# A complete continuous-variable quantum computation architecture:
from cluster state generation to fault-tolerant accomplishment

Peilin Du  Jing Zhangmissing<EMAIL_ADDRESS>Tiancai Zhangmissing Rongguo Yang
<EMAIL_ADDRESS>Jiangrui Gaomissing State Key Laboratory of Quantum Optics and
Quantum Optics Devices, Shanxi University, Taiyuan, 030006, China
College of Physics and Electronic Engineering, Shanxi University,Taiyuan,
030006, China
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan,
030006, China

###### Abstract

Continuous-variable measurement-based quantum computation, which requires
deterministically generated large-scale cluster state, is a promising
candidate for practical, scalable, universal, and fault-tolerant quantum
computation. In this work, a complete architecture including cluster state
preparation, gate implementations, and error correction, is demonstrated.
First, a scheme for generating two-dimensional large-scale continuous-variable
cluster state by multiplexing both the temporal and spatial domains is
proposed. Then, the corresponding gate implementations for universal quantum
computation by gate teleportation are discussed and the actual gate noise from
the generated cluster state and Gottesman-Kitaev-Preskill (GKP) state are
considered. After that, the quantum error correction can be further achieved
by utilizing the square-lattice GKP code. Finally, a fault-tolerent quantum
computation can be realized by introducing bias into the square-lattice GKP
code (to protect against phase-flips) and concatenating a classical repetition
code (to handle the residual bit-flip errors), with a squeezing threshold of
12.3 dB. Our work provides a possible option for a complete fault-tolerent
quantum computation architecture in the future.

††preprint: APS/123-QED

##  I introduction

In measurement-based quantum computation (MBQC), gate implementations can be
flexibly achieved by measurements with adaptively chosen basis, obviating
those rigorous requirements in circuit-based quantum computation. Continuous-
variable (CV) MBQC, which requires deterministically generated large-scale
cluster state, is a promising candidate for practical, scalable, universal,
and fault-tolerant quantum computation[1, 2, 3]. In general, a complete CV
MBQC architecture mainly includes three steps: cluster state generation, gate
implementation (universal quantum computation), and (fault-tolerant) quantum
error correction.

As the prerequisite of CV MBQC, large-scale CV cluster state must be prepared
first. In the past ten years, different kinds of large-scale CV cluster states
have been experimentally generated by time [4, 5, 6, 7] and frequency
multiplexing[8, 9]. Also many theoretical schemes for generating large-scale
cluster states by multiplexing in both temporal-frequency[10, 11, 12],
temporal-spatial[13], and frequency-spatial[14, 15] domains have been proposed
recently. Considering the structure of the generated cluster states, at least
two dimensions are required to achieve universal quantum computation in CV
MBQC[16], one for computation and another for manipulation.

After preparing the key quantum resource, large-scale cluster state(s), one
can proceed to the gate implementation step. There were abundant theoretical
demonstrations of single-mode Gaussian [17, 4], two-mode Gaussian [10, 18, 19,
20, 21] and single-mode non-Gaussian [22, 23] gates, by projective
measurements based on gate teleportation or macronode structure. In
experiment, single-mode Gaussian operation and further scalable quantum
operations were demonstrated [24], also a small quantum circuit consisting of
10 single-mode gates and 2 two-mode gates was executed on a three-mode input
state[25], both multiplexing in the time domain.

Unfortunately, during gate implementations, Gaussian noise is inevitably added
to the quantum information during computation, because only finite squeezing
can be achieved in labs. To combat this noise, one can encode logical qubits
into physical bosonic modes, using the cat-code[26], the binomial code[27], or
the Gottesman-Kitaev-Preskill (GKP) code[28]. Thus, the Gaussian noise can be
converted into Pauli errors in the encoded qubits and can be partially
corrected. However, residual errors can be further treated by combining these
bosonic codes with certain higher-level qubit error correction codes, such as
the repetition code[29, 30, 31], the [[4, 2, 2]] code[32], or the surface
code[33, 34, 35, 36], towards fault-tolerant MBQC.

A fault-tolerant MBQC scheme using the surface-GKP code was proposed and the
corresponding squeezing threshold was 18.6 dB[36], without considering the
generation of entangled state resources. The required squeezing was reduced by
harnessing both the digital and analog information contained in the GKP
qubits[34] based on an ideal cluster state without any gate noise. Recently, a
fault-tolerant MBQC architecture based on a specially prepared three-
dimensional (3D) cluster state using topological surface-4-GKP was proposed
and the corresponding squeezing threshold was 12.7 dB[21]. However, the
considered gate noise in this scheme was assumed to be some certain values,
rather than the actual gate noise from the generated cluster states. In this
work, by combing repetition code with biased-GKP code, we validate a complete
fault-tolerant CV MBQC architecture, considering the actual gate noise caused
by finite squeezing in the preparation of cluster state and GKP state.

The paper is organized as follows. In Sec. II, the whole quantum computation
architecture is introduced in a clear way. In Sec. III, the scheme for
generating the spatiotemporal two-dimensional (2D) cluster state is
demonstrated in detail. In Sec. IV, the universal MBQC framework based on the
generated 2D CV cluster state is deduced and analyzed. In Sec. V, the quantum
error correction with GKP code is discussed, and in Sec. VI, bias is
introduced to GKP code and repetition code is added for fault tolerance. Our
main results are concluded and further discussed in Sec. VII.

##  II The complete fault-tolerant CV quantum computation architecture

Figure 1: The schematic diagram of the complete continuous-variable quantum
computation architecture.

To give an overall impression, the complete CV quantum computation
architecture is shown in Fig.1, which consists of three parts: cluster state
generation (green area), gate implementation (blue area), and quantum error
correction (pink area). In the cluster state generation part, a cluster state
with entanglement structure of bilayer square lattice is generated by
multiplexing in both the temporal and spatial domains, which only requires one
optical parametric oscillator (OPO). In the quantum computation part, single-
mode and two-mode gates, which are necessary for universal quantum
computation, can be efficiently implemented by homodyne detecting each
generated entangled modes with proper measurement angle. Through an optical
switch, the input state can be injected, and the result state can be output
after a gate teleportation process. In the quantum correction part, the output
result state are projected onto the ancillary GKP states
$\left|+\right\rangle_{GKP}$ and $\left|0\right\rangle_{GKP}$ through the
$\hat{C}_{X}$ and $\hat{C}_{Z}$ gate, respectively. Results of measuring the
ancillary GKP state can be used for error correction. In addition, fault-
tolerant quantum computation can be further achieved by combing biased GKP
code with repetition code. All the details of each part will be given in the
following sections.

##  III The spatiotemporal cluster state generation

###  III.1 scheme for generating the 2D cluster state

As is shown in Fig.2(a), the 2D cluster state can be generated by 4 steps. In
step I, the pumped beam composed of Hermite Gaussian (HG) modes HG02 and HG20
injects into the cavity, in which there is a pair of $\alpha$-cut type-II KTP
crystals for simultaneous resonance of all modes and a half-wave plate (HWP)
for Gouy-phase compensation. Then two pairs of entangled HG01 and HG10 modes
with orthogonal polarization, i.e., HGi,10, HGs,10, HGi,01, and HGs,01 ($i$
and $s$ means idler and signal, respectively), are generated[37, 38].

The interaction Hamiltonian of the OPO cavity system is

$\hat{H}=i\hbar\xi\sum\left\\{\mathbf{G}\hat{a}_{l}^{j\dagger}\hat{a}_{l}^{j\dagger}\right\\}+H.C.$
(1)

where $\xi$ is the second-order nonlinear coefficient, and $\mathbf{G}$ is the
adjacent matrix of $H$ graph, whose matrix element is $1$ when the parametric
process exists, and $0$ otherwise. $\hat{a}_{l}^{j\dagger}$ and $a_{l}^{j}$
($j\in\left\\{i,s\right\\}$ represents the polarization of the mode,
$l\in\left\\{01,10\right\\}$ represents the spatial mode) are the creation and
annihilation operators of the fundamental fields, respectively. According to
the properties of HG modes, one can get a square entangled state of
(HG${}^{s}_{45^{\circ}}$, HG${}^{s}_{135^{\circ}}$,
HG${}^{i}_{10}$,HG${}^{i}_{01}$), by coupling HG${}^{s}_{10}$ and
HG${}^{s}_{01}$ modes as[39],

$\displaystyle HG^{s}_{45^{\circ}}=HG^{s}_{01}+HG^{s}_{10},$ $\displaystyle
HG^{s}_{135^{\circ}}=HG^{s}_{01}-HG^{s}_{10},$ (2)

The continuous-wave signal and idler beams can be divided into time bins of
time period $t$, where $1/t$ is narrower than the bandwidth of the OPO cavity.
Thus, a series of square entangled states separated by time interval $t$ can
be deterministically created, and square entangled state in each time bin is
independent of each other.

In step II, the square entangled state is devided into two parts:
(HG${}^{s}_{45^{\circ}}$, HG${}^{s}_{135^{\circ}}$, horizontal polarization)
and (HG${}^{i}_{01}$, HG${}^{i}_{10}$, vertical polarization) , by a polarized
beam splitter (PBS). Then the signal beams (HG${}^{s}_{45^{\circ}}$,
HG${}^{s}_{135^{\circ}}$) are spatially rotated into signal beams
(HG${}^{s}_{10}$, HG${}^{s}_{01}$) by a dove prism. Thus the entangled modes
become (HG${}^{s}_{10}$, HG${}^{s}_{01}$,HG${}^{i}_{10}$, HG${}^{i}_{01}$). In
step III, HG10 and HG01 modes in signal and idler beams are separated by
transverse-mode beam splitters (TBSs), and then HG${}^{s}_{01}$ and
HG${}^{s}_{01}$ are delayed by $t$ and $10t$, respectively. After the fiber
delay, the HG${}^{s}_{01}$ mode of $k$ time is synchronized with the
HG${}^{s}_{10}$ mode of ($k+1$) time, while the HG${}^{i}_{01}$ mode of $k$
time is synchronized with the HG${}^{i}_{10}$ mode of ($k+10$) time. The delay
lines multiplex these modes in time so that they form a 2D (but still
disconnected) grid layout with cylindrical boundary.

In step IV, the staggered HG01 and HG10 modes in signal and idler beams are
coupled by a spatial “beam splitter” which consists of a TBS and a dove prism.
Thus, a series of square entangled states, which form a 2D grid layout with
cylindrical boundary, are connected at this step. The generated 2D cluster
state has the continuous cylindrical structure, as is shown in Fig.2(b). Here
each node (transparent gray ball) in the structure contains two modes:
(HG${}^{i}_{01}$ (green), HG${}^{i}_{10}$ (red)), or (HG${}^{s}_{01}$ (blue),
HG${}^{s}_{10}$ (purple)), and each edge (black line) represents a dual-rail
structured connection. Fig.2(c) gives the detailed graph of unrolling this
cylindrical structure, i.e., the bilayer square lattice structure, which is a
universal resource for quantum computation. Here the blue and red lines
correspond to the entanglement weights of $1/2\sqrt{2}$ and $-1/2\sqrt{2}$,
respectively. Note that the delay between temporal modes arriving at each
detector is set by the shorter delay $t$. The longer delay line $Nt$ (here
$N=10$) determines the circumference ($N$ sites) of the cylinder. Therefore,
an arbitrary large 2D cluster state can be generated by choosing proper longer
delay $Nt$, which is limited by the coherence time of the light source.

Figure 2: (a) The scheme for generating 2D spatiotemporal cluster state. (b)
Whole structure. (c) Detailed unrolled structure. Here the blue and red lines
correspond to the entanglement weights of $1/2\sqrt{2}$ and $-1/2\sqrt{2}$,
respectively.

###  III.2 Nullifier and VLF full-inseparability criterion

To analyze the squeezing requirement of the 2D entangled state, we need to
discuss the nullifier based on the van Loock–Furusawa (VLF) full-
inseparability criterion[40].

When the $\mathbf{G}$ matrix is self-inverse and bipartite, the complex
adjacency matrix $\mathbf{Z}$ of Gaussian pure state can be written as

$\mathbf{Z}=i\cosh\left(2r\right)\mathbf{I}-i\sinh\left(2r\right)\mathbf{G}$
(3)

where $\mathbf{I}$ represents an identity matrix and $r$ is the squeezing
parameter. This $H$ graph state can be transformed into a cluster state by
operating a $\frac{\pi}{2}$ phase-space rotation on half of its modes or a
$\frac{\pi}{4}$ phase-space rotation on all of its modes. Since this phase-
space rotation can be absorbed into the measurement basis when measuring each
mode of the state, a self-inverse bipartite $H$ graph state can be considered
as a cluster state. Therefore, the generated entangled state can be considered
as a cluster state.

An $n$-mode Gaussian pure state $\left|\psi\right\rangle$ can be efficiently
characterized by $n$ independent nullifiers, which are linear combinations of
the quadrature operators, with $\left|\psi\right\rangle$ corresponding to
their mutual zero-eigenstate. Nullifiers also play an important role in
verifying genuine multipartite inseparability for cluster states transformed
from self-inverse bipartite $H$ graph states. Such cluster states are
approximately nullified by linear combinations of quadratures that are all of
position- or momentum- type. For our 2D large-scale spatiotemporal cluster
state, the nullifiers consist of six modes can be obtained as,

$\displaystyle
N_{k}^{x_{1}}=\left(\hat{x}_{10,k}^{i}-\hat{x}_{01,k}^{i}\right)$
$\displaystyle-\frac{1}{\sqrt{2}}\left(\hat{x}_{10,k+1}^{s}+\hat{x}_{01,k+1}^{s}\right)$
(4)
$\displaystyle+\frac{1}{\sqrt{2}}\left(\hat{x}_{10,k}^{s}-\hat{x}_{01,k}^{s}\right)$
$\displaystyle
N_{k}^{x_{2}}=\left(\hat{x}_{10,k+10}^{i}+\hat{x}_{01,k+10}^{i}\right)$
$\displaystyle-\frac{1}{\sqrt{2}}\left(\hat{x}_{10,k+1}^{s}+\hat{x}_{01,k+1}^{s}\right)$
(5)
$\displaystyle-\frac{1}{\sqrt{2}}\left(\hat{x}_{10,k}^{s}-\hat{x}_{01,k}^{s}\right)$
$\displaystyle
N_{k}^{p_{1}}=\left(\hat{p}_{10,k}^{i}-\hat{p}_{01,k}^{i}\right)$
$\displaystyle+\frac{1}{\sqrt{2}}\left(\hat{p}_{10,k+1}^{s}+\hat{p}_{01,k+1}^{s}\right)$
(6)
$\displaystyle-\frac{1}{\sqrt{2}}\left(\hat{p}_{10,k}^{s}-\hat{p}_{01,k}^{s}\right)$
$\displaystyle
N_{k}^{p_{2}}=\left(\hat{p}_{10,k+10}^{i}+\hat{p}_{01,k+10}^{i}\right)$
$\displaystyle+\frac{1}{\sqrt{2}}\left(\hat{p}_{10,k+1}^{s}+\hat{p}_{01,k+1}^{s}\right)$
(7)
$\displaystyle+\frac{1}{\sqrt{2}}\left(\hat{p}_{10,k}^{s}-\hat{p}_{01,k}^{s}\right)$

In fact, nullifiers span a vector space, and any linear combination of
nullifiers is still a nullifier. Therefore the generated 2D cluster state can
also be specified with the following set of nullifiers,

$\displaystyle\frac{N_{k+10}^{x_{1}}+N_{k}^{x_{2}}}{2}=\hat{x}_{10,k+10}^{i}-\frac{1}{2\sqrt{2}}\left(\hat{x}_{10,k+11}^{s}+\hat{x}_{01,k+11}^{s}\right)$
$\displaystyle-\frac{1}{2\sqrt{2}}\left(\hat{x}_{01,k+10}^{s}-\hat{x}_{10,k+10}^{s}\right)-\frac{1}{2\sqrt{2}}\left(\hat{x}_{10,k+1}^{s}+\hat{x}_{01,k+1}^{s}\right)$
$\displaystyle-\frac{1}{2\sqrt{2}}\left(\hat{x}_{10,k}^{s}-\hat{x}_{01,k}^{s}\right)$
(8)
$\displaystyle\frac{-N_{k+10}^{x_{1}}+N_{k}^{x_{2}}}{2}=\hat{x}_{10,k+10}^{i}+\frac{1}{2\sqrt{2}}\left(\hat{x}_{10,k+11}^{s}+\hat{x}_{01,k+11}^{s}\right)$
$\displaystyle-\frac{1}{2\sqrt{2}}\left(\hat{x}_{10,k+10}^{s}-\hat{x}_{01,k+10}^{s}\right)-\frac{1}{2\sqrt{2}}\left(\hat{x}_{10,k+1}^{s}+\hat{x}_{01,k+1}^{s}\right)$
$\displaystyle-\frac{1}{2\sqrt{2}}\left(\hat{x}_{10,k}^{s}-\hat{x}_{01,k}^{s}\right)$
(9)
$\displaystyle\frac{N_{k+10}^{p_{1}}+N_{k}^{p_{2}}}{2}=\hat{p}_{10,k+10}^{i}+\frac{1}{2\sqrt{2}}\left(\hat{p}_{10,k+11}^{s}+\hat{p}_{01,k+11}^{s}\right)$
$\displaystyle+\frac{1}{2\sqrt{2}}\left(\hat{p}_{01,k+10}^{s}-\hat{p}_{10,k+10}^{s}\right)+\frac{1}{2\sqrt{2}}\left(\hat{p}_{10,k+1}^{s}+\hat{p}_{01,k+1}^{s}\right)$
$\displaystyle+\frac{1}{2\sqrt{2}}\left(\hat{p}_{10,k}^{s}-\hat{p}_{01,k}^{s}\right)$
(10)
$\displaystyle\frac{-N_{k+10}^{p_{1}}+N_{k}^{p_{2}}}{2}=\hat{p}_{10,k+10}^{i}-\frac{1}{2\sqrt{2}}\left(\hat{p}_{10,k+11}^{s}+\hat{p}_{01,k+11}^{s}\right)$
$\displaystyle+\frac{1}{2\sqrt{2}}\left(\hat{p}_{10,k+10}^{s}-\hat{p}_{01,k+10}^{s}\right)+\frac{1}{2\sqrt{2}}\left(\hat{p}_{10,k+1}^{s}+\hat{p}_{01,k+1}^{s}\right)$
$\displaystyle+\frac{1}{2\sqrt{2}}\left(\hat{p}_{10,k}^{s}-\hat{p}_{01,k}^{s}\right)$
(11)

these nullifiers match well with the entanglement structure in Fig.2(c). When
the red (HG${}^{i}_{10}$) and green (HG${}^{i}_{01}$) balls in the solid
rectangular box are focused on, it is found that they connect with other four
sets of blue (HG${}^{s}_{01}$) and purple (HG${}^{s}_{10}$) balls in the
dashed rectangular boxes. The nullifiers of quadrature amplitude and phase of
the red and green balls are corresponding to the Eq.(8)-(11), respectively.

Figure 3: The nullifier of quadrature $\hat{x}$ (a) and $\hat{p}$ (b) versus
the squeezing parameter, for the generated 2D cluster state.

For a minimum unit of the 2D cluster state, there are 6 basic modes and 31
possible bipartitions, based on which the following inequalities as a
sufficient condition due to full-inseparability can be obtained as

$\left\langle\left(\hat{N}_{k}^{x_{1}}\right)^{2}\right\rangle<\frac{1}{4\sqrt{2}}\qquad\left\langle\left(\hat{N}_{k}^{p_{1}}\right)^{2}\right\rangle<\frac{1}{4\sqrt{2}}.$
(12)

While the variances of nullifiers $\hat{N}_{k}^{x_{1}}$ and
$\hat{N}_{k}^{p_{1}}$ can be derived:

$\displaystyle\left\langle\left(\hat{N}_{k}^{x_{1}}\right)^{2}\right\rangle=\frac{1}{2}e^{-2r_{s}},$
(13)
$\displaystyle\left\langle\left(\hat{N}_{k}^{p_{1}}\right)^{2}\right\rangle=\frac{1}{2}e^{-2r_{i}}.$

Therefore, the above conditions can determine the required minimum squeezing
for generating 2D cluster states. As is shown in Fig.(3), when the squeezing
is larger than 4.5 dB, the sufficient condition for full-inseparability will
be satisfied.

##  IV universal quantum computation

After generating a large-scale 2D cluster state, how to use it in MBQC is now
under our consideration. An arbitrary single-mode or two-mode Gaussian gate
acting on bosonic modes can be realized by using the gate teleportation. While
the gate teleportation is diagrammatically depicted in Fig.4. Here the input
state $\left|\Psi\right\rangle_{in}$ and one entangled mode 2 are coupled
through a beam splitter. By choosing a set of measurement basis
$\left(\theta_{in},\theta_{2}\right)$, the output modes are measured as
$m_{in}$ and $m_{2}$, which are transmitted to the other entangled mode 3 to
get the output gate teleportation result
$\hat{U}\left|\Psi\right\rangle_{in}$. The resulting transformation $\hat{U}$
of the quadratures in the Heisenberg picture can be expressed as

$\begin{pmatrix}\hat{x}_{out}^{\prime}\\\
\hat{p}_{out}^{\prime}\end{pmatrix}=\mathcal{G}\begin{pmatrix}\hat{x}_{in}\\\
\hat{p}_{in}\end{pmatrix}+\mathcal{N}\begin{pmatrix}\hat{p}_{2}\\\
\hat{p}_{3}\end{pmatrix}+\mathcal{D}\begin{pmatrix}{m}_{in}\\\
{m}_{2}.\end{pmatrix}$ (14)

Figure 4: The schematic diagram of gate teleportation circuit. The
$\left|\Psi\right\rangle_{in}$ is input state, mode 2 and mode 3 are a pair of
entangled state.

The first term represents the Gaussian gate implementation, where the Gaussian
gate matrix $\mathcal{G}$ is a symplectic matrix. The second term represents
gate noise added to the quadratures due to finite squeezing in the cluster
state, and it will vanish when $r\to\infty$. The last term is the
computational byproduct in the form of a displacement. Here $\mathcal{N}$ and
$\mathcal{D}$ are the noise and displacement matrices, respectively. When all
gates are Gaussian, the displacement byproduct can be compensated in the
measurement result of the final output state.

Figure 5: Simplified graphical-calculus representation of quantum computation
based on the bilayer square lattice CV cluster state. Here the blue lines
correspond to the entanglement weights of $\tanh(2r)$.

To analyze the quantum computation scheme based on our generated 2D cluster
state, the graphical-calculus representation with simplified entanglement
structure is shown in Fig.5[41]. The resource squeezing is $\delta=e^{-2r}$,
and the corresponding squeezing contained in the generated 2D cluster state is
$\varepsilon=\operatorname{sech(2r)}$. Here the blue lines correspond to the
entanglement weights of $\tanh(2r)$, the red dashed arrows represent the
process of Eq.(2), while the red and green solid arrows represent the action
of the spatial “beam splitter” for idler and signal beam, respectively. The
four modes in the black dashed rectangle arrive the detectors simultaneously
at $k+10$ time.

The input states shown by the black nodes can be switched into the circuit and
it can be teleported along the quantum wires (light-blue area) by using gate
teleportation in sequence. The neighbor quantum wires can be connected through
the control wires (light-green area). The modes in quantum wires that carry
quantum information to be processed are called wire modes. And the modes in
control wires that control the connectivity between the neighbor wires are
called control modes. GKP code used in the error correction step is the most
popular bosonic code and its universal Clifford gate set can be realized by
the corresponding universal Gaussian gate set,

$\underset{Gaussian\enspace
gates}{\underbrace{{\left\\{\hat{I},\hat{F},\hat{P}\left(1\right),\hat{C}_{Z}\left(1\right)\right\\}}}}\longmapsto\underset{Clifford\enspace
gates}{\underbrace{{\left\\{\mathbf{I},\mathbf{F},\mathbf{P\left(1\right)},\mathbf{{C}_{Z}\left(1\right)}\right\\}}}}$

where $\hat{I},\hat{F},\hat{P}(1),\hat{C}_{Z}(1)$ represent the identity,
Fourier, shear, and controlled-Z gates, respectively. In the Heisenberg
picture, these Gaussian gates can be described by symplectic matrices:

$\displaystyle I=\left(\begin{matrix}1&0\\\ 0&1\end{matrix}\right)\qquad
F=\left(\begin{matrix}0&1\\\ -1&0\end{matrix}\right)$ $\displaystyle
P(1)=\left(\begin{matrix}1&0\\\ 1&1\end{matrix}\right)\qquad
C_{Z}(1)=\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&1&1&0\\\
1&0&0&1\end{matrix}\right).$ (15)

Therefore, to accomplish the universal computation, the implementation of this
gate set based on the generated 2D cluster state is discussed in the
following.

###  IV.1 single-mode gate

To implement single-mode gates, neighbor quantum wires must be decoupled from
each other, i.e., the measurement angles of control modes HG${}^{s}_{01,k}$
and HG${}^{s}_{10,k}$ are set as
$\left(-1\right)^{k}\theta_{c}$=$\pm\frac{\pi}{4}$[10, 41]. The corresponding
quantum wires for single-mode gate implementations are shown in Fig.6(a).
Since the measurements angles of control modes are fixed, one-step single-mode
gate can be implemented by two-step gate teleportation shown in the black
dashed box. The corresponding equivalent quantum circuit of one-step single-
mode gate is shown in Fig.6(b), where the gray (dark) detectors represent the
homodyne measurements on wire (control) modes with changeable (fixed)
measurement angles. The implemented one-step single-mode gate
$\hat{\mathcal{G}}$ can be expressed as

$\displaystyle\hat{\mathcal{G}}=\hat{S}\left(\pm
2\Gamma^{2}\right)\hat{R}\left(\frac{\theta_{+}}{2}\right)\hat{S}\left(\tan\frac{\theta_{+}}{2}\right)\hat{R}\left(\frac{\theta_{+}}{2}\right)$
(16)

Figure 6: The logic level (a) and the equivalent quantum circuit (b) of
implementing one-step single-mode gate quantum computation on the bilayer-
square-lattice cluster state.

where $\theta_{\pm}=\theta_{1}\pm\theta_{2}$, $\theta_{1}$ and $\theta_{2}$
are the measurement angles of the wire modes. The sign in the first term of
Eq.(16) is positive or negative, depends on the measurement angles of the
control modes. The one-step single-mode identity gate can be implemented by
choosing

$\left(\begin{matrix}\theta_{1}\\\
\theta_{2}\end{matrix}\right)_{\hat{I}}=\left(\begin{matrix}\left(-1\right)^{k+1}\arctan\left(\tanh^{-2}\left(2r\right)\right)\\\
\left(-1\right)^{k}\arctan\left(\tanh^{-2}\left(2r\right)\right)\end{matrix}\right)$
(17)

However, Gaussian noise is inevitably added to the quantum information during
computation due to infinite squeezing. Thus, the resulting gate noise
$\mathcal{N}$ for $\hat{I}$ gate is[42]

$\mathcal{N}=\left(\begin{matrix}\frac{1}{2\Gamma^{2}}&\frac{1}{2\Gamma}&-\frac{1}{2\Gamma}&0\\\
0&-\Gamma&-\Gamma&1\end{matrix}\right)$ (18)

where $\Gamma=\tanh(2r)$. According to Eq.(18), the gate noises that added to
the output quadrature $\hat{x}$ and $\hat{p}$ are $\mathcal{N}_{x}\varepsilon$
and $\mathcal{N}_{p}\varepsilon$, respectively, where

$\displaystyle\mathcal{N}_{x}$ $\displaystyle=$
$\displaystyle\frac{1}{tanh^{4}(2r)}+\frac{1}{tanh^{2}(2r)}$ (19)
$\displaystyle\mathcal{N}_{p}$ $\displaystyle=$ $\displaystyle
tanh^{2}(2r)+1.$ (20)

Moreover, an arbitrary two-step single-mode gate $\hat{\mathcal{G}}$ can be
realized by further implementing two one-step single-mode gates
$\hat{\mathcal{G}_{1}}\hat{\mathcal{G}_{2}}$ in sequence[17], as shown in the
black solid box of Fig.6(a). Thus, the single-mode $\hat{F}$ and
$\hat{P}\left(1\right)$ gates can be implemented by two one-step single-mode
gates, and the measurement angles of the first and second steps are
($\theta_{1},\theta_{2}$) and ($\theta_{3},\theta_{4}$), respectively. For
single-mode Fourier gate,

$\left(\begin{matrix}\theta_{1}\\\ \theta_{2}\\\ \theta_{3}\\\
\theta_{4}\end{matrix}\right)_{\hat{F}}=\left(\begin{matrix}\frac{\pi}{2}\\\
0\\\ \arctan\left(\tanh^{-2}\left(2r\right)\right)\\\
-\arctan\left(\tanh^{-2}\left(2r\right)\right)\end{matrix}\right),$ (21)

and the gate noises that added to the output quadrature $\hat{x}$ and
$\hat{p}$ are equal, which is
$\left(\mathcal{N}_{x}+\mathcal{N}_{p}\right)\varepsilon$. For single-mode
shear gate,

$\left(\begin{matrix}\theta_{1}\\\ \theta_{2}\\\ \theta_{3}\\\
\theta_{4}\end{matrix}\right)_{\hat{P}\left(1\right)}=\left(\begin{matrix}0\\\
\arctan 2\\\ \frac{\pi}{2}\\\ 0\end{matrix}\right),$ (22)

while the gate noises that added to the output quadrature $\hat{x}$ and
$\hat{p}$ are $2\mathcal{N}_{x}\varepsilon$ and $2\mathcal{N}_{p}\varepsilon$,
respectively. Thus, by implementing one-step single-mode gate at most twice,
we can accomplish all single-mode gates in the universal Gaussian gate set.

###  IV.2 two-mode gate

A two-mode gate can be implemented by lifting the measurement restrictions on
certain control modes, as is shown by the black dashed box part in Fig.7(a).
And the corresponding equivalent quantum circuit of the two-mode gate
implementation is shown in Fig.7(b). Here a controlled-Z gate, as the two-mode
gate in the universal gate set, can be obtained as

Figure 7: The logic level (a) and the equivalent quantum circuit (b) of
implementing two-mode gate quantum computation on the bilayer-square-lattice
cluster state.

$\displaystyle\mathcal{\hat{G}}$ $\displaystyle=$
$\displaystyle\left[\hat{R}\left(\mp\frac{3\pi}{4}\right)\otimes\hat{R}\left(\pm\frac{\pi}{4}\right)\right]\hat{C}_{Z}\left(g\right)$
(23) $\displaystyle=$
$\displaystyle\left[\hat{R}\left(\pm\frac{3\pi}{4}\right)\otimes\hat{R}\left(\mp\frac{\pi}{4}\right)\right]\hat{C}_{X}\left(g\right),$

with the chosen measurement angles

$\displaystyle\left(\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5},\theta_{6}\right)=$
$\displaystyle\left(-\frac{\pi}{8},\frac{3\pi}{8},-\frac{\pi}{8},\frac{3\pi}{8},\frac{\pi}{4}\pm
arctan\frac{2}{g},\frac{\pi}{4}\mp arctan\frac{2}{g}\right),$

where $g$ is the tunable coupling strength of the controlled-Z (controlled-X)
gate, which is followed by the byproduct (known, fixed)
$\hat{R}\left(\mp\frac{3\pi}{4}\right)\otimes\hat{R}\left(\pm\frac{\pi}{4}\right)$.
By applying the inverse single-mode gates immediately after this gate, this
byproduct part can be compensated in the next step.

When $g=1$, the resulting gate noise $\mathcal{N}$ for $\hat{C}_{Z}(1)$ gate
is

$\mathcal{N}=\left(\begin{matrix}\frac{1}{-\Gamma^{2}}&\frac{1}{2\Gamma^{2}}&-\frac{3c}{4\Gamma}&-\frac{c}{4\Gamma}&-\frac{\sqrt{2}c}{2\Gamma}&-\frac{\sqrt{2}c}{4\Gamma}&0&0\\\
-\frac{1}{2\Gamma^{2}}&\frac{1}{\Gamma^{2}}&\frac{3c}{4\Gamma}&-\frac{c}{4\Gamma}&-\frac{\sqrt{2}c}{4\Gamma}&-\frac{\sqrt{2}c}{2\Gamma}&0&0\\\
0&-\frac{1}{2}&\frac{\Gamma}{4}&\frac{\Gamma}{4}&-\frac{\sqrt{2}\Gamma}{2}&\frac{\sqrt{2}\Gamma}{4}&1&0\\\
\frac{1}{2}&0&-\frac{\Gamma}{4}&\frac{\Gamma}{4}&\frac{\sqrt{2}\Gamma}{4}&-\frac{\sqrt{2}\Gamma}{2}&0&1\end{matrix}\right),$
(25)

where $c=\operatorname{csch}\left(2r\right)$ and the gate noises that added to
the output quadratures $\hat{x}_{1}$($\hat{x}_{2}$) and
$\hat{p}_{1}$($\hat{p}_{2}$) are $\mathcal{N}_{x_{1}}\varepsilon$
($\mathcal{N}_{x_{2}}\varepsilon$), and $\mathcal{N}_{p_{1}}\varepsilon$
($\mathcal{N}_{p_{2}}\varepsilon$), respectively, with

$\displaystyle\mathcal{N}_{{x}_{1}}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{{x}_{2}}=\frac{5}{2}\left(\frac{1}{tanh^{4}(2r)}+\frac{1}{tanh^{2}(2r)}\right)$
(26) $\displaystyle\mathcal{N}_{{p}_{1}}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{{p}_{2}}=\frac{5}{2}\left(tanh^{2}(2r)+1\right).$
(27)

As yet, all gates in the universal Gaussian gate set have been implemented by
gate teleportation. To quantify the effect of the quantum gate noises, the
residual squeezing after implementing the one-step single-mode, two-step
single-mode, two-mode gate, compared with the resource squeezing and the
squeezing contained in the generated 2D cluster state, are shown in Fig.8.
Here the identity gate, the shear gate, and the controlled-Z gate, are chosen
on behalf of the one-step single-mode, the two-step single-mode, and the two-
mode gates, respectively. Note that the squeezing in the cluster state modes
($\varepsilon=\operatorname{sech(2r)}$) is reduced about 3 dB compared with
the resource squeezing ($\delta=e^{-2r}$) in high squeezing level, which is
the cost of preparing the cluster state[43]. And the residual squeezing of
one-step single-mode gate is also reduced about 3 dB compared with the
squeezing contained in the generated 2D cluster state
($\mathcal{N}_{x},\mathcal{N}_{p}\rightarrow 2\varepsilon$). It is easier to
understand that the residual squeezing of the two-step single-mode gate is
still 3 dB lower than that of the one-step single-mode due to the
implementation of two one-step single-mode gates in sequence. For the residual
squeezing of the two-mode gate, it is a little bit lower than that of the two-
step single-mode, due to the additional noise from the control modes. For
example, when the resource squeezing is 10 dB, the corresponding squeezing
contained in the generated 2D entangled state is 7 dB, the residual squeezing
of the one-step single-mode, the two-step single-mode, and the two-mode gates
are 4 dB, 1 dB and 0 dB, respectively. This means that a resource squeezing of
10 dB can only support one two-mode gate at most, which is far from enough for
quantum computation. Therefore, the quantum error correlation must be taken
into account to treat with the Gaussian noises by transfer them to Pauli
errors and then correct the errors.

Figure 8: Resource remaining squeezing after quantum gates operation.

###  IV.3 non-gaussian gate

As is known, non-Gaussian operations or resources are required, to achieve
universal quantum computation. And the abovementioned single-mode gates and
two-mode gates involve only Gaussian measurements (homodyne detection) on
Gaussian cluster states. In many CV quantum computing proposals, the non-
Gaussian cubic phase gate $\hat{K}\left(\chi\right)=e^{i\chi\hat{x}^{3}/3}$ is
considered, as the complement of the universal Gaussian gate set, and such a
non-Gaussian gate can be implemented by incorporating non-Gaussian elements
into the measurement devices[18].

However, the inherent non-Gaussianity of the GKP states is sufficient to
achieve universal quantum computation in the GKP-encoded qubit subspace using
solely Gaussian transformations. Although the cubic phase gate based on non-
Gaussian measurement (photo counting) is available, it is more convenient to
use the GKP states to meet the requirement for non-Gaussianity in the
following quantum error correction part.

##  V Quantum correction

It is now clear that quantum computation based cluster state will inevitably
suffer from gate noise that accumulates throughout the computation. Therefore,
the efficient GKP encoding is chosen to correct quadrature error between
implemented gates. In fact, GKP state is often considered not only as a non-
Gaussian resource but also as an ancillary resource for error correction[28].

Here, we consider the square-lattice GKP state which can be defined as the
two-dimensional subspace of a bosonic Hilbert space that is stabilized by the
two stabilizers

$\hat{S}_{x}\equiv\exp\left[i2\sqrt{\pi}\hat{q}\right],\qquad\hat{S}_{p}\equiv\exp\left[i2\sqrt{\pi}\hat{p}\right].$
(28)

The computational basis states of the ideal GKP states are given by

$\displaystyle\left|0\right\rangle_{GKP}=\sum_{n\in\mathbb{Z}}\left|\hat{x}=2n\sqrt{\pi}\right\rangle,$
$\displaystyle\left|1\right\rangle_{GKP}=\sum_{n\in\mathbb{Z}}\left|\hat{x}=\left(2n+1\right)\sqrt{\pi}\right\rangle,$
(29)

and the alternative basis states are

$\displaystyle\left|+\right\rangle_{GKP}=\sum_{n\in\mathbb{Z}}\left|\hat{p}=2n\sqrt{\pi}\right\rangle,$
$\displaystyle\left|-\right\rangle_{GKP}=\sum_{n\in\mathbb{Z}}\left|\hat{p}=\left(2n+1\right)\sqrt{\pi}\right\rangle.$
(30)

How the ancillary GKP states are used to correct quadrature error of output
data GKP state without destroying information, is shown in Fig.9. After each
implemented gate, the quadrature amplitude (phase) of data GKP state
$\left|D\right\rangle_{GKP}$ is projected onto that of ancillary state
$\left|+\right\rangle_{GKP}$ ($\left|0\right\rangle_{GKP}$) by controlled-X
(controlled-Z) gate, which can be expressed by

$\displaystyle\hat{C}_{Z}=\exp\left[-i\hat{x}_{D}\hat{x}_{\left|+\right\rangle}\right]$
$\displaystyle\hat{C}_{X}=\exp\left[i\hat{p}_{D}\hat{p}_{\left|0\right\rangle}\right]$
(31)

where $\hat{x}_{D}$ ($\hat{p}_{D}$) and $\hat{x}_{\left|+\right\rangle}$
($\hat{p}_{\left|0\right\rangle}$) represent quadrature amplitude (phase) of
data and ancillary GKP state, respectively. Thus, the projected quadrature
amplitude and phase can be considered as:

$\displaystyle\hat{x}^{\prime}_{\left|0\right\rangle}=\hat{x}_{\left|0\right\rangle}+\hat{p}_{D}$
$\displaystyle\hat{p}^{\prime}_{\left|+\right\rangle}=\hat{p}_{\left|+\right\rangle}+\hat{x}_{D}.$
(32)

Homodyne measuring these quadrature operators modulo $\sqrt{\pi}$ gives
results $m_{x}$ and $m_{p}$. Since the eigenvalues of the first terms of
Eq.(32) are integral multiple of $\sqrt{\pi}$, the above measurements can be
simplified to:

$\displaystyle m_{x}=\hat{p}_{D}mod{\sqrt{\pi}}$ $\displaystyle
m_{p}=\hat{x}_{D}mod{\sqrt{\pi}}.$ (33)

If there is no gate noise in quadrature operators of the data state,
$m_{x}=m_{p}=0$. However, the gate noises add shift errors $\xi_{x}$ and
$\xi_{p}$ to the quadrature amplitude and phase, respectively:

$\displaystyle\hat{x}_{D}\rightarrow\hat{x}_{D}+\xi_{x}$
$\displaystyle\hat{p}_{D}\rightarrow\hat{p}_{D}+\xi_{p}.$ (34)

$\xi_{x(p)}$ obeys Gaussian random distribution with zero mean and variance
$\sigma_{x(p)}^{2}$, which can be expressed as:

Figure 9: The circuit of GKP quantum correction for correcting gate noise
after each implemented gate.

$\displaystyle\xi_{x}=R_{\sqrt{\pi}}\left(\xi_{x}\right)+n_{x}\sqrt{\pi}$
$\displaystyle\xi_{p}=R_{\sqrt{\pi}}\left(\xi_{p}\right)+n_{p}\sqrt{\pi}$ (35)

where $n_{x},n_{p}\in\mathbb{Z}$, and

$\displaystyle
n_{x(p)}=argmin_{n\in\mathbb{Z}}\left|\xi_{x(p)}-n\sqrt{\pi}\right|$ (36)

Thus, the measurement outcomes $m_{x}$ and $m_{p}$ become

$\displaystyle
m_{x}=R_{\sqrt{\pi}}\left(\xi_{p}\right)=\xi_{p}-n_{p}\sqrt{\pi}$
$\displaystyle
m_{p}=R_{\sqrt{\pi}}\left(\xi_{x}\right)=\xi_{x}-n_{x}\sqrt{\pi}.$ (37)

If $n_{x}$ and $n_{p}$ are even numbers, one can correct these shift errors by
acting displacement operators $exp\left[-im_{x}\hat{x}\right]$ and
$exp\left[im_{p}\hat{p}\right]$ on the quadrature amplitude and phase of the
data state, respectively. The overall shift operations during the above
process can be expressed by

$\displaystyle
e^{i\left(\xi_{p}-m_{x}\right)\hat{x}}=e^{in_{p}\sqrt{\pi}\hat{x}}=\left(Z\right)^{n_{p}}$
$\displaystyle
e^{-i\left(\xi_{x}-m_{p}\right)\hat{p}}=e^{-in_{x}\sqrt{\pi}\hat{p}}=\left(X\right)^{n_{x}}.$
(38)

Note that this GKP error correction does not work anymore for odd numbers and
a Pauli Z (Pauli X) error will be added on the data state. Hence, the success
probability of the GKP error correction can be obtained as

$\displaystyle p_{succ}^{sq}$ $\displaystyle=\sum_{n_{x},n_{p}\in
2\mathbb{Z}}\int_{\left(n_{x}-\frac{1}{2}\right)\sqrt{\pi}}^{\left(n_{x}+\frac{1}{2}\right)\sqrt{\pi}}d\xi_{x}\frac{1}{\sqrt{2\pi\sigma_{x}^{2}}}e^{\left[-\frac{\xi_{x}^{2}}{2\sigma_{x}^{2}}\right]}$
(39)
$\displaystyle\ast\int_{\left(n_{p}-\frac{1}{2}\right)\sqrt{\pi}}^{\left(n_{p}+\frac{1}{2}\right)\sqrt{\pi}}d\xi_{p}\frac{1}{\sqrt{2\pi\sigma_{p}^{2}}}e^{\left[-\frac{\xi_{p}^{2}}{2\sigma_{p}^{2}}\right]},$

where
$\sigma_{x(p)}^{2}=\left\\{\mathcal{G}X\right\\}_{x(p)}+\mathcal{N}_{x(p)}\varepsilon$,
with $X$ representing the quadrature variance matrix of the initial data
state. After gate implementing (identity, shear, and controlled-Z), the
quadrature variance matrix of the data state, i.e., $\mathcal{G}X_{\sigma}$,
can be expressed by:

$\displaystyle IX_{\sigma}=\left(\begin{matrix}\sigma_{0}\\\ \sigma_{0}\\\
\end{matrix}\right)\qquad P(1)X_{\sigma}=\left(\begin{matrix}\sigma_{0}\\\
2\sigma_{0}\\\ \end{matrix}\right)$ $\displaystyle
C_{Z}(1)X_{\sigma}=\left(\begin{matrix}\sigma_{0}\\\ \sigma_{0}\\\
2\sigma_{0}\\\ 2\sigma_{0}\end{matrix}\right).$ (40)

where $\sigma_{0}=e^{-2r}/{2}$, which is the quadrature squeezing of the
initial data state. In our noise model, measure noise and idle noise are
ignored, since measurements have been implemented pretty high efficiency and
all modes are measured in the optical MBQC platform.

Figure 10: Error probabilities ($p_{e}^{sq}$) for different implemented gate
during quantum correction. The $P^{1}_{S}$ ($P^{2}_{S}$) and $P_{T}$ represent
the error probability of one-step (two-step) single-mode and two-mode gates,
respectively.

Here, the quadrature squeezing of the date state $\sigma_{0}$ and the resource
squeezing $\delta$ are assumed to be equal. The error probabilities
($p_{e}^{sq}=1-p_{succ}^{sq}$) for one-step single-mode, two-step single-mode
and two-mode gates versus squeezing, are shown in Fig.10. It is clear that the
high the squeezing level the low the error probability. For single-mode gate
implementation, two-step leads to higher error probability due to bigger gate
noise, compared with one-step. While two-mode gate implementation brings
higher error probability due to more complex correction process, compared with
two-step single-mode gate implementation. For example, if squeezing is 15 dB,
which is the present maximum squeezing obtained in experiment[44], the error
probabilities of the one-step and two-step single-mode, two-mode gates are
about 0.008, 0.05 and 0.08, respectively. Although GKP encoding is effective
to most Gaussian random shift errors, it is still far from enough for
practical quantum computation. Therefore, to further reduce the error
probability, the Pauli errors added to the data state by GKP encoding must be
corrected, for achieving fault-tolerant quantum computation.

##  VI fault-tolerant quantum computation

To reduce the error probability to a necessary ultra-low level, the square-
lattice GKP state is extended to rectangular-lattice GKP state, i.e., a bias
is introduced, to protect against phase-flip errors. Combing with a classical
$n$-qubit repetition encoding, bit-flip errors can be corrected as well.

###  VI.1 Rectangular-lattice GKP code

Up to now, square-lattice GKP states are most widely studied due to their
balanced protection in each quadrature. However, in principle, alternative-
lattice grid states can be defined freely, as long as the stabiliser
generators commute. Here rectangular-lattice GKP state is taken into account,
and the balanced protection is broken, i.e., biasing the Pauli errors of the
data state. Thus, the basis states of the rectangular-lattice GKP states can
be expressed as

$\displaystyle\left|0\right\rangle_{GKP\left(R\right)}=\sum_{n\in\mathbb{Z}}\left|\hat{x}=2n\sqrt{\pi/R}\right\rangle,$
$\displaystyle\left|1\right\rangle_{GKP\left(R\right)}=\sum_{n\in\mathbb{Z}}\left|\hat{x}=\left(2n+1\right)\sqrt{\pi/R}\right\rangle,$
(41)

and the alternative basis states are

$\displaystyle\left|+\right\rangle_{GKP\left(R\right)}=\sum_{n\in\mathbb{Z}}\left|\hat{p}=2n\sqrt{\pi
R}\right\rangle,$
$\displaystyle\left|-\right\rangle_{GKP\left(R\right)}=\sum_{n\in\mathbb{Z}}\left|\hat{p}=\left(2n+1\right)\sqrt{\pi
R}\right\rangle,$ (42)

where $R$ is the aspect ratio of the lattice, with $R=1$ corresponding to the
square-lattice GKP states. The success probability of the biased GKP code can
be given by

$\displaystyle p_{succ}^{re}$ $\displaystyle=\sum_{n_{x},n_{p}\in
2\mathbb{Z}}\int_{\left(n_{x}-\frac{1}{2}\right)\sqrt{\frac{\pi}{R}}}^{\left(n_{x}+\frac{1}{2}\right)\sqrt{\frac{\pi}{R}}}d\xi_{x}\frac{1}{\sqrt{2\pi\sigma_{x}^{2}}}e^{\left[-\frac{\xi_{x}^{2}}{2\sigma_{x}^{2}}\right]}$
(43) $\displaystyle\ast\int_{\left(n_{p}-\frac{1}{2}\right)\sqrt{\pi
R}}^{\left(n_{p}+\frac{1}{2}\right)\sqrt{\pi
R}}d\xi_{p}\frac{1}{\sqrt{2\pi\sigma_{p}^{2}}}e^{\left[-\frac{\xi_{p}^{2}}{2\sigma_{p}^{2}}\right]}$

and the probabilities of Pauli X and Pauli Z errors are

$\displaystyle p_{X}=1-\sum_{n_{x}\in
2\mathbb{Z}}\int_{\left(n_{x}-\frac{1}{2}\right)\sqrt{\frac{\pi}{R}}}^{\left(n_{x}+\frac{1}{2}\right)\sqrt{\frac{\pi}{R}}}d\xi_{x}\frac{1}{\sqrt{2\pi\sigma_{x}^{2}}}e^{\left[-\frac{\xi_{x}^{2}}{2\sigma_{x}^{2}}\right]}$
(44) $\displaystyle p_{Z}=1-\sum_{n_{p}\in
2\mathbb{Z}}\int_{\left(n_{p}-\frac{1}{2}\right)\sqrt{\pi
R}}^{\left(n_{p}+\frac{1}{2}\right)\sqrt{\pi
R}}d\xi_{p}\frac{1}{\sqrt{2\pi\sigma_{p}^{2}}}e^{\left[-\frac{\xi_{p}^{2}}{2\sigma_{p}^{2}}\right]}$
(45)

It is noted that biased GKP state has different tolerances for Pauli Z and
Pauli X errors. While Pauli Z and Pauli Y errors can be suppressed by
increasing $R$, at the expense of increasing the Pauli X error rate, which can
be solved by combing with a classical $n$-qubit repetition encoding.

###  VI.2 Concatenated repetition code

In our scheme, repetition code is concatenated after biased GKP code. An
$n$-qubit repetition code is given by

$\left|\bar{0}\right\rangle=\left|0\right\rangle_{GKP\left(R\right)}^{\otimes
n},\qquad\left|\bar{1}\right\rangle=\left|1\right\rangle_{GKP\left(R\right)}^{\otimes
n}.$ (46)

Here data GKP states are used to represent $\left|\bar{0}\right\rangle$ and
$\left|\bar{1}\right\rangle$ and each data state has a bit-flip error rate
$p_{X}$. For this repetition code, the probability of $j$ bit-flip errors
obeys the binomial distribution

$P\left(j\right)=\left(\begin{matrix}n\\\
j\end{matrix}\right)\left(p_{X}\right)^{j}\left(1-p_{X}\right)^{n-j}$ (47)

The decoding process of repetition code is a simple majority vote and it can
correct up to $k$ bit-flip errors, where
$k=\left\lfloor\left(n-1\right)/2\right\rfloor$. Therefore, the success
probability $p\left(j\leq k\right)$ of correcting bit-flips can be obtained
as,

$P^{X}_{succ}\left(j\leq
k\right)=\sum_{j=0}^{k}\binom{n}{j}\left(p_{X}\right)^{j}\left(1-p_{X}\right)^{n-j},$
(48)

which is just a partial binomial sum containing all combinations of outcomes.
The bit-flip error probability is then given by $1-P^{X}_{succ}$.

The repetition code does not have built-in protection against any phase-flip
errors, and we cannot completely bias away quadrature phase displacement
errors. Since even number of phase-flip errors will not change the logical
state, here only odd number condition is concerned with. For an $n$-qubit
code, the probability that even numbers of Pauli Z errors occurs, can be
expressed by

$\displaystyle P^{Z}_{succ}\left(2j\right)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{\left\lfloor
n/2\right\rfloor}\binom{n}{2j}\left(p_{Z}\right)^{2j}\left(1-p_{Z}\right)^{n-2j}$
(49) $\displaystyle=$ $\displaystyle\frac{1+\left(1-2p_{Z}\right)^{n}}{2},$

so the phase-flip error probability is $1-P^{Z}_{succ}$.

###  VI.3 Calculation results

(a)

(b)

(c)

Figure 11: Error probabilities ($P_{e}$) using the biased GKP repetition code
with the repetition code $n$=1, 3, 11, 31, 51, and 101 for the one-step
single-mode (a), two-step single-mode (b), two-mode (c) gates, respectively.

We combine each error to give the overall code performance of the biaesd GKP
repetition code. In our fault-tolerant quantum computation scheme, the inner
rectangular-lattice GKP code is used to treat with the Gaussian random shift
noise that acting on data GKP state by transforming Gaussian noise into Pauli
errors, meanwhile the Pauli Z error is effectively suppressed. Then outer
repetition code is ready to combat the unsuppressed Pauli X error. Considering
the residual Pauli Z errors and the uncorrected Pauli X errors, the overall
code error probability is then given by

$P_{e}=1-P^{X}_{succ}\left(j\leq k\right)P^{Z}_{succ}\left(2j\right).$ (50)

It is worth mentioned that $R$ must be optimised depending on $n$.

Figure 12: Error probabilities ($P_{e}$) using the biased GKP repetition
encoding with the repetition code $n$=101 for the one-step single-mode
($P_{S}^{1}$), two-step single-mode ($P_{S}^{2}$), two-mode ($P_{T}$) gates,
respectively.

In our scheme, the fault-tolerant thresholds of one-step single-mode, two-step
single-mode and two-mode gate implementations are 9.4 dB, 11.5 dB, and 12.3
dB, respectively, as is shown in Fig.11(a), (b), and (c). These figures can be
categorised into three distinct regions that divided by the red and black
dashed lines, according to the code behaviour. The intersection of $n$=1 and
$n$=101 corresponds the fault-tolerant threshold. In the right region, the
biased GKP repetition code performs worse than the no-bias (square) no-
repetition condition ($n$=1). In the middle region, the biased GKP repetition
code starts to show the advantages and all repetition codes gradually perform
better than the no-bias no-repetition condition. In the left region, the
biased GKP repetition code shows absolute superiority, even for very small
repetition numbers ($n$=3). It is found that the advantage of the biased GKP
repetition code will be obvious if the squeezing level is high enough.
Moreover, the error probability can be further deduced by increasing either
the repetition number or the squeezing level. Since the universal quantum
computation contains the implementations of the above three kinds of quantum
gates, the fault tolerance threshold of the universal quantum computation
based on our scheme is 12.3 dB. The behaviors of the above three gate
implementations by using biased GKP repetition code ($n$=101) are shown in
Fig.12, which have at least two orders of magnitude improvement compared with
that in Fig.(10).

##  VII Conclusion

In this work, we investigate a complete CV quantum computation architecture,
which includes 2D spatiotemporal cluster state generation, universal gate
implementations, error correction, and fault tolerance (with squeezing
threshold 12.3 dB). Different from other works, here we obtain the squeezing
threshold by taking into account the actual gate noise caused by finite
squeezing of the generated 2D cluster state and GKP state in experiment. The
computational performance can be further improved by increasing either the
squeezing level or the repetition number, so one can finally obtain an ultra-
low error probability by using the biased GKP repetition code, which has
advantages of easier decoding and less stabiliser measurements requirement at
the qubit level. In our next work, we would like to further optimize each
steps in our quantum computation architecture, such as extending the cluster
state from 2D to 3D, reducing the gate noise by streamlining the universal
quantum gates[45], considering the analog information during GKP error
correction[30], etc. Our work provides a possible thought for realizing a
fault-tolerant measurement-based CV quantum computation in experiment.

## acknomledgments

This work was supported by National Key Research and Development Program of
China (); National Natural Science Foundation of China (NSFC) (); Natural
Science Foundation of Shanxi Province ().

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# Using the AIDA Language to
Formally Organize Scientific Claims

Tobias Kuhn  Department of Computer Science, Vrije Universiteit Amsterdam, The
Netherlands

###### Abstract

Scientific communication still mainly relies on natural language written in
scientific papers, which makes the described knowledge very difficult to
access with automatic means. We can therefore only make limited use of formal
knowledge organization methods to support researchers and other interested
parties with features such as automatic aggregations, fact checking,
consistency checking, question answering, and powerful semantic search.
Existing approaches to solve this problem by improving the scientific
communication methods have either very restricted coverage, require formal
logic skills on the side of the researchers, or depend on unreliable machine
learning for the formalization of knowledge. Here, I propose an approach to
this problem that is general, intuitive, and flexible. It is based on a unique
kind of controlled natural language, called AIDA, consisting of English
sentences that are atomic, independent, declarative, and absolute. Such
sentences can then serve as nodes in a network of scientific claims linked to
publications, researchers, and domain elements. I present here some small
studies on preliminary applications of this language. The results indicate
that it is well accepted by users and provides a good basis for the creation
of a knowledge graph of scientific findings.

## Introduction

It is increasingly difficult for scientists to keep up with the rapidly
growing literature, and science is in a genuine communication crisis [1]. Text
mining has been the favorite approach to address this problem, but results
remain far from perfect even for basic tasks, such as entity recognition [2]
and extraction of simple relations [3], and such approaches have therefore
failed to mimic human capacity to understand texts describing complex results.
As an alternative solution, annotation approaches [4] have been proposed that
require humans in the loop to manually attach formal representations and links
to existing articles. Such annotations are, however, quite complicated to
create, typically apply restrictions on what can be expressed, and can often
be understood only in the larger context of the underlying article text.

In earlier work [5], I sketched an alternative approach that goes beyond
annotation, is simple and intuitive, is fully general, and leads to
representations that can be linked to scientific articles but are completely
independent entities. Specifically, this approach is based on the simple idea
that we can use single English sentences to capture scientific findings and
hypotheses, which then form the nodes in a network of scientific knowledge. My
previous work introduced the concept of AIDA sentences [5], which are defined
as English sentences that are: Atomic: a sentence describing one thought that
cannot be further broken down in a practical way; Independent: a sentence that
can stand on its own, without external references like “this effect” or “we”;
Declarative: a complete sentence ending with a full stop that could (at least
in theory) be true or false; and Absolute: a sentence describing the core of a
claim ignoring the (un)certainty about its truth and ignoring how it was
discovered (without phrases such as “probably” or “evaluation showed that”).

The language of AIDA sentences thereby forms a Controlled Natural Language
(CNL) [6]. Its approach is based on the assumption that virtually every
scientific hypothesis can be represented as such a sentence. In the future, we
could ask from researchers to publish their results in such a format from the
start, and these AIDA sentences can then serve as nodes in a growing network
of scientific claims and as a basis for the formal representation of their
content, following our proposed vision of genuine semantic publishing [7].

Some examples of AIDA sentences are shown here:

  * •

“A combination of system and searcher biases lead search engine users to
settle on the incorrect answer to yes/no-questions around half of the time.”
(from https://doi.org/10.1145/2484028.2484053)

  * •

“Teenagers reply on average faster to emails than adults.” (from
https://doi.org/10.1145/2736277.2741130)

  * •

“Deep learning is a powerful and accurate method for automatic speech
recognition.” (from https://doi.org/10.1109/ASRU.2011.6163930,
https://doi.org/10.1109/MSP.2012.2205597, and
https://doi.org/10.1109/ICASSP.2013.6639347)

These examples illustrate the benefits of the different requirements of the
AIDA approach. Atomicity ensures that each sentence is as short and concise as
possible. Independence allows us to interpret and understand these sentences
without further context (the sentences above can be understood without looking
at the references we provided). Declarativeness gives them a common form and
allows us to categorize them as true or false, or any degree of uncertainty in
between. Absoluteness, finally, contributes to normalizing the sentences,
thereby allowing us to use the same identifier (i.e. AIDA sentence) for
reported results that only differ in their uncertainty or method of discovery,
as exemplified by the last sentence above. These degrees of certainty and
these methods of discovery are of course important too, but they are
relatively easy to record with classical formal methods and formally linked to
AIDA sentences. Various ontologies have in fact been proposed for these
aspects (e.g. [8] and [9]).

These properties make an AIDA sentence highly reusable, and we can treat it as
an anchor to formally link, for example, papers that claim or refute it. We
can also link the sentences among each other, such as stating that a given
AIDA sentence is more specific or more general than another one, or has the
same meaning. Moreover, we can allow for these AIDA sentences themselves to be
partially or fully specified in a formal logic language like RDF, thereby
allowing for a full continuum from informal over semi-formal to fully
formalized statements, as proposed in our earlier work [10].

In previous work, we also presented two studies on the manual and automatic
creation, respectively, of AIDA sentences in the biomedical field [5]. These
studies showed that manual creation of AIDA sentences by untrained researchers
as well as their automatic creation from an existing biomedical data source
can be performed in either case in an effective and accurate manner. In both
cases, about 70% of the created AIDA sentences received a perfect quality
score. In a follow-up study, we worked on the extraction of AIDA sentences
from paper abstracts with a simple rule-based approach [11].

AIDA is certainly not the first controlled natural language that aims to
improve the way how science is conducted and communicated. In fact, very first
English-based CNL, Basic English, was designed around 1930 to improve the
global communication in science [12], among other spheres such as politics and
economy. However, Basic English had no relation to formal methods and
automatic knowledge organization, but only dealt with inter-human
communication. Formally precise CNLs have been proposed more recently to
improve the communication of scientific results [13] and to provide intuitive
yet powerful query interfaces to researchers [14, 15]. These approaches have
however quite narrow application ranges that are limited by the expressiveness
of the underlying logic formalism and the coverage of the used vocabularies or
ontologies. AIDA is unique in the sense that it aims to support formal
knowledge representation, while focusing on expressiveness rather than
precision [6].

## Data

In past few years, I have been building a small dataset of hand-curated AIDA
sentences, which serves as the basis for the small studies to be introduced
below. The two studies in the biomedical field introduced above [5] provide us
with the first batch of AIDA sentences. The manual AIDA extraction study
created 51 AIDA sentences and the automatic extraction created another 189 of
them, therefore 240 AIDA sentences in total. They all come with an identifier
of the associated publication they were extracted from.

Figure 1: Part of the network of AIDA sentences and publications from the
Alzheimer’s case study. The links between AIDA claims connect a more specific
claim (bottom) to a more general one (top).

The next sets of AIDA sentences come from two small case studies on meta-
reviews in different domains. The first of these meta-reviews is a Cochrane
Library report entitled “Cholinesterase inhibitors for Alzheimer’s disease”
(https://doi.org/10.1002/14651858.CD005593), summarizing evidence and findings
from a number of publications on the topic. I manually created a network of
AIDA sentences and the relevant publications based on the information found in
the meta-review report. Figure 1 shows the main part of the resulting network,
with three levels of AIDA sentences (more general ones at the top), and the
lowest level being linked to the individual publications. The full data
contains 62 AIDA sentences and can be found
online.111https://github.com/tkuhn/aida/blob/master/usecases/alzheimers.md

The second case study targeted a less formal meta-review report published by
SPARC Europe222https://sparceurope.org/what-we-do/open-access/sparc-europe-
open-access-resources/open-access-citation-advantage-service-oaca/ on the
question of whether Open Access publications enjoy a citation advantage. This
general question can be expressed as the AIDA sentence “Open Access
publications receive on average more citations than similar publications that
are not Open Access”. Most covered publications, however, investigate a
narrower claim such as “Open Access publications _in astronomy and physics_
receive …”. Overall, this second case study created AIDA sentences for 70
publications (only one paper could not be found and had to be excluded). The
details of this study can be found online as
well.333https://github.com/tkuhn/aida/blob/master/usecases/openaccess.md

Finally, I have been building a personal collection of AIDA sentences for some
of the scientific publications I have read. This collection consists at the
moment of 287 AIDA sentences from a wide variety of scientific disciplines.
The examples shown in the introduction of this paper are from this collection.

The combined collection therefore consists at the moment of 659 AIDA sentences
(650 at the time the network study to be described below was conducted; I have
added nine entries to my personal collection since). All these AIDA sentences
were manually curated. Automatically extracted sentences were only added after
a manual check for accuracy and AIDA compliance.

## User Study

I felt that AIDA sentences could turn out to be a beneficial technology also
in the classroom setting. I have therefore used AIDA sentences for a Master
course that I have been giving at VU Amsterdam during the fall semesters since
2015. In this course, entitled “Knowledge and Media”, students are required to
read 20 given papers on topics around knowledge organization. In order to help
them remember and understand what they have read, I provided them with AIDA
sentences for each of these 20 publications and for some additional
publications that I mentioned in the lecture slides. These sentences are also
part of the personal AIDA collection introduced above.

In the end of the course, I asked the students to give me feedback on their
opinions with respect to AIDA sentences. Specifically, I asked them the
following questions with the shown answer options:

> 1\. AIDA Sentences: Were the AIDA sentences, as presented during the
> lectures and on the slides, helpful for you to understand and remember the
> content of the papers?
>
>   * •
>
> Yes, the AIDA sentences were helpful.
>
>   * •
>
> Maybe. I am not sure whether the AIDA sentences were helpful.
>
>   * •
>
> No, the AIDA sentences were not helpful.
>
>

> 2\. AIDA sentences compared to classical text summaries: Did you find the
> AIDA sentences, as presented during the lectures and on the slides, to be
> more or less useful than classical text summaries?
>
>   * •
>
> I found the AIDA sentences to be more useful than classical text summaries.
>
>   * •
>
> I found the AIDA sentences to be about as useful as classical text
> summaries.
>
>   * •
>
> I found the AIDA sentences to be less useful than classical text summaries.
>
>

In total, 128 students participated over the three years since 2015. The exact
set of AIDA sentences varied slightly between the years as I tried to optimize
the mix of papers and removed unpopular ones from the list. The students’
responses are shown in Figure 2.

[max=35,step=10,width=10cm,scale=0.8] [label=about as useful (48.4%)
2015,color=green!50]29 [label=helpful (70.3%) 2016,color=green!50]31
[label=2017,color=green!50]30 [label=2015,color=blue!30]8 [label=maybe helpful
(28.1%) 2016,color=blue!30]17 [label=2017,color=blue!30]11
[label=2015,color=red!50]0 [label=not helpful 0(1.6%) 2016,color=red!50]1
[label=2017,color=red!50]1

[max=35,step=10,width=10cm,scale=0.8] [label=2015,color=green!50]19
[label=more useful (45.3%) 2016,color=green!50]16
[label=2017,color=green!50]23 [label=2015,color=blue!30]15 [label=about as
useful (48.4%) 2016,color=blue!30]28 [label=2017,color=blue!30]19
[label=2015,color=red!50]3 [label=less useful 0(6.3%) 2016,color=red!50]5
[label=2017,color=red!50]0 number of participants

Figure 2: Responses from the participants on whether AIDA sentences were
helpful (top) and on how AIDA sentences compared to classical text summaries
(bottom)

We see that 70.3% of all participating students thought the AIDA sentences
were helpful to understand and remember the content of the papers. There is
some variation over the years, but the _helpful_ group formed a clear majority
in every single year, and over the three years only two students (out of 128)
responded with _not helpful_. If we try to boil this down to a single number
by assigning positive answers the value $+1$, _maybe_ answers the value $0$,
and negative answers the value $-1$, we get an average response of $+0.69$,
which is far in the positive range.

These results indicate that AIDA sentences are indeed helpful to a certain
extent, but they do not tell us whether this extent is large or small. The
answers of the students to the second question, comparing AIDA sentences to
classical text summaries, can give us some insights on this. The positive
answers (i.e. that AIDA sentences are more useful) still form the majority in
the years 2015 and 2017, but not in 2016 and the overall dataset, at 45.3% of
the overall responses. The majority in 2016 as well as the overall majority
responded with the neutral answer (i.e. that AIDA sentences are about as
useful as classical summaries). On the other hand, only eight out of the 128
respondents would have preferred classical text summaries. If we quantify the
overall effect again in a single number in the same manner as before, we get a
value of $+0.39$, which is still clearly in the positive range, even though
considerably less so than for the first question.

These results indicate that AIDA sentences are indeed an intuitive and
accurate method to structure scientific findings. Importantly, a vast majority
of students found the AIDA sentences to be not less useful than classical
summaries, despite the fact that they did not even get to experience some of
the core benefits of AIDA sentences, namely advanced knowledge access powered
by the formal interlinking features.

## Linking and Network Study

While AIDA sentences do not require us to formally represent their domain-
level content, they can serve as a tool and structure to support this process.
The benefit of the AIDA approach is that this formalization does not have to
happen right away (or at all), can be done by a different person (or
algorithm) at a later point in time, and allows for any degree of partial
formalization. And in the meantime, formal links on the meta-level of
statements can be established. The study presented in this section
investigates to what extent a simple kind of such post-hoc partial
formalization and linking can lead to a connected knowledge graph of
scientific findings.

The aim of this study is to automatically link the AIDA sentences from the
datasets above to the Linked Open Data cloud [16]. For that, I applied DBpedia
Spotlight [17], which is an annotation tool automatically linking text in
natural language to DBpedia [18] identifiers (which map to Wikipedia pages). I
used the DBpedia Spotlight API444http://www.dbpedia-spotlight.org/api, with
the default confidence parameter of $0.5$. This annotation process of the 650
AIDA sentences of our combined dataset led to overall 1726 DBpedia mappings,
i.e. on average 2.66 per AIDA sentence.

In order to assess the quality of these links, I first performed a manual
evaluation of a random sample of 10% of the resulting annotations (173 out of
1726). The result showed that these annotations were correct in 94.2% of the
cases, i.e. a bit more than one error per twenty annotations. The errors range
from minor differences, such as annotating the word “Japanese” in “Japanese
population” with the DBpedia term for “Japanese language”, to completely
unrelated concepts, such as mapping the mention of the gene identifier
“CHES-1” to the DBpedia entry for the “Workshop on Cryptographic Hardware and
Embedded Systems”, which happens to share the same acronym.

The correct annotations also show a broad range, with general and
straightforward annotations on the one end, such as “probability” in this
example from the Alzheimer’s study:

> The treatment of Alzheimer’s disease with one of the three cholinesterase
> inhibitors donepezil, galantamine or rivastigmine has a higher probability
> of at least one adverse event of anorexia before the end of the treatment as
> compared to a placebo treatment.

On the more specific and more complex end, the phrase “cholinesterase
inhibitors” of the example above is correctly mapped to the DBpedia term
“Acetylcholinesterase inhibitor”, which is a slightly different name for the
same thing. Moreover, also “donepezil”, “galantamine”, “rivastigmine”,
“adverse event”, “anorexia”, and “placebo” in the example above are correctly
mapped to their DBpedia identifiers.

Figure 3: Visualization of the network structure of AIDA sentences with their
associated publications (left), additionally augmented with some known domain
identifiers (middle), and with automatically detected DBpedia connections
(right). The largest connected component is shown in blue.

As a next step, we can look into the degree to which we achieved the goal of a
connected network of scientific findings. Well-connected networks allow for
effectively browsing, searching, aggregating, and clustering this conceptual
space of scientific knowledge.

We can first look at the structure of the network consisting of AIDA sentences
and the publications they link to. Unsurprisingly, this leads to a network of
many small disconnected components, as shown on the left hand side of Figure
3. The network consists of 989 nodes (615 of them unique AIDA sentences, with
identical sentences merged from the initial set of 650 sentences) forming 332
network components (i.e. internally connected sub-networks without connections
to other components). The AIDA sentences from the Alzheimer’s study form the
largest component with 62 AIDA nodes (10.1%), which are connected via the node
for the respective meta-review publication. Already before applying our
automatic linking, some of the AIDA sentences came with links to domain
entities: The sentences from our previous study on the automatic extraction in
the biomedical domain came with formal links to gene and organism identifiers.
Adding these links to the network produces a large new component, comprising
of 149 AIDA sentences (24.2%), shown in the middle part of Figure 3. The
network is, however, still dominated by many small disconnected components.

On this background, we can now assess the impact of our automatic DBpedia
linking. It introduced additional 711 nodes to the network, representing 711
distinct DBpedia concepts. The biggest component has now grown to include
almost half of all AIDA nodes (48.1%, or 296 out of 615) and the number of
components is reduced to 66. 80.1% of the components of the initial network
therefore have been merged by the added links. The network starts to show a
more complex structure, as can be seen on the right hand part of Figure 3.
About half of the AIDA sentences, therefore, can now be found by browsing
through the largest component. These network results do not form a direct
proof but can be seen as an indication that a dense and useful knowledge
structure starts appearing when applying such automatic linking methods.

All data, code, and results from the presented studies can be found
online.555https://github.com/tkuhn/aida

## Conclusions

AIDA sentences are designed to improve organization and communication of
scientific knowledge by establishing an intuitive and general formalism to
identify and link scientific claims. This formalism, in turn, can be used as a
basis for further formalization in the “upward” as well as “downward”
direction. Upward formalization can establish the different kinds of
relationships and aggregations with formal relations expressing things like
“is more general than” or “follows from”. Downward formalization can take AIDA
sentences as a anchor to attach partial or complete domain-level
representations, for example involving relations like “is a disease affected
by gene” or “correlates in human populations with”. Formal representations get
complicated at the very low as well as the very high level. The AIDA approach
boils down to the idea that the most practical method might be to start from
the middle layer of individual statements and to grow the formalization from
there in both directions, and then to see how far we get.

The preliminary results presented here indeed indicate that the approach is
promising. Students confirmed that AIDA sentences summarize research findings
in a useful manner. Our linking and network study moreover showed that such
findings can be automatically connected to Linked Data identifiers at good
accuracy and that this process leads to a dense and broad network of
scientific findings.

As future work, I plan to work on providing a publishing infrastructure for
such AIDA sentences, based on the concept and infrastructure of a Linked Data
format called nanopublications [19]. With these techniques, AIDA sentences
could be published, corrected, reviewed, linked, searched, and aggregated in a
fully decentralized and open manner. Researchers could then publish their
latest findings as AIDA sentences right away, and link it to the existing body
of scientific findings.

## References

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  * [5] T. Kuhn, P. E. Barbano, M. L. Nagy, and M. Krauthammer, “Broadening the scope of nanopublications,” in _Proceedings of ESWC_. Springer, 2013, pp. 487–501.
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