\zeta$, as shown
in Fig. 16(b). The yellow areas with horizontal lines show the probability
that the results of the measurement are discarded by introducing $\zeta$. The
vertical line areas show the probability that our method fails. By using the
HRM, the probability of misidentifying the bit value, namely bit- and phase-
flip errors, decreases with increasing $\zeta$ at the cost of reducing the
success probability of the measurement.

Figure 16: (a) The conventional measurement of the GKP qubit, where the
Gaussian probability distribution following the displacement error of the GKP
qubit has variance $\sigma^{2}$. (b) The highly-reliable measurement (HRM).
Reproduced from [130]. Figure 17: (a) The small-scale cluster state
construction by using the Bell measurement with the HRM. (b) The three-
dimensional (3D) cluster construction from the small-scale cluster state,
which is referred to as the hexagonal cluster state, by using the Bell
measurement without the HRM [130].

One of the applications of the HRM is the reliable construction of large-scale
cluster states. Figures 17(a) and (b) show the schematic of the three-
dimensional (3D) cluster state construction with the HRM, where the large-
scale 3D cluster state can be used for topologically protected one-way QC with
the surface code [79, 80]. In Ref. [130], the three mode cluster state,
composed of a node qubit and two leaf qubits, is firstly prepared by the CZ
gate, as shown in Fig. 17(a). Then, the elementary cluster state, referred to
as the hexagonal cluster state, is prepared from the three mode cluster states
by using the Bell measurement with the HRM, where the HRM reduces the
probability of misidentifying the bit values in the Bell measurement. The
reason for using the Bell measurement is that the entanglement generation
using the Bell measurement does not increase the variance of the node qubit,
while that using the CZ gate increases the variances due to the propagation of
the displacement errors between qubits. Once the hexagonal cluster states are
prepared probabilistically, the large-scale 3D cluster state is generated
deterministically by using the Bell measurement without the HRM, as shown in
Fig. 17(b). In Ref. [130], the fault-tolerant threshold of the squeezing level
can be relaxed to less than 10 dB by combining the reliable 3D cluster state
construction and the analog QEC. Fault-tolerant QC employing the HRM has been
studied in several works [236, 272]. Furthermore, the Gauss-Markov theorem in
statistics is useful to improve the threshold introduced in Refs. [236, 273,
272]. In addition, the HRM has been applied to loss-tolerant QECCs with the
GKP qubits, e.g., Varnava’s code [274] to achieve long-distance quantum
communication [243] and the quantum parity code [222] to realize the efficient
decoding of the QEC [239].

####  4.3.5 Architecture for fault-tolerant QC with the GKP qubits

Promising architectures for fault-tolerant QC with the GKP qubit have been
proposed recently in a superconducting circuit [24, 35, 36] and an optical
setup [37, 39, 38]. In an optical setup, there are roughly two types of
architectures for implementing large-scale QC. The first one is the time-
multiplexing approach (see for example Refs. [275, 38]), which is described in
Sec. 3.2.2. The second one is the integrated approach (see Refs. [37, 39] as
an example). In the following, we see these two approaches.

In the first time-multiplexing approach, a time-multiplexed cluster state for
the resource of one-way QC is prepared by using a compact experimental setup.
In Ref. [275], the noise analysis of the quantum gate and the threshold for
fault-tolerant QC have been studied under an experimentally feasible noise
model, employing an efficient way to perform one-way QC, called the macro node
protocol [276, 131]. In Ref. [38], the generation scheme of the 3D cluster
state has been proposed by using time-multiplexing, where topologically
protected QEC using the surface code on the 3D cluster state was considered.
The threshold in this architecture was found to be 13.2 dB. Experimental
realization of one-way QC based on the time-multiplexing approach is described
in Sec. 3.2.2.

In the second integrated approach in Refs. [37, 39], the 3D cluster state is
generated from integrated photonic devices. This architecture circumvents one
disadvantage of the first approach, namely lossy long optical delay lines
required for increasing the number of qubits, while the size of photonic
devices becomes larger as the number of the logical qubits and the code
distance of a surface code increase. Ref. [37] is the first work considering
the probabilistic nature of the GKP qubit preparation schemes in one-way QC
with GKP qubits. This probabilistic nature has been a crucial problem in
large-scale QC. In Ref. [37], the node for the event of the failure of the GKP
qubit preparation is replaced by the node of the squeezed vacuum state. Then,
one-way QC is performed on the 3D cluster state whose nodes are partially
replaced by the squeezed vacuum states. This scheme allows us to implement
large-scale QC even with non-deterministic GKP qubit preparation. On the other
hand, it requires an optical switches for reconfigurability and inline
squeezing for implementing the CZ gate.

These experimental requirements are removed by the scheme in Ref. [39].
Additionally, Ref. [39] proposed to construct the 3D cluster states from so-
called qunaught states, which are also useful for quantum sensing applications
[277]. The logical 0 and 1 states for the ideal qunaught state are given by
$\ket{{0}}_{\rm naught}=\sum_{m=-\infty}^{\infty}\ket{2m\sqrt{2\pi}}_{q}$ and
$\ket{{1}}_{\rm naught}=\sum_{m=-\infty}^{\infty}\ket{(2m+1)\sqrt{2\pi}}_{q}$,
respectively. For the 3D cluster state construction, an entangled pair of the
GKP qubits is prepared from two approximate qunaught states with $\sigma_{\rm
gkp}^{2}$ by using a beam-splitter coupling, where the variance of the GKP
qubit composing the entangled pair is ($\sigma_{\rm gkp}^{2},\sigma_{\rm
gkp}^{2}$) in $q$ and $p$ quadratures. In the case of the entangled pair
preparation by two GKP qubits with $\sigma_{\rm gkp}^{2}$ and the CZ gate, the
variance of the GKP qubit is ($\sigma_{\rm gkp}^{2},2\sigma_{\rm gkp}^{2}$).
Thus, the qunaught state reduces the noise in the entangled pair, which is
expected to improve the threshold of the squeezing level.

####  4.3.6 Preparation of the GKP qubit in an optical setup

In superconducting circuit and ion trap systems, the GKP qubit has been
realized recently [212, 213], and the efficient preparation of the GKP qubit
has been extensively studied [278, 279, 280, 281, 75, 245, 282]. In an optical
setup, to the best of our knowledge, the GKP qubit has not been realized,
although there are several approaches such as the breeding protocol [283,
279], the use of photon-number resolving detectors [284, 282], the use of the
interaction between light and matter qubits [285, 286, 287], the use of the
cross-Kerr interaction [288, 289], and so on. Here we see the first two
approaches, which can prepare the GKP qubit in an all-optical setup.

The breeding protocol proposed by Vasconcelos et al. [283] is the method to
generate the GKP qubit by linear optics from squeezed cat states, where a cat
state is defined as a superposition of two coherent states with opposite
phases. In the original protocol [283], two squeezed cat states are interfered
by a beam splitter and the homodyne measurement is performed on one of the
modes in the $p$ quadrature, as shown in Fig. 18(a). When the measurement
outcome is close to zero, the first step of the protocol succeeds. Assuming
that the wave function of the cat state has two Gaussian peaks illustrated in
Fig. 18(a), the output state has three Gaussian peaks. Then, by repeating the
breeding, the number of the Gaussian peaks grows, and the output states after
each of the steps become similar to the GKP qubit. The total success
probability of the GKP qubit generation for the original protocol is too low
due to the postselection in the homodyne measurement. Ref. [278] overcomes
this problem by applying a phase estimation, where the postselection in the
measurement is not required, but only the specific feedforward operation for
phases depending on measurement outcomes is required. The scheme in Ref. [278]
is intended for the GKP qubit generation in microwave modes in superconducting
circuits. In Ref. [279], the breeding protocol with a phase estimation was
developed by using linear optics. The breeding protocol has been
experimentally demonstrated in Ref. [290], while the scheme in Ref. [290] was
not aimed to increase the number of Gaussian peaks, but to enlarge the
amplitude of the cat state.

Figure 18: Preparation of the GKP qubit in an optical setup. (a) Breeding
protocol [283]. (b) Optical quantum state synthesizer referred to as the
Gaussian boson sampling (GBS)-based device [284, 245].

Another method is to use the optical quantum state synthesizer consisting of
linear optics and photon-number resolving detectors, which is referred to as
the Gaussian boson sampling (GBS)-based device [284, 245]. Figure 18(b) shows
the schematic diagram for the optical quantum state synthesizer, where $l$
input vacuum states are initially displaced and squeezed, then interacted by a
beam-splitter network, and finally those except for the output state are
measured by photon-number resolving detectors. Depending on the pattern of the
detected photon numbers and circuit parameters for linear optics, the output
state is prepared as $\ket{\psi}_{\rm out}\approx\hat{U}\sum_{i=0}^{n_{\rm
max}}\frac{c_{i}}{N}\ket{i},$ where $n_{\rm max}$, $c_{i}$, and $N$, are the
truncated photon number in the Fock basis, the coefficient of the Fock state
$\ket{i}$, and a normalization factor, respectively, and $\hat{U}$ is composed
of single-mode Gaussian operations (e.g., squeezing, displacement, and
rotation) [284]. The truncated photon number is given by $n_{\rm
max}=\sum_{i=2}^{l}{m_{i}}$, where ${m_{i}}$ is the detected photon number in
the $i$-th detector. The coefficients $c_{i}$ depend on $m_{i}$ and the
circuit parameters such as transmissivities of the beam splitters and the
amount of the squeezing and displacement operations [284]. Those parameters
are optimized by an optimization algorithm running on a classical computer so
that the output state $\ket{\psi}_{\rm out}$ becomes close to the target non-
Gaussian state. To prepare an arbitrary non-Gaussian state with $n_{\rm max}$,
we need to optimize coefficients of the generated state up to $n_{\rm max}$
independently. The number of independent parameters, which we can access to
optimize them, has been conjectured as ${(l+2)(l-1)}/{2}$ [284] and scales
polynomially with $l$. Thus, the GBS-based device with $l$ input states is
expected to prepare an arbitrary state up to $\ket{n_{\rm max}}$ with $n_{\rm
max}=(l+2)(l-1)/2-1$.

For fault-tolerant QC, the GBS-based device needs to generate well-
approximated GKP qubits, which require sufficiently large $n_{\rm max}$.
However, the complexity for the classical numerical simulation of the GBS-
based device scales exponentially with $l$. This limits the number of
coefficients we can optimize, considering $n_{\rm max}$ scales polynomially
with $l$. The complexity mainly comes from the calculation of a $loop$
$hafnian$ which is contained in the class of #P-complete problems [291, 292].
In fact, the complexity of the calculation of a hafnian has been used for
achieving quantum supremacy over a classical computer by a protocol called GBS
[7, 293]. Recently, the efficient backcasting search has been developed to
solve the problem of time complexity for arbitrary non-Gaussian state
generation [282]. Also, Ref. [282] has shown a specific configuration of an
optical circuit based on the GBS in order to generate the GKP qubit required
for fault-tolerant QC.

##  5 Conclusion

By virtue of many advantages compared with other physical systems, optical
CVQC has attracted much interest in the last decade as a promising candidate
for realizing universal and fault-tolerant QC. In this review, we have
highlighted several topics of recent experimental and theoretical progress in
terms of universality, scalability, and fault tolerance in optical CVQC. As we
have seen, the technologies of optical multiplexing, bandwidth broadening, and
integrated photonic chips pave the way towards scaling up photonic quantum
computers. Furthermore, there are many recent theoretical efforts towards
universal and fault-tolerant CVQC with bosonic codes, including reductions of
the threshold for the squeezing level and proposals for the architecture of
CVQC. Among the many available bosonic codes, the GKP qubit has been shown to
be the optimal code for CVQC because it not only has superior error tolerance
but also requires only Gaussian operations for universality. Thus, the
promising research direction towards large-scale CVQC would be to develop
efficient light sources for optical GKP qubits and process these states with
time-multiplexed CVQC architectures, which already have been demonstrated to
perform Gaussian operations in a scalable manner.

Of course, there remains a long way towards the realization of large-scale QC.
Specifically, non-Gaussian gates required for universality have not been
demonstrated experimentally. In addition, the GKP qubits with a sufficient
squeezing level for large-scale QC has not been generated yet in an optical
setup. Furthermore, those need to be combined with the optical technologies in
a scalable and low-loss configuration manner. Nevertheless, we believe that
the emerging ideas described in this review will open promising paths to these
problems. These ideas will also open up new possibilities for architecture
design, hardware integration, and operating software for CVQC, which finally
lead to large-scale QC. We hope that this review will attract many researchers
to the field of quantum information with CVs.

## Acknowledgments

This work was partly supported by JST [Moonshot R$\&$D][Grant No. JPMJMS2064],
JST [Moonshot R$\&$D][Grant No. JPMJMS2061], donations from Nichia
Corporation, MEXT Leading Initiative for Excellent Young Researchers, Katsu
Research Encouragement Award of the University of Tokyo, Toray Science
Foundation (19-6006), Research Foundation for Opto-Science and Technology, and
the Kayamori Foundation of Informational Science Advancement. The authors
would like to thank Yutaro Enomoto, Takahiro Kashiwazaki, Takaya Matsuura,
Takahiro Mitani, Hayata Yamasaki for their helpful comments on the manuscript.

## References

## References

  * [1] Arute F, Arya K, Babbush R, Bacon D, Bardin J C, Barends R, Biswas R, Boixo S, Brandao F G S L, Buell D A, Burkett B, Chen Y, Chen Z, Chiaro B, Collins R, Courtney W, Dunsworth A, Farhi E, Foxen B, Fowler A, Gidney C, Giustina M, Graff R, Guerin K, Habegger S, Harrigan M P, Hartmann M J, Ho A, Hoffmann M, Huang T, Humble T S, Isakov S V, Jeffrey E, Jiang Z, Kafri D, Kechedzhi K, Kelly J, Klimov P V, Knysh S, Korotkov A, Kostritsa F, Landhuis D, Lindmark M, Lucero E, Lyakh D, Mandra S, McClean J R, McEwen M, Megrant A, Mi X, Michielsen K, Mohseni M, Mutus J, Naaman O, Neeley M, Neill C, Niu M Y, Ostby E, Petukhov A, Platt J C, Quintana C, Rieffel E G, Roushan P, Rubin N C, Sank D, Satzinger K J, Smelyanskiy V, Sung K J, Trevithick M D, Vainsencher A, Villalonga B, White T, Yao Z J, Yeh P, Zalcman A, Neven H and Martinis J M 2019 Nature 574 505
  * [2] Gong M, Wang S, Zha C, Chen M C, Huang H L, Wu Y, Zhu Q, Zhao Y, Li S, Guo S, Qian H, Ye Y, Chen F, Ying C, Yu J, Fan D, Wu D, Su H, Deng H, Rong H, Zhang K, Cao S, Lin J, Xu Y, Sun L, Guo C, Li N, Liang F, Bastidas V M, Nemoto K, Munro W J, Huo Y H, Lu C Y, Peng C Z, Zhu X and Pan J W 2021 Science 372 948–952
  * [3] Pagano G, Bapat A, Becker P, Collins K S, De A, Hess P W, Kaplan H B, Kyprianidis A, Tan W L, Baldwin C, Brady L T, Deshpande A, Liu F, Jordan S, Gorshkov A V and Monroe C 2020 Proceedings of the National Academy of Sciences 117 25396–25401
  * [4] Pino J M, Dreiling J M, Figgatt C, Gaebler J P, Moses S A, Allman M S, Baldwin C H, Foss-Feig M, Hayes D, Mayer K, Ryan-Anderson C and Neyenhuis B 2021 Nature 592 209
  * [5] Watson T F, Philips S G J, Kawakami E, Ward D R, Scarlino P, Veldhorst M, Savage D E, Lagally M G, Friesen M, Coppersmith S N, Eriksson M A and Vandersypen L M K 2018 Nature 555 633
  * [6] Hendrickx N W, Lawrie W I L, Russ M, van Riggelen F, de Snoo S L, Schouten R N, Sammak A, Scappucci G and Veldhorst M 2021 Nature 591 580