# Quantum memory error correction computation based on Chamon model

Jian Zhao  Yu-Chun Wu<EMAIL_ADDRESS>Guo-Ping Guo

###### Abstract

Quantum error correction codes play a central role in the realisation of
fault-tolerant quantum computing. Chamon model is a 3D generalization of the
toric code. The error correction computation on this model has not been
explored so far. In this work, the Chamon model is turned to a non-CSS error
correction code. Logical qubits are built by the construct of logical Pauli
operators. The property of logical operators reveals the expressions of code
distance. According to the topological properties of Chamon models, an error
elimination algorithm is proposed. Based on the error elimination algorithm,
we propose a global randomized error correction algorithm to decode Chamon
models in every single-qubit depolarized channel. This decoding algorithm is
improved by adding the pretreatment process, termed the probabilistic greedy
local algorithm, which adapts to different kinds of high-dimensional models.
The estimated threshold error rate for numerical experiment can be raised to
$4.92\%$.

###### keywords:

Chamon model , quantum error correction computation , decoding algorithm

###### PACS:

03.65.Ud, 03.67.Mn

###### MSC:

81-08 , 94B35

[inst1]Key Laboratory of Quantum Information, Chinese Academy of Sciences,
School of Physics, University of Science and Technology of China, Hefei,
Anhui, 230026, P. R. China [inst2]CAS Center For Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui, 230026, P. R. China [inst3]Hefei National Laboratory,
University of Science and Technology of China, Hefei 230088, P. R. China

[inst4]Institute of Artificial Intelligence, Hefei Comprehensive National
Science Center, Hefei, Anhui, 230088, P. R. China [inst5]Origin Quantum
Computing Hefei, Anhui 230026, P. R. China

##  1 Introduction

When processing certain computing tasks, it is reasonable to presume that the
potential effect of quantum computers may outperform classical computers[1].
Quantum computing promises exponentially or quadratically faster processing of
factoring problems[2, 3], quantum simulating problems [4] and searching
problems[5]. However, due to the objective existence of quantum noise, the
realization of practical quantum computers suffers challenge.

Quantum error correction codes play a central role in the realisation of
fault-tolerant quantum computing[6, 7]. The type of encoding determines the
arrangement of qubits in the hardware and the control of quantum gates at the
software level. Many errors correcting codes have been realised, including
Shor code[8], surface code[9, 10, 11], color code[12, 13], repetition code[14,
15, 16] and bosons quantum error-correcting code[17]. The work of high-
dimensional error-correcting codes mainly stays at the theoretical level, such
as $4$-D toric code[18], high-dimensional color code[19] and cubic code[20].
These proposed high-dimensional error correction codes with geometric topology
structures belong to CSS codes, whose stabilizers can be divided into two
classes.

A natural question is whether it is possible to analyse fault-tolerant quantum
computation in non-CSS codes with topological structures. The $XZZX$ code as a
variant of the surface code has been proposed [21]. We turn attention to high-
dimensional non-CSS code out of curiosity.

The original Chamon model is proposed to present a many-body Hamiltonian of
the system that cannot reach their ground states when the environment
temperature is closed to absolute zero, termed topologically ordered ground
states[22]. It can be regarded as a $3$-dimensional generalization of toric
codes with six-qubit nearest-neighbor interactions on a $3$-dimensional
lattice. Bravyi investigated thoroughly the types of the excitations in this
model and demonstrated the ground state degeneracy[23]. Recently, Shirley has
studied the fermionic gauge theory and fracton topological order in Chamon
models[24]. Nevertheless, it has not been considered so far for error
correction quantum computation based on Chamon models. For filling in gaps and
inspired by previous work, we turn Chamon model to a kind of error correction
code.

In this paper, a construction method of logical qubits is proposed in Chamon
model with arbitrary scales. The expression of code distance in Chamon models
is derived. According to the topological properties of Chamon models, this
paper presents a specific error correction algorithm. An improved error
correction algorithm is also proposed, which slightly raise the threshold to
$4.92\%$.

For given $N$ data qubits, one can construct a stabilizer code with $[N,k,d]$,
where $k$ is the number of constructed logical qubits and $d$ is called code
distance[25]. The sizes of $k$ and $d$ respectively reflect the ability to
accommodate the number of logical qubits and the ability to fault
tolerance[26]. In Chamon models, there exists a trade-off between $k$ and $d$.
Its asymptotic properties are manifested that when the number of physical
qubits is given $\mathcal{O}(n^{3})$, the maximum number of logical qubits can
reach $\mathcal{O}(n)$, with $d=\mathcal{O}(n)$; if the number of logical
qubits are sacrificed to a constant, the code distance $d$ can reach
$\mathcal{O}(n^{2})$. For example, if encoding constant logical qubits,
$\mathcal{O}(d^{2})$ physical qubits are needed in surface code, but in Chamon
models we only need to use $\mathcal{O}(d^{3/2})$ qubits.

Theoretically, the code distance in Chamon models with dimensions
$\alpha_{x},\alpha_{y},\alpha_{z}$ is stil an unresolved problem. For Chamon
models with $N$ qubits, the code distance $d$ was estimated by a rough upper
bound $\mathcal{O}(\sqrt{N})$[23]. We solved this problem in two cases that
the code distance
$d=\min\\{\alpha_{y}\alpha_{z},\alpha_{z}\alpha_{x},\alpha_{x}\alpha_{y}\\}$
when $\alpha_{x},\alpha_{y},\alpha_{z}$ are pairwise coprime and $d=2\alpha$
when $\alpha_{x},\alpha_{y},\alpha_{z}=\alpha$.

The discussion of paper is organized as follows. In Sec.2, the definition of
Chamon model is introduced. In Sec.3, by the constructions of logical
operators, Chamon model is turned to $4\alpha$ logical qubits and the
properties of logical operators are derived. The code distances of different
Chamon models are analysed in Sec.4. When three dimensions are coprime, the
logical operators become half-chain operators forming a loop in toric
representations, which is put in A. In Sec.5, for decoding Chamon models, the
global error correction algorithm is constructed and the threshold is
estimated, $p_{th}=4.45\%$. Consiering local properties of sparse qubits
errors, the decoding algorithm is improved, such that the threshold becomes
$4.92\%$. To organize the material, the proof of Lemma 3 is put in B and the
procedure of the probabilistic greedy local algorithm is expounded in C. Last
in Sec.6, we draw the conclusions of this paper.

Notation. The paper uses capital Roman letters $A$, $B$,$\ldots$, for matrices
or operators, lower case Roman letters $x$, $y$,$\ldots$, for vectors, and
Greek letters $\alpha$, $\beta$,$\ldots$, for scalars. In three-dimensional
space, denote the unit vectors by $e_{x}^{\pm}=(\pm 1,0,0)$,
$e_{y}^{\pm}=(0,\pm 1,0)$, $e_{z}^{\pm}=(0,0,\pm 1)$. Specially, denote the
Pauli operators by $X,Z$, and $Y=ZX$.

##  2 Chamon model

In this section, first introduce the definition of Chamon model, then analyse
the degrees of freedom in code space. The degrees of freedom determines the
maximum number of logical qubits used.

###  2.1 Building Chamon model

On three-dimensional cubic Bravais lattice, qubits are placed on the face
centers each neighbouring two cubes and the vertices each shared in eight
cubes. On each site there is only one qubit. The stabilizer acts on six
qubits, which is closest to either the midpoint of the fixed side of a cube or
the body center of a cube. According to the directions of the cube side,
called $x$-direction,$y$-direction and $z$-direction, these six qubits are
divided into three classes. Each class includes two qubits, whose sites
connecting a line parallel to one of the three directions. The stabilizers
action type is corresponding $X,Y$ and $Z$.

For convenience and rigor, it is necessary to introduce some notations. Given
a positive integer $\alpha$, the set
$\mathcal{Z}_{\alpha}=\\{0,1,\dots,\alpha-1\\}$ equipped with a
modulo-$\alpha$ addition operation forms a group. The Chamon model is
constructed by the group
$\mathcal{A}=\mathcal{Z}_{2\alpha_{x}}\times\mathcal{Z}_{2\alpha_{y}}\times\mathcal{Z}_{2\alpha_{z}}$.

######  Definition 1.

Given a group
$\mathcal{A}=\mathcal{Z}_{2\alpha_{x}}\times\mathcal{Z}_{2\alpha_{y}}\times\mathcal{Z}_{2\alpha_{z}}$,
$\alpha_{x},\alpha_{y},\alpha_{z}\in\mathcal{Z}^{+}$, let
$\mathcal{D}=\\{(x,y,z)\in\mathcal{A}|(x+y+z)/2\in\mathcal{Z}\\}$. Data qubits
are placed on each point of $\mathcal{D}$, called data qubits set. For all
$s=(s_{x},s_{y},s_{z})\in\mathcal{A}\backslash\mathcal{D}$, denote $s+e^{\pm}$
briefly by $e^{\pm}$, then the stabilizers group is generated by

$\displaystyle
S_{s}=X_{e_{x}^{+}}X_{e_{x}^{-}}Y_{e_{y}^{+}}Y_{e_{y}^{-}}Z_{e_{z}^{+}}Z_{e_{z}^{-}}.$
(1)

Denote the stabilizers group by $\mathcal{S}_{\mathcal{D}}=\langle
S_{s}\rangle$, $s\in\mathcal{A}\backslash\mathcal{D}$. Then the data qubits
set $\mathcal{D}$ and the corresponding stabilizers group
$\mathcal{S}_{\mathcal{D}}$ form the Chamon model.

In Chamon model, each qubit maps to a stabilizer one by one according to the
definition. Specifically, measure qubits can be put a unit length from every
data qubit along the positive x-axis; see Fig.1.

######  Remark 1.

In the special case when $\alpha_{y}=1$, all the stabilizers in Chamon model
$S_{s}$ are coupled with only four data qubits. The reason is the two coupled
qubits in $Y$-direction are in fact the same for each fixed stabilizer. Hence
$Y_{e_{y}^{+}}=Y_{e_{y}^{-}}$. Then the stabilizers are degenerated to an
interesting case:

$\displaystyle S_{s}=X_{e_{x}^{+}}X_{e_{x}^{-}}Z_{e_{z}^{+}}Z_{e_{z}^{-}},$

which is exactly regarded as the two-layer surface code independently with
$XZZX$ type stabilizers [21]. So is the case that $\alpha_{x}=1$ or
$\alpha_{z}=1$ due to the rotation symmetry. For the rest of this paper, only
consider general situations where $\alpha_{x},\alpha_{y},\alpha_{z}>1$.

Figure 1: The Chamon model. The stabilizer $S_{(1,0,0)}$ acts six blue qubits.
Data qubits are placed on the face centers and the vertices of the cubes. Each
side of the cubes represents one stabilizer. To be seen clearly some data
qubits are omitted.

The definition of stabilizers is rational. Note that all the stabilizers
always commute. It is trivial that $S_{s}$ and $S_{s^{\prime}}$ do not have
qubits shared. Once the stabilizers share qubits in common, they either share
only one data qubit applied to the same acting type, in which case they
commute apparently, or share two qubits, in which case the operators acting on
each shared qubit anti-commute. Thus for all the stabilizers,
$S_{s}S_{s^{\prime}}$=$S_{s^{\prime}}S_{s}$.

###  2.2 Analysing the Eigenspace

Chamon model as a kind of stabilizer code is a subspace of code space
$\mathcal{H}^{\otimes|\mathcal{D}|}$. For any quantum state
$|\psi\rangle\in\mathcal{H}^{\otimes|\mathcal{D}|}$, all stabilizers $S_{s}$
force the Hilbert space into a common eigen-subspace. Based on the measurement
results obtained by the stabilizers, the initial state is collapsed on a fixed
eigenstate. For lack of consideration for the independence of the generated
stabilizers, the number of stabilizers in Eq.(1) is
$|\mathcal{D}|=4\alpha_{x}\alpha_{y}\alpha_{z}$, $i=0,\dots,|\mathcal{D}|-1$.
The measurement outcomes can be denoted by a $|\mathcal{D}|$ dimension two-
valued vector $r=(r_{0},\dots,r_{|\mathcal{D}|-1})$,
$S_{i}|\phi\rangle=r_{i}|\phi\rangle$, $r_{i}\in\\{-1,1\\}$. Note that the
label $S_{i}$ has been remarked. For a fixed measurement $r$, the eigenspace
is denoted by

$\displaystyle\mathcal{E}_{r}=\left\\{|\phi\rangle\in\mathcal{H}^{\otimes|D|}\big{|}S_{i}|\phi\rangle=r_{i}|\phi\rangle,S_{i}\in\mathcal{S}_{\mathcal{D}},i=0,\dots,|\mathcal{D}|-1\right\\}.$

Denote the greatest common divisor of $\alpha_{x},\alpha_{y},\alpha_{z}$ by
$\alpha$. Bravyi has proved that the dimension of the eigenspace
$\mathcal{E}_{r}$ is $2^{4\alpha}$ [23]. Our purpose is to build $4\alpha$
logical qubits in any eigenspace.

##  3 Logical qubits

In this section, by the constructions of proper logical operators, the logical
qubits are built in any Chamon model.

###  3.1 The special case in $\alpha=1$

When $\alpha=1$, at most Chamon model can form $4\alpha=4$ logical qubits. The
purpose is to construct a kind of logical $X$ operators and logical $Z$
operators, denoted by $X_{L}^{(i)}$ and $Z_{L}^{(i)}$, satisfying

$\displaystyle{X_{L}^{(i)}}^{2}=$ $\displaystyle{Z_{L}^{(i)}}^{2}=I$
$\displaystyle X_{L}^{(i)}Z_{L}^{(j)}=$
$\displaystyle(-1)^{\delta_{ij}}Z_{L}^{(j)}X_{L}^{(i)},$ (2)

where $\delta_{ij}=0$ when $i\neq j$ or $\delta_{ij}=1$ when $i=j$,
$i,j=0,1,2,3$. Given an eigenspace $\mathcal{E}_{r}$, all the logical
operators must commute with $S_{i}$, which ensures that the logical operation
is performed inside the eigenspace. Identical to Pauli relationships in data
qubits, the logical $Y$ operators are defined as $Y_{L}^{(i)}\triangleq
Z_{L}^{(i)}X_{L}^{(i)}$.

It is not unique to construct proper logical operators $X_{L}$ and $Z_{L}$. A
naive thought is the constructed $X_{L}^{(i)}$ can be generated by only bit
flips $X$ on data qubits, $Z_{L}^{(i)}$ generated by only phase flips $Z$.
Next, introduce the construction process of the logical operators through
classifications of data qubits. Note that the qubits in data qubits set
$\mathcal{D}$ can be divided into four subsets, which contain one vertex
qubits set and 3 face-centered qubits sets. These qubits sets can be generated
by the points $g_{0}=(0,0,0)$, $g_{1}=(1,1,0)$, $g_{2}=(1,0,1)$ and
$g_{3}=(0,1,1)$ respectively. Define the data qubits set
$\mathcal{D}_{x}^{g_{i}},\mathcal{D}_{z}^{g_{i}}\subset\mathcal{D}$ as

$\displaystyle\mathcal{D}_{x}^{g_{i}}=\\{d|d=g_{i}+2\lambda_{x}e_{z}^{+}+2\lambda_{y}e_{y}^{+},\lambda_{x},\lambda_{y}\in\mathcal{Z}\\}$
$\displaystyle\mathcal{D}_{z}^{g_{i}}=\\{d|d=g_{i}+2\lambda_{x}e_{x}^{+}+2\lambda_{y}e_{y}^{+},\lambda_{z},\lambda_{y}\in\mathcal{Z}\\},$
(3)

where $i=0,1,2,3$. Clearly, the intersection of $\mathcal{D}_{x}^{g_{i}}$ and
$\mathcal{D}_{x}^{g_{j}}$ is empty for $i\neq j$. The union of
$\mathcal{D}_{x}^{g_{0}}$ and $\mathcal{D}_{x}^{g_{3}}$ is
$\\{d\in\mathcal{D}|d_{x}=0\\}$, denoted by $\mathcal{D}_{x=0}$ and
$\mathcal{D}_{x}^{g_{1}}\cup D_{x}^{g_{2}}=\mathcal{D}_{x=1}$, both
constituting a lattice plane parallel to $yOz$ plane, respectively. The same
considerations apply to $\mathcal{D}_{z}^{g_{i}}$. Define
$X|_{\mathcal{D}^{\prime}}=\prod_{j\in\mathcal{D}^{\prime}}X_{j}$, then

######  Theorem 1.

Given a Chamon model with $\alpha=1$, the operators
$X|_{\mathcal{D}_{x}^{g_{i}}}$ and $Z|_{\mathcal{D}_{z}^{g_{i}}}$ make the
Chamon model $4$ logical qubits.

###### Proof.

Firstly, for a fixed $i$,
$X^{2}|_{\mathcal{D}_{x}^{g_{i}}}=I=Z^{2}|_{\mathcal{D}_{z}^{g_{i}}}$. Note
that
$\mathcal{D}_{x}^{g_{i}}\cap\mathcal{D}_{z}^{g_{i}}=\\{d|d=g_{i}+2\lambda_{y}e_{y}^{+},\lambda_{y}\in\mathcal{Z}\\}$,
which contains $\alpha_{y}$ elements. Based on the assumption that
$\alpha_{y}$ is odd, hence the operators $X|_{\mathcal{D}_{x}^{g_{i}}}$ and
$Z|_{\mathcal{D}_{z}^{g_{i}}}$ anti-commute. Since all the stabilizers commute
with each other, $X|_{\mathcal{D}_{x}^{g_{i}}}$ and
$Z|_{\mathcal{D}_{z}^{g_{i}}}$ cannot be generated by stabilizers. One can
easily check $X|_{\mathcal{D}_{x}^{g_{i}}}$ and $Z|_{\mathcal{D}_{z}^{g_{i}}}$
commute with each stabilizer generator. Thus the Chamon model can be used as
logical qubits. Further since $D_{x}^{g_{i}}\cap D_{z}^{g_{j}}=\varnothing$,
the operators $X|_{\mathcal{D}_{x}^{g_{i}}}$ and
$Z|_{\mathcal{D}_{z}^{g_{j}}}$ pairwise commute, $\forall i\neq j$. It implies
the operators satisfy the conditions in Eq.(3.1). The commutation relations
hold the independence of logical operators
$X|_{\mathcal{D}_{x}^{g_{i}}},Z|_{\mathcal{D}_{z}^{g_{i}}}$, that is they
cannot be generated by other logical operators
$X|_{\mathcal{D}_{x}^{g_{j}}},Z|_{\mathcal{D}_{z}^{g_{j}}}$, $i\neq j$. We
have shown the Chamon model indeed forms 4 logical qubits.

∎

For analysing the property of logical operators here introduce some concepts.
Consider the centralizer of $\mathcal{S}$ in Pauli group denoted by

$\displaystyle\mathcal{C}(\mathcal{S})=\langle\mathcal{S},X|_{\mathcal{D}_{x}^{g_{i}}},Z|_{\mathcal{D}_{z}^{g_{i}}}\rangle,$

where $i=0,1,2,3$. Let us concern more about all the logical operators with
$\mathcal{S}$ invariance. Thus introduce the quotient group generated by

$\displaystyle\mathcal{C}(\mathcal{S})/\mathcal{S}=\langle
X|_{\mathcal{D}_{x}^{g_{i}}},Z|_{\mathcal{D}_{z}^{g_{i}}}\rangle,$

where denote the equivalence class of logical operators by the same notations.

Note that the logical operators can be translated. For example, for
$g_{0}=(0,0,0)$ and $g_{0}^{\prime}=(2,0,0)$, clearly,
$X|_{\mathcal{D}_{x}^{g_{0}}}=X|_{\mathcal{D}_{x}^{g_{0}^{\prime}}}$, due to

$\displaystyle\prod_{S_{j}\in\mathcal{S}^{\prime}}S_{j}X|_{\mathcal{D}_{x}^{g_{0}}}=X|_{\mathcal{D}_{x}^{g_{0}^{\prime}}},$

where
$\mathcal{S}^{\prime}=\\{S_{s}\in\mathcal{S}|s=e_{x}^{+}+2\lambda_{y}e_{y}^{+}+2\lambda_{z}e_{z}^{+},\lambda_{y},\lambda_{z}\in\mathcal{Z}\\}$.
Thus, It can be concluded that the logical operators
$X|_{\mathcal{D}_{x}^{g_{i}}},Z|_{\mathcal{D}_{z}^{g_{i}}}$ can be translated
across the x-axis and z-axis directions, respectively.

Coupled with half of the qubits on a certain plane, the operators
$X|_{\mathcal{D}_{x}^{g_{i}}}$ and $Z|_{\mathcal{D}_{z}^{g_{i}}}$ are termed
half-plane operators. It is clear that translated half-plane operators defined
in Thm.1 are still half-planar.

###  3.2 The general case

In this section, the logical $X$ operators and logical $Z$ operators in
$\mathcal{C}(\mathcal{S})/\mathcal{S}$ are introduced generally.

Before the constructions, we first visualize the geometry configurations of
the logical operators by an example with $\alpha_{x}=\alpha_{y}=\alpha_{z}=3$.
The greatest common divisor is $\alpha=3$. Let $g_{0,1}=(0,2,0)$ and define a
square half-plane operator in the $yOz$ plane as

$\displaystyle\mathcal{D}_{x}^{{0,1}}=\\{d|d=g_{0,1}+\lambda(e_{z}^{+}+e_{y}^{+})+\lambda^{\prime}(e_{z}^{+}+e_{y}^{-}),\lambda,\lambda^{\prime}\in\mathcal{Z}_{\alpha}\\}.$

Then apply $X$ operator on each qubit located in $\mathcal{D}_{x}^{{0,1}}$,
denoted by $X|_{\mathcal{D}_{x}^{{0,1}}}$. It is coupled with $9$ qubits,
which form a square occupying half of the $yOz$ plane. One can check it
commutes with all stabilizers; see the operator $X_{L0}^{1}$ in Fig.2.

Applying this idea to the case when $\alpha_{x}=\alpha_{y}=\alpha_{z}=\alpha$,
define $g_{i,j}=g_{i}+2je_{y}^{+}$, and let

$\displaystyle\mathcal{D}_{x}^{i,j}=$
$\displaystyle\\{d|d=g_{i,j}+\lambda(\mu_{ij}e_{z}^{+}+e_{y}^{+})+\lambda^{\prime}(\mu_{ij}e_{z}^{+}+e_{y}^{-}),\lambda,\lambda^{\prime}\in\mathcal{Z}_{\alpha}\\},$
$\displaystyle\mathcal{D}_{z}^{i,j}=$
$\displaystyle\\{d|d=g_{i,j}+\lambda(\nu_{ij}e_{x}^{+}+e_{y}^{+})+\lambda^{\prime}(\nu_{ij}e_{x}^{+}+e_{y}^{-}),\lambda,\lambda^{\prime}\in\mathcal{Z}_{\alpha}\\},$
(4)

where $i=0,1,2,3$ and $j=0,1,2,\dots,\alpha-1$,
$\mu_{ij}=(-1)^{[i/2]},\nu_{ij}=(-1)^{\lceil i/2\rceil({\rm mod}\hskip
1.39998pt2)}$. Let $X_{Li}^{j}=X|_{\mathcal{D}_{x}^{i,j}}$ and
$Z_{Li}^{j}=Z|_{\mathcal{D}_{x}^{i,j}}$, termed square logical operators.

######  Lemma 1.

Given a Chamon model with $\alpha_{x}=\alpha_{y}=\alpha_{z}=\alpha$, the
operators $X_{Li}^{j}$ and $Z_{Li}^{j}$ make the Chamon model $4\alpha$
logical qubits.

###### Proof.

Due to $D_{x}^{i,j}\cap D_{z}^{i,j}=\\{g_{i,j}\\}$, it implies that the
operators $X_{Li}^{j}$ and $Z_{Li}^{j}$ anti-commute. And these square half-
plane operators commute with all the stabilizers. Lastly to verify that
$X_{Li}^{j}$ and $Z_{Li^{\prime}}^{j^{\prime}}$ commute, one can check
$D_{x}^{i,j}\cap D_{z}^{i^{\prime},j^{\prime}}=\varnothing$,
$\forall(i,j)\neq(i^{\prime},j^{\prime})$. Thus, $X_{Li}^{j}$ and $Z_{Li}^{j}$
as logical operators make the Chamon model $4\alpha$ logical qubits.  ∎

A specific example is presented with $\alpha_{x},\alpha_{y},\alpha_{z}=3$; see
Fig.2. Note that each logical operator occupies a square half-plane.

Figure 2: The Chamon model with $\alpha_{x},\alpha_{y},\alpha_{z}=3$. Data
qubits are set with square borders in different colors, corresponding to the
logical operators. Among them, the blue border corresponds to the case $i=0$,
a lighter blue for $j=1$, darker for $j=2$. Also, the sites $g_{i,j}$ are
filled with the corresponding colors. Solid borders denote the logical $X$
operators and dotted borders denote the logical $Z$ operators. The part of the
logical operators is still omitted.

The construction of logical operators generally is available in previous ideas
of square half-plane operators. Denote the greatest common divisor of
$\alpha_{x},\alpha_{y},\alpha_{z}$ by
$\alpha=g(\alpha_{x},\alpha_{y},\alpha_{z})$. Then
$\alpha_{x}=\beta_{x}\alpha$, $\alpha_{y}=\beta_{y}\alpha$,
$\alpha_{z}=\beta_{z}\alpha$, where $\beta_{x},\beta_{y}$ and $\beta_{z}$
satisfy $g(\beta_{x},\beta_{y},\beta_{z})=1$. There must exist an odd number
in $\beta_{x},\beta_{y}$ and $\beta_{z}$. Without loss of generality, suppose
$\beta_{y}$ is odd. An intuitive thought is that the plane parallel to $yOz$
plane in Chamon model can be divided into $\beta_{y}\beta_{z}$ congruent
squares. Apply the corresponding $X$ operators on every square and find they
constitute logical $X$ operators; similarly, it is applied to logical $Z$
operators.

Based on the previous analysis, it is necessary for applying to the general
case to modify the definition in Eq.(3.2). Let

$\displaystyle
D_{x}^{i,j}=\\{d|d=g+\lambda(\mu_{ij}e_{z}^{+}+e_{y}^{+})+\lambda^{\prime}(\mu_{ij}e_{z}^{+}+e_{y}^{-}),$
$\displaystyle\lambda,\lambda^{\prime}\in\mathcal{Z}_{\alpha},g\in\mathcal{G}_{i,j}^{x}\\},$
(5)
$\displaystyle\mathcal{D}_{z}^{i,j}=\\{d|d=g+\lambda(\nu_{ij}e_{x}^{+}+e_{y}^{+})+\lambda^{\prime}(\nu_{ij}e_{x}^{+}+e_{y}^{-}),$
$\displaystyle\lambda,\lambda^{\prime}\in\mathcal{Z}_{\alpha},g\in\mathcal{G}_{i,j}^{z}\\},$
(6)

where

$\displaystyle\mathcal{G}_{i,j}^{x}=$
$\displaystyle\\{g|g=g_{i,j}+2\lambda\alpha e_{y}^{+}+2\lambda^{\prime}\alpha
e_{z}^{+},\lambda\in\mathcal{Z}_{\beta_{y}},\lambda^{\prime}\in\mathcal{Z}_{\beta_{z}}\\},$
$\displaystyle\mathcal{G}_{i,j}^{z}=$
$\displaystyle\\{g|g=g_{i,j}+2\lambda\alpha e_{y}^{+}+2\lambda^{\prime}\alpha
e_{x}^{+},\lambda\in\mathcal{Z}_{\beta_{y}},\lambda^{\prime}\in\mathcal{Z}_{\beta_{x}}\\}.$

######  Theorem 2.

Given a Chamon model with $\alpha=g(\alpha_{x},\alpha_{y},\alpha_{z})$, the
operators $X_{Li}^{j}$ and $Z_{Li}^{j}$ make the Chamon model $4\alpha$
logical qubits.

Let us go on to study other properties of square logical $X,Z$ operators.
Firstly, observe the relation of the nearest neighbor square operators. In
Fig.2, $X_{L0}^{1}$ and $X_{L3}^{1}$ share 6 qubits and the support of
$X_{L3}^{1}$ can be translated from $X_{L0}^{1}$, that is

$\displaystyle\mathcal{D}_{x}^{3,1}=\mathcal{D}_{x}^{0,1}+e_{z}^{+}+e_{y}^{+},$
(7)

where $\mathcal{D}^{\prime}=\mathcal{D}+x$ indicates
$\mathcal{D}^{\prime}=\\{d+x,d\in\mathcal{D}\\}$. It implies
$X_{L0}^{1}X_{L3}^{1}=X|_{\mathcal{D}}$,
$\mathcal{D}=\\{d|d=g_{0,1}+\lambda(e_{z}^{+}+e_{y}^{-}),\lambda\in\mathcal{Z}_{2\alpha}\\}$,
which forms a rigid chain of length $2\alpha$. Generally,

######  Corollary 1.

There exists a logical $X$ operator forming a rigid chain of length
$2\beta_{y}\beta_{z}\alpha$, similarly, the logical $Z$ operator forming a
rigid chain of length $2\beta_{y}\beta_{x}\alpha$.

Firstly for any square $X$ operator in the plane parallel to $yOz$, explain it
is a logical operator and give the expression generated by $X_{Li}^{j}$. Let

$\displaystyle\mathcal{D}_{x}^{0,2}=$
$\displaystyle\mathcal{D}_{x}^{0,1}+2e_{y}^{+},$
$\displaystyle\mathcal{D}^{\prime}\triangleq$
$\displaystyle\mathcal{D}_{x}^{0,1}+e_{z}^{-}+e_{y}^{+}.$ (8)

One can check $X_{L0}^{2}X|_{\mathcal{D}^{\prime}}=X_{L0}^{1}X_{L3}^{1}$
because they represent the same rigid chain. It implies that the square
operator $X|_{\mathcal{D}^{\prime}}$ can be generated by other 3 logical
operators, $X|_{\mathcal{D}^{\prime}}=X_{L0}^{1}X_{L3}^{1}X_{L0}^{2}$. Define
the square operators as
$X_{L0}^{1},X_{L3}^{1},X_{L0}^{2}X|_{\mathcal{D}^{\prime}}$ the adjacent $4$
square operators, the support of which satisfies the adjacent relations in
Eq.(7)(3.2). Thus, in the $yOz$ plane, for any square $X$ operator, it can
always be generated by $X_{L0}^{j},X_{L3}^{j}$, $j=0,\dots,\alpha-1$. In other
words, the product of any adjacent $4$ square operators is unit operator with
$\mathcal{S}$ invariance. Take the square half-plane logical X operator as an
example, and define square half-plane operator as
$X_{sq}(g)=X|_{\mathcal{D}_{x}^{\mu_{l}\nu_{l}}(g)}$,

$\displaystyle\mathcal{D}_{x}^{\mu_{l}\nu_{l}}(g)=\\{d|d=g+\lambda\mu_{l}(e_{z}^{+}+e_{y}^{+})+\lambda^{\prime}\nu_{l}(e_{z}^{+}+e_{y}^{-}),$
$\displaystyle\lambda,\lambda^{\prime}\in\mathcal{Z}_{\alpha},g\in\mathcal{D}\\},$

where $\mu_{l},\nu_{l}\in\\{1,-1\\}$. In the fact, the whole relations of
adjacent $4$ square operators are as follows,

######  Corollary 2.

With $\mathcal{S}$ invariance,

$\displaystyle
X_{sq}(s+e_{x}^{+})X_{sq}(s+e_{x}^{-})X_{sq}(s+e_{y}^{+})X_{sq}(s+e_{y}^{-})=I,$
$\displaystyle
X_{sq}(s+e_{z}^{+})X_{sq}(s+e_{z}^{-})X_{sq}(s+e_{x}^{+})X_{sq}(s+e_{x}^{-})=I,$
$\displaystyle
X_{sq}(s+e_{z}^{+})X_{sq}(s+e_{z}^{-})X_{sq}(s+e_{y}^{+})X_{sq}(s+e_{y}^{-})=I,$
(9)

where $s\in\mathcal{A}/\mathcal{D}$.

###### Proof.

The product of these operators either forms the envelope surface of the
stabilizers or form identical transformation. We can check the two former
transformations are exactly the product of all stabilizers in the envelope.  ∎

This result can be extended similarly to square half-plane $Y,Z$ operators.
Further, it reveals that any square half-plane operator can be generated by
$4\alpha$ square half-plane logical Pauli operators defined in Thm.2.

Let us review the half-plane operators in the special case $\alpha=1$.
Generally, when $\alpha\neq 1$, still denote the corresponding half-plane
operator by $X|_{\mathcal{D}_{x}^{g_{i}}},Z|_{\mathcal{D}_{z}^{g_{i}}}$, where
$\mathcal{D}_{x}^{g_{i}},\mathcal{D}_{z}^{g_{i}}$ have the same expressions as
Eq.(3.1). The relations between half-plane operators and square logical
operators are

######  Corollary 3.

Each half-plane operator is a logical operator and can be generated by square
logical operators. Specifically,

$\displaystyle X|_{\mathcal{D}_{x}^{g_{i}}}=\prod_{j=0}^{\alpha-1}X_{Li}^{j},$
$\displaystyle Z|_{\mathcal{D}_{z}^{g_{i}}}=\prod_{j=0}^{\alpha-1}Z_{Li}^{j}.$
(10)

###### Proof.

Consider the product of square operators $X_{L0}^{j}$, $j=0,\dots,\alpha-1$.
The support of this product operator is constrained in the $yOz$ plane. In
this plane, the data qubits in $\mathcal{D}_{x}^{g_{0}}$ are applied flip
operations odd times. The others in $\mathcal{D}_{x}^{g_{3}}$ are all flipped
even times, which implies $X|_{\mathcal{D}_{x}^{g_{0}}}=\prod X_{L0}^{j}$.
Similarly, the situations in Eq.(3) can be proved. Note that half-plane
logical operators can be translated, which means any half-plane operators can
be generated by square logical operators with $\mathcal{S}$ invariance.  ∎

The result shows that for any Chamon model, not only can $4$ logical qubits be
constructed, but in each logical qubit space there are $\alpha$ logical qubits
encoded.

##  4 The distance of Chamon model

In this section, introduce the code distance of Chamon models to analyze the
error correction capability. And there exists a trade-off between the code
distance and the number of logical qubits in one Chamon model.

######  Definition 2.

For a stabilizer code $(\mathcal{D},\mathcal{S})$, the weight of a logical
operator $L$ is defined as the minimum number of qubit bit flips or phase
flips needed in $L$ with $\mathcal{S}$ invariance, denoted by $|L|$. Then the
code distance $d$ is defined as the minimum weights of logical operators, that
is

$\displaystyle d=\min_{L\neq I,L\in\mathcal{C}(\mathcal{S})/\mathcal{S}}|L|.$

In some cases, the explicit expressions of code distance are given.

######  Theorem 3.

Suppose $\alpha_{x},\alpha_{y}$ and $\alpha_{z}$ are pairwise coprime, then
for Chamon models with linear dimensions $\alpha_{x},\alpha_{y},\alpha_{z}$,
the expression of code distance $d$ is described as

$\displaystyle
d=\min\\{\alpha_{x}\alpha_{y},\alpha_{y}\alpha_{z},\alpha_{z}\alpha_{x}\\}.$
(11)

The proof of Thm.3 is based on the toric representations; see A.

######  Theorem 4.

Given a Chamon model with linear dimensions
$\alpha_{x}=\alpha_{y}=\alpha_{z}=\alpha$, the expression of code distance $d$
is described as

$\displaystyle d=2\alpha.$ (12)

###### Proof.

Consider one data qubit flip, which always affects the nearest 4 stabilizers
in a plane, locally. There are two methods to modify the stabilizers
measurements by applying flips on the nearest qubits. One is along 3 diagonal
lines and the other is along coordinate axis. Since the logical operators
commute with all stabilizers, the affected stabilizers measurements need to be
eliminated, which means the support of logical operator needs to form a loop.
If the loop is along coordinate axis, the operator becomes half-plane operator
with weight $\alpha^{2}$.

Another scenario has been discussed in Corollary 1. One can always find the
logical chain operator arranged diagonally with weight $2\alpha$. Thus the
minimum weight is $2\alpha$.  ∎

For a general Chamon model with dimensions $\alpha_{x},\alpha_{y}$ and
$\alpha_{z}$. According to the greatest common divisor $\alpha$, the logical
operator generators can always be divided into fragments distributed in
different squares. It becomes too complicated to guarantee that the logical
operator with minimum weight can be regularly divided. However, one can easily
construct a logical chain operators according to Corollary 1, showing the code
distance $d\leq\min\\{2\beta_{y}\beta_{z}\alpha,2\beta_{y}\beta_{x}\alpha\\}$.

It reveals the relationship between the number of physical qubits, the number
of constructed logical qubits and the code distance, and there is a balance
between these parameters. When the number of physical qubits is given, the
larger the number of logical qubits, the smaller the code distance in general.
Its asymptotic properties are manifested that when the number of physical
qubits is given $\mathcal{O}(n^{3})$, the maximum number of logical qubits can
be $\mathcal{O}(n)$, and the code distance is also $\mathcal{O}(n)$; if
sacrificing the number of logical qubits to be a constant, the code distance
can reach $\mathcal{O}(n^{2})$.

##  5 Decoding Chamon model

Our goal is to use the Chamon model for fault-tolerant quantum computing. In
this section, let us introduce noise to Chamon models and propose an error
correction algorithm.

The errors influence of the Chamon model is complex. If analyzing the errors
of the measurement qubits and the errors propagation between the measurement
qubits and the data qubits, we even cannot recover from the errors. Thus, here
only analyze the memory errors in Chamon model with the perfect measurement
process.

###  5.1 Noise model

For analysing the memory errors on each data qubit, naturally introduce
depolarized channel to Chamon models. Suppose the error probability of data
qubits is $p$, then the representation for depolarized channel is

$\displaystyle\mathcal{\epsilon}(\rho)=$
$\displaystyle\frac{p^{\prime}I}{2}+(1-p^{\prime})\rho$ $\displaystyle=$
$\displaystyle(1-p)\rho+\frac{p}{3}(X\rho X+Y\rho Y+Z\rho Z),$ (13)

where $p=3p^{\prime}/4$. This channel includes $X$ flips, $Y$ flips and $Z$
flips and all the error rates are equal to $p/3$.

To obtain the error correction capability in Chamon models, it is necessary to
consider the meaning of the logical error rate and threshold error rate. The
success or failure of an error correction process is accidental, which is not
suitable for measuring the error correction ability of stabilizer codes.
Usually, use the Monte Carlo method to estimate the error probability of
logical qubits. If $N$ experiments are performed and the number of decoding
failed is $N_{r}$, then the logical error rate is defined as

$\displaystyle P_{L}=f(p)=\lim_{N\to\infty}N_{r}/N.$

It can be seen that $f$ is an increasing function. For a fixed stabilizer
code, the asymptotic properties of the logical error rate $P_{L}$ as a
function of $p$ depend on the code distance $d$. It shows that when $d$ is
relatively large and $p$ gets smaller, error correction process has a higher
probability of success, that is $P_{L}$ becomes smaller. Strictly speaking,
for sufficiently large $d$, sufficiently small $p$, $P_{L}$ can be arbitrarily
small.

######  Definition 3.

$\forall\varepsilon>0$, $\exists$ $d_{0},p^{\prime}_{th}$ such that when
$d>d_{0}$ and $p<p^{\prime}_{th}$, $|P_{L}-0|<\varepsilon$. The threshold
error rate is defined as $p_{th}=\sup\\{p^{\prime}_{th}\\}$.

###  5.2 Decoding algorithm

For estimating the threshold, a decoding algorithm is proposed to calculate
logical error rate. The core idea of our error correction algorithm is first
to eliminate errors globally and then search the logical state space.
Considering the local properties of data qubit errors, the algorithm is
improved by adding a pretreatment process.

The step of eliminating errors is challenging in the Chamon model, which is
distinct from general topological codes. For example, the most common minimum-
weight perfect-matching algorithm is applied widely to decode surface codes,
in which any given match corresponds to one way to eliminate errors. However,
the error distribution of the stabilizers in Chamon model is chaotic so that
error elimination cannot be obtained directly.

####  5.2.1 Error elimination algorithm

Define a recover Pauli operator, denoted by $P_{\rm r}$, which can eliminate
all measurement errors. In this section, a kind of expression of $P_{\rm r}$
is obtained by the proposed error elimination algorithm. Though it may not
help to explore the right logical state, the errors of the stabilizers are
eliminated.

The algorithm can be listed as $4$ steps, which suits the case when
$g(\alpha_{x},\alpha_{y})=1$.

Step 1. The first step is to apply $Z$-flips in the $2\alpha_{z}$ planes
perpendicular to the $z$ axis, and the order is towards the positive $x$ axis.
The details are as follows. In each plane, denoted briefly by $z=z_{i}$,
$x=x_{j}$ starts from $0$ to $2\alpha_{x}-3$, each time $x_{j}$ increasing one
each time, $i=0,\cdots,2\alpha-1$. Every fixed $z=z_{i}$ and $x=x_{j}$ forms a
line with $\alpha_{y}$ stabilizers. The coordinates of these stabilizers are
related to $x,z$. Start from the stabilizer with the minimum $y$ coordinate
and observe the measurement result. If the measurement result of the $k$-th
stabilizer, the coordinate denoted by $s_{i,j,k}$, changes on this line, apply
$Z$ flip to the nearest data qubit in the positive $x$-axis direction, denoted
by $Z_{s_{i,j,k}+e_{x}^{+}}$. The measurement results need to be updated after
each operation. Last, record all qubits $Z$-flips and get

$\displaystyle P_{1}=\otimes Z_{s_{i,j,k}+e_{x}^{+}}$

This step makes the errors restricted to the plane of $x=2\alpha_{x}-2$ and
$x=2\alpha_{x}-1$.

Step 2. Apply $X$-flips in the remaining two planes in Step 1, and the order
is towards the positive $z$ axis. Specifically, for any $x=x_{j}$, $z=z_{i}$
starts from $0$ to $2\alpha_{z}-3$,$j=2\alpha_{x}-2,2\alpha-1$. Similar to
Step 1, in this way $z=z_{i}$ and $x=x_{j}$ forms a line with $\alpha_{y}$
stabilizers. Start from the stabilizer with the minimum $y$ coordinate and
observe the measurement result. If the measurement of the $k$-th stabilizer
changes, apply $X$ flip to the nearest data qubit in the positive $z$-axis
direction and update the measurement results. Record the whole bit flips, that
is

$\displaystyle P_{2}=\otimes X_{s_{i,j,k}+e_{z}^{+}}$

This step makes the errors distributed in the stabilizers set
$\mathcal{S}|_{\\{x=2\alpha_{x}-2,2\alpha_{x}-1\\}\cap\\{z=2\alpha_{z}-2,2\alpha_{z}-1\\}}$.
Intuitively, the errors are restricted to four lines parallel to the $y$ axis.

Step 3. Eliminate errors on every line. Before the process of Step 3, we
supplement some properties. Similar to data qubits, the stabilizers can also
be classified into $4$ classes, denoted by $\mathcal{S}^{s_{i}}$,

$\displaystyle\mathcal{S}^{s_{i}}=\\{S_{s}|s=s_{i}+2\lambda_{x}e_{x}^{+}+2\lambda_{y}e_{y}^{+}+2\lambda_{z}e_{z}^{+},$
$\displaystyle(\lambda_{x},\lambda_{y},\lambda_{z})\in\mathcal{Z}_{\alpha_{x}}\times\mathcal{Z}_{\alpha_{y}}\times\mathcal{Z}_{\alpha_{z}}\\},$
(14)

where $s_{0}=(0,1,0),s_{1}=(1,0,0),s_{2}=(0,0,1),s_{3}=(1,1,1).$ In Chamon
models, one can check that

######  Lemma 2.

For any $i=0,1,2,3$, the number of the stabilizers with changed measurements
in $S^{s_{i}}$ is even.

Note that all the stabilizers on the line parallel to the $y$ axis belong to
exactly one class of stabilizers. By Lemma 2, the number of stabilizers whose
measurements change on each line is even.

Before explaining the process of Step 3, there is one property left to prove.
On the line parallel to the $y$-axis, consider the two stabilizers
$S_{s_{(1)}},S_{s_{(2)}}\in\mathcal{S}^{s_{i}}$. Define two stabilizers the
second-nearest neighbor stabilizers, if they obey

$\displaystyle s_{(1)}-s_{(2)}=4e_{y}^{\pm}.$

Then another useful property is shown as follows:

######  Lemma 3.

Any two second-nearest neighbor stabilizers are denoted as $S_{s_{(1)}}$ and
$S_{s_{(2)}}$, and the midpoint of $s_{(1)}$ and $s_{(2)}$ is denoted as
$s_{(0)}$. In the Chamon model, if $\alpha_{x},\alpha_{y}$ are coprime, then
there exists some $Z$ flips, whose support parallel to the $xOy$ plane,
denoted as $P_{Z}(s_{(0)})$, so that the measurements of $S_{s_{(1)}}$ and
$S_{s_{(2)}}$ are changed, and all else are unchanged.

The proof of Lemma 3 is put in B. Here let us set an example illustrating the
idea of this proof.

(a) (b) (c)

Figure 3: The product of chain $Z$ operators changes the outcomes of the
second-nearest neighbor stabilizers. (a-b) In the direction of
$e_{x}^{+}+e_{y}^{-}$ or $e_{x}^{+}+e_{y}^{+}$, the chain $Z$ operator, whose
support colored black, changes only $4$ adjacent stabilizers, colored blue.
(c)The product of two chain $Z$ operators, denoted by $P_{Z}$, guarantees the
outcomes change of the second-nearest neighbor stabilizers.

######  Example 1.

Set $\alpha_{x}=3$, $\alpha_{y}=5$, as shown in Fig.3. Consider four adjacent
stabilizers, belonging to two classes of stabilizers. There is a chain $Z$
operator, acting on the data qubits arranged diagonally in the direction of
$e_{x}^{+}+e_{y}^{+}$, which can just change the outcomes of these
stabilizers, as shown in Fig.3(a). The support has $12$ data qubits.
Similarly, consider the symmetrical case, as shown in Fig.3(b). The chain $Z$
operators in the direction of $e_{x}^{+}+e_{y}^{-}$ can also change the
measurement results of the four adjacent stabilizers. Finally, the product of
these two chain operators results in the changes of the second-nearest
neighbor stabilizers measurements, as shown in Fig.3(c).

Now start to describe Step 3. On each line formed by $\alpha_{y}$ stabilizers,
sort the stabilizers along the $y$ axis, from $0$ to $\alpha_{y}-1$. Consider
the $k$-th stabilizer, whose coordinate denoted by $s_{k}$,
$k=0,\alpha_{y}-3$. $k$ increases sequentially from $0$, and if the $k$-th
stabilizer measurement changes, then apply $P_{Z}(s_{k}+2e_{y}^{+})$. This
step recovers the measurements of the stabilizers located at the $0$-th to the
$(\alpha_{y}-3)$-th. That is, on each line there are at most two points with
errors, whose coordinates are $s_{\alpha_{y}-2}$ and $s_{\alpha_{y}-1}$.

Before Step 4, briefly analyze the situation of the two stabilizers
measurements on each line. Two stabilizers belong to one class and by Lemma 2
the measurement results either both change or both have not changed. If the
stabilizer measurements do not change, the errors on this line have been
eliminated, or perform Step 4.

Step 4. Repeat Step 3 from $s_{\alpha_{y}-2}$ with step-length 2.

The reasons for eliminating all errors are as follows. Observing the
$(\alpha_{y}-2)$-th stabilizer, if the measurement result changes, the
measurement result of the $(\alpha_{y}-1)$-th stabilizer also changes.
Therefore, according to Step 3, apply the second-nearest neighbor operator
$P_{Z}(s_{\alpha_{y}-1})$. The $0$-th and the $(\alpha_{y}-1)$-th two
stabilizers measurement results still change. Since $\alpha_{y}-1$ is an even
number, thus all errors are eliminated by applying a series of the second-
nearest neighbor operators $P_{Z}(s_{\alpha_{y}-2})\cdots P_{Z}(s_{3})\cdot
P_{Z}(s_{1})$.

Here is an example to illustrate the process in Step 4.

######  Example 2.

Considering a Chamon model with $\alpha_{y}=5$, after the first three steps,
suppose there are two errors on a line parallel to the $y$ axis. These two
errors are on the $k$-th stabilizer, $k=3,4$, as shown in Fig. 4(a). After the
second-nearest neighbor Pauli $Z$ operator $P_{Z}(s_{4})$, the errors are
distributed in the $0$-th and the $4$-th stabilizers, as shown in Fig. 4(b).
The error propagates from the $0$-th stabilizer to the $k$-th stabilizer,
where $k$ is even, until $k=\alpha_{y}-1=4$. Thus, errors on this line are
eliminated, as shown in Fig. 4(b) and Fig. 4(c).

(a) (b) (c)

Figure 4: Step 4 of error elimination algorithm. (a) Translate the error in
$s_{3}$ to $s_{0}$ by applying $P_{Z}(s_{4})$; (b-c) Translate the error in
$s_{0}$ to $s_{4}$ by applying $P_{Z}(s_{3})\cdot P_{Z}(s_{1})$.

Denote the product of all the second-nearest-neighbor Pauli $Z$ operators in
Step 3 and Step 4 as $P_{34}=\prod_{i}P_{Z}(s_{i})$. The error elimination
algorithm shows that the recover Pauli operator is $P_{\rm
r}=P_{34}P_{2}P_{1}$.

Note that this algorithm requires that there exist two dimensions coprime, so
the algorithm has a wide range of applications to Chamon models. In the
construction process of Chamon model, given the scales of the total data
qubits, such as $\mathcal{O}(n^{3})$, one can build a Chamon model in which
three dimensions are $\mathcal{O}(n)$ and pairwise coprime.

####  5.2.2 Global randomized error correction algorithm

In this section, the global randomized error correction algorithm is proposed
for Chamon models with $\alpha_{x},\alpha_{y},\alpha_{z}$ pairwise coprime. In
a numerical experiment, the basic idea is first to randomize the error
elimination algorithm $L$ times to obtain corresponding recover Pauli
operators, denoted by $P_{{\rm r}l}$, $l=0,\cdots,L-1$. Then apply half-plane
logical operators $X|_{\mathcal{D}_{x}^{g_{i}}}$ and
$Z|_{\mathcal{D}_{z}^{g_{i}}}$ to $P_{{\rm r}l}$ and choose the final recover
Pauli operator $P_{\rm r}$ with the minimum weight.

The process of randomization is described in three ways.
(1) The starting point is chosen randomly. In Step 1, randomize the starting
data qubits. The original algorithm starts from the data qubit located at
$(0,0,0)$. In fact, any position of data qubit can be used as a starting
point. After $P_{1}$, the errors are included on two adjacent planes. In Step
2, similarly, randomize the starting data qubits. The same idea suits Step
$3-4$. This is all caused by the translation invariance of the Chamon model as
a 3D torus.
(2) The order of Step 1 and Step 2 can also be adjusted. One can apply $X$
operators first, and then follow the $Z$ operators.
(3) From Step 3 to Step 4, performing the second-nearest neighbor Pauli $Z$
operator needs to limit $\alpha_{x}$ and $\alpha_{y}$ coprime. Similarly,
Since $\alpha_{y},\alpha_{z}$ are coprime, the second-nearest neighbor $X$
operator can be selected.

Finally, let us summarize the global randomized algorithm. Firstly, introduce
the standard depolarizing channel to Chamon models with perfect measurements.
Then apply the recover Pauli operator $P_{\rm r}$ to correct the data qubits
errors. For each recovered state, test for logical $X$ failure by checking
whether there are an odd or even number of $X$ flips on any planes
perpendicular to the $z$-axis. There will be an odd number of $X$-flips in the
plane perpendicular to the $z$-axis if a half-plane logical $X$ operator has
been applied, indicating a logical $X$ error. By performing many simulations,
the probability of logical $X$ error $P_{L}$ versus $p$ can be plotted, as
shown in Fig. 5. The threshold error rate is about $4.45\%$. The dimensions of
the three models denoted by $(\alpha_{x},\alpha_{y},\alpha_{z})$ are
$(2,3,2)$, $(2,3,5)$, $(2,3,7)$, $(2,3,11)$ and $(5,7,11)$, respectively.

Figure 5: Logical $X$ error rate $P_{L}$ as a function of data qubit $X$-flip
probability $p$.  Here shows various Chamon models denoted by
$\alpha_{x},\alpha_{y},\alpha_{z}$ with different colors and the threshold
error rate is $4.45\%$ for the global error correction algorithm.

Note that the algorithm does not reflect the impact of local qubits errors,
but considers all stabilizers measurements. Thus, this error correction
algorithm is global. A natural question is whether the error correction
capability of the Chamon model can be improved when local errors form is
considered. Next, introduce this idea to the Chamon models and find the
threshold can be improved slightly.

###  5.3 Improved error correction algorithm

In this section, the global error correction algorithm is improved by a
pretreatment process, termed the probabilistic greedy local algorithm.

Although the global algorithm can eliminate errors definitely, when recovering
the original logical state, it may be led to randomness due to global
searching. The basic idea for the probabilistic greedy local algorithm is to
decrease the number of false stabilizers greedily by using the property that
any qubit memory error can affect the nearest $4$ stabilizers. Details of
specific procedures are put in C. Hence the process of the improved algorithm
is listed as follows:

  * (a)

Execute the probabilistic greedy local algorithm. If the errors are all
eliminated, skip to Step (c).

  * (b)

Execute the global randomized error elimination algorithm.

  * (c)

Apply half-plane logical operators $X|_{\mathcal{D}_{x}^{g_{i}}}$ and
$Z|_{\mathcal{D}_{z}^{g_{i}}}$ to search the recover Pauli operator with the
minimum weight.

  * (d)

Use parity check method to compute $P_{L}$.

Using the improved error correction algorithm, the probability of logical $X$
error $P_{L}$ versus $p$ can be plotted, as shown in Fig. 6. The threshold
error rate of this algorithm is about $4.92\%$, slightly higher than that
computed by the global error correction algorithm.

Figure 6: Logical $X$ error rate $P_{L}$ as a function of $X$-flip probability
$p$. Chamon models in different dimensions are denoted by
$\alpha_{x},\alpha_{y},\alpha_{z}$ with different colors. The threshold error
rate for the improved error correction algorithm is about $4.92\%$.

It is clear that the improvement of the local error correction algorithm to
the global algorithm is not essential. When $p$ is small, for a given data
qubit error rate $p$, it can be found that the logical error rate $P_{L}$ of
the improved error correction algorithm will be less. However, around the
threshold, the two algorithms exhibit essentially similar performance. The
reason is that the effect of the stabilizers is excluded from consideration in
the process of recovering Pauli operators. However, in the consideration of
stabilizers, decoding becomes an exponentially searching problem. The
improvement of the decoding algorithms for Chamon models is also our future
research direction.

##  6 Conclusion

The Chamon model in this paper becomes a kind of error correction code.
Compared to previous work, this paper thoroughly explains how $4\alpha$
logical qubits are constructed. The logical operators and the related
properties are given for any Chamon model. The threshold for Chamon model is
obtained under the perfect measurements.

In the decoding process, the proposed probabilistic greedy local algorithm
adapts to different kinds of high-dimensional models. And the advantage of
this retreatment process lies in its low cost without consideration of global
properties for error correction codes. A possible future research topic is to
generalize the error correction algorithm to other high-dimensional codes.

The global decoding algorithm depends on the linear dimensions for Chamon
models, which implies topological properties in Chamon models. It cannot be
suitable for the case that any two dimensions are not coprime. Hence it
remains to be studied how to generalize the applicable situations. On the
other hand, the local error correction algorithm also has room for
improvement. For example, one can design an algorithm more exquisitely by
considering the second-nearest interactions to improve the performance of the
algorithm.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Grant No. 12034018) and Innovation Program for Quantum Science and Technology
(Grant No. 2021ZD0302300).

##  Appendix A Chamon model in toric representations

In this section, the Chamon model is represented in a torus. The support of
logical operators in this representation is a non-contractible loop on the
torus. The stabilizers can be regarded as a circle with some discrete points.
Our goal is to explore the shortest loop on the torus, corresponding to the
logical operator with the minimum weight.

###  A.1 Stabilizers and data qubits in toric representations[23]

First, introduce the idea of the representations of stabilizers. The
stabilizer and the corresponding 6 data qubits are represented by a regular
hexagon, in which the center represents the stabilizer, and each vertex
represents data qubits. A regular hexagon has three longest diagonals, whose
two endpoints represent the same measurement type, forming the three positive
directions of $x,y,z$, as shown in Fig. 7.

Figure 7: The stabilizer and corresponding hexagon representations.
$S=X_{a}X_{f}Y_{A}Y_{B}Z_{\delta_{1}}Z_{\delta_{2}}$.

Then consider the stabilizers on the $xOy$ plane. Note that these stabilizers
on the $xOy$ plane can be arranged diagonally in the $e_{x}+e_{y}$ direction.
It can be obtained that every two adjacent stabilizers share two data qubits.
Suppose $\alpha_{x}$ and $\alpha_{y}$ are coprime, then these diagonally
arranged stabilizers fill the whole $xOy$ plane under the periodic boundary
conditions. In the $e_{x}+e_{y}$ direction, two adjacent stabilizers share two
data qubits, corresponding to two regular hexagons sharing an edge. Fig. 8
shows hexagon representation when $\alpha_{x}=2,\alpha_{y}=3$. Note that one
data qubit corresponds to two vertices of the regular hexagons, because each
data qubit on the plane is coupled with four stabilizers, and its vertex is
shared by two regular hexagons. Thus the data qubits and stabilizers on the
$xOy$ plane can be represented by a row of regular hexagons.

(a)

(b)

Figure 8: The representations of stabilizers and data qubits on $xOy$ plane.
(a) Data qubits are marked with dots and letters, and stabilizers are
represented by blue edges. The data qubits used by the half-plane logical $Z$
operator in the $xOy$ plane are marked with different colors. (b) Stabilizers
are represented by the centers of regular hexagons and data qubits are
represented by vertices.

Last, generalize the method on $xOy$ plane to the entire Chamon model. A
natural idea is that the data qubits and the stabilizers in each plane
perpendicular to the $z$-axis can be represented as a row of hexagons
similarly. Thus intuitively there are $2\alpha_{z}$ rows of regular hexagons.

In this tessellation way, each vertex shares three hexagon faces, and each
face shares 6 vertices. This is inconsistent with the data qubits being
measured by the nearest 6 stabilizers, which is twice the number of shared
faces. Therefore, a straightforward and concise implementation is just to map
each data qubit to two vertices. Due to the periodic boundary conditions of
directions of $e_{z}$ and $e_{x}+e_{y}$, mathematically the Chamon model can
be represented as a torus. Here illustrate the toric representation of a
Chamon model with an example.

######  Example 3.

Given a Chamon model with $\alpha_{x}=2,\alpha_{y}=3$ and $\alpha_{z}=5$, as
shown in Fig.9, there are a total of $2\alpha_{x}\alpha_{y}$ generators of
stabilizers in the plane $z=i$, corresponding to a row of
$2\alpha_{x}\alpha_{y}$ regular hexagons, $i=0,\cdots,2\alpha_{z}-1$. When
$i=0$, the data qubits on the plane inherit the notation of Fig.8. There are a
total of $2\alpha_{z}=10$ layers, thus a total of
$4\alpha_{x}\alpha_{y}\alpha_{z}$ regular hexagons.

Figure 9: The Chamon model in toric representations. The logical operators are
displayed in three color chains, in which the blue chain and red chain
represent the logical $Z$ operator and logical $X$ operator, respectively. The
yellow one depends on the parity of $\alpha_{x}$ and $\alpha_{z}$. The
composite of colors indicates the overlapping of chains.

###  A.2 Logical Pauli operators in toric representations

Suppose $\alpha_{x},\alpha_{y}$ and $\alpha_{z}$ are pairwise coprime for
simplicity. Recall the half-plane logical $Z$ operator
$Z_{Li}=Z|_{\mathcal{D}_{z}^{g_{i}}}$, the definition of
$\mathcal{D}_{z}^{g_{i}}$ as in Eq.(3.1). As in Fig.8(b), in toric
representations, the half-plane logical operator forms a loop non-
contractible. A trivial chain is the $Z$ operator acting the data qubits
arranged diagonally, then obtain $Z_{L}=Z_{A}Z_{a}Z_{C}Z_{c}\cdots$, that is
$Z_{L0}Z_{L1}$. This kind of full plane logical operator, termed the chain
operator in toric representations, also forms a non-contractible loop.

Now give the proof of the Thm.3. Our goal is changed to explore the loop with
the minimum weights. Similarly, the half-plane logical $Z$ operator is called
a half-chain operator, coupled with half of data qubits. In the toric
representations, the length of this chain is $2\alpha_{x}\alpha_{y}$. In this
direction, any chains with less than this length will break and fail to form a
loop. Due to each data qubit corresponding to two points on the chain, the
weight of this logical Z operator is $\alpha_{x}\alpha_{y}$. As in Fig.9,
there are two other direction half-chain operators. It can be proved that one
data qubit corresponds to two points in each chain, and the weights of the
logcial operators are $\alpha_{y}\alpha_{z}$ and $\alpha_{z}\alpha_{x}$,
respectively when $\alpha_{x},\alpha_{y}$ and $\alpha_{z}$ are coprime. In
three directions half-chain operators are the logical operators with minimum
weight in each direction. Thus, the code distance is the minimum number of
$\alpha_{x}\alpha_{y},\alpha_{y}\alpha_{z},\alpha_{z}\alpha_{x}$ for Chamon
model with pairwise coprime linear dimensions.

##  Appendix B The proof of Lemma 3

From translation symmetry, let $s_{(1)}=(0,1,0)$, and set $s_{(2)}=(0,5,0)$.
Our goal is to find a set of data qubits on the $xOy$ plane, such that after
applying $Z$ operators to all data qubits in the set, only the outcomes of
$S_{s_{(1)}}$ and $S_{s_{(2)}}$ change. By $\alpha_{x}$ and $\alpha_{y}$
coprime, let

$\displaystyle n_{xy}=\min_{n\in Z^{+}}\\{n|n\alpha_{x}({\rm
mod}\alpha_{y})=1\\}.$

In Example 1, $n_{xy}=2$. The geometric meaning of $n_{xy}$ in the chain $Z$
operator is the number of basic units, whose length is $2\alpha_{x}$. Then the
weights of two chain Pauli $Z$ operators are $2\alpha_{x}n_{xy}$. Define the
supports of two chain Pauli $Z$ operators as

$\displaystyle\mathcal{D}_{z}^{++}=$
$\displaystyle\\{d|d=s_{(0)}+e_{x}^{+}+\lambda(e_{x}^{+}+e_{y}^{+}),\lambda\in\mathcal{Z}_{2\alpha_{x}n_{xy}-1}\\},$
$\displaystyle\mathcal{D}_{z}^{+-}=$
$\displaystyle\\{d|d=s_{(0)}+e_{x}^{+}+\lambda(e_{x}^{+}-e_{y}^{+}),\lambda\in\mathcal{Z}_{2\alpha_{x}n_{xy}-1}\\}.$

Then the Pauli $Z$ operator satisfying the Lemma 3 is constructed, namely

$\displaystyle P_{Z}(s_{(0)})=Z|_{\mathcal{D}_{z}^{++}}\cdot
Z|_{\mathcal{D}_{z}^{+-}}.$

Due to the translation invariance, any two second-nearest neighbor stabilizers
correspond to a Pauli $Z$ operator. In the proof of this lemma, not only is
the existence of $P_{Z}(s_{(0)})$ proved but the explicit expression is
obtained.

##  Appendix C The probabilistic greedy local algorithm

The procedures of the greedy local algorithm for Chamon model are as follows:
Step 1. Assignment. A three-dimensional vector, recorded as
$w=(w_{x},w_{y},w_{z})$, is attached to each data qubit, representing three
weights, where $w_{x}$ is defined as the number of errors of four adjacent
stabilizers located in the plane perpendicular to the $x$ axis, so $1\leq
w_{x}\leq 4$. Each data qubit also corresponds to an auxiliary three-
dimensional vector with initialized value, denoted by $v=(0,0,0)$. Similarly,
$w_{y}$ and $w_{z}$ have the same meanings.
Step 2. Eliminating $w_{p}=4$. Read the weights $w_{p}$ of data qubits in
order, $p=x,y,z$. If $w_{p}=4$, then apply the Pauli operator $P$ to the
corresponding data qubit and update the weights of itself and the adjacent
data qubits. Return to Step 2, unless all weights of data qubits $w_{p}<4$.
Check whether the stabilizers errors are eliminated. If they have been
eliminated, the algorithm succeeds, or continue to Step 3.
Step 3. Eliminating $w_{p}=3$. Read the weights $w_{p}$ of data qubits in
order, $p=x,y,z$. If $w_{p}=3$, then apply the Pauli operator $P$ to the
corresponding data qubit, update weights and return to Step 2. If all weights
of data qubits $w_{p}<3$, then check whether the stabilizers errors are
eliminated. If errors have been eliminated, the algorithm succeeds, or
continue to Step 4.
Step 4. Eliminating $w_{p}=2$. Read the weights $w_{p}$ of data qubits in
order, $p=x,y,z$. If $w_{p}=2$ and $v_{p}=0$, then apply the Pauli operator
$P$ to the corresponding data qubit, update weights, record the corresponding
$v_{p}=1$, and return to Step 2. If not, claim the algorithm fails.

If the algorithm ends at Step 2 or Step 3, the algorithm succeeds, or the
algorithm fails. Especially, the greedy local algorithm cannot handle the case
that all the weights $v_{p}\leq 1$.

In Step 2 and Step 3, the number of error stabilizers has been decreasing.
Thus, the computational complexity is $\mathcal{O}(N^{2})$, where $N$ is the
number of data qubits. In Step 4, the record of $v$ can avoid the infinite
loop of the algorithm. And in the worst case, all the data qubits have
$v_{p}=1$, which means Step 4 is executed $\mathcal{O}(N)$ times. Thus the
complexity for the probabilistic greedy local algorithm is
$\mathcal{O}(N^{3})$.

## References

  * [1] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, S. Lloyd, Quantum machine learning, Nature 549 (7671) (2017) 195–202.
  * [2] P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in: Proceedings 35th annual symposium on foundations of computer science, Ieee, 1994, pp. 124–134.
  * [3] T. Monz, D. Nigg, E. A. Martinez, M. F. Brandl, P. Schindler, R. Rines, S. X. Wang, I. L. Chuang, R. Blatt, Realization of a scalable shor algorithm, Science 351 (6277) (2016) 1068–1070.
  * [4] S. Lloyd, Universal quantum simulators, Science 273 (5278) (1996) 1073–1078.
  * [5] L. K. Grover, A fast quantum mechanical algorithm for database search, in: Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, 1996, pp. 212–219.
  * [6] B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics 87 (2) (2015) 307.
  * [7] J. Roffe, Quantum error correction: an introductory guide, Contemporary Physics 60 (3) (2019) 226–245.
  * [8] Y.-H. Luo, M.-C. Chen, M. Erhard, H.-S. Zhong, D. Wu, H.-Y. Tang, Q. Zhao, X.-L. Wang, K. Fujii, L. Li, et al., Quantum teleportation of physical qubits into logical code spaces, Proceedings of the National Academy of Sciences 118 (36) (2021).
  * [9] C. K. Andersen, A. Remm, S. Lazar, S. Krinner, N. Lacroix, G. J. Norris, M. Gabureac, C. Eichler, A. Wallraff, Repeated quantum error detection in a surface code, Nature Physics 16 (8) (2020) 875–880.
  * [10] G. Q. AI, Exponential suppression of bit or phase errors with cyclic error correction, Nature 595 (7867) (2021) 383.
  * [11] A. Erhard, H. Poulsen Nautrup, M. Meth, L. Postler, R. Stricker, M. Stadler, V. Negnevitsky, M. Ringbauer, P. Schindler, H. J. Briegel, et al., Entangling logical qubits with lattice surgery, Nature 589 (7841) (2021) 220–224.
  * [12] D. Nigg, M. Mueller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, R. Blatt, Quantum computations on a topologically encoded qubit, Science 345 (6194) (2014) 302–305.
  * [13] C. Ryan-Anderson, J. Bohnet, K. Lee, D. Gresh, A. Hankin, J. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. Brown, et al., Realization of real-time fault-tolerant quantum error correction, Physical Review X 11 (4) (2021) 041058.
  * [14] J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett, R. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, et al., Realization of quantum error correction, Nature 432 (7017) (2004) 602–605.
  * [15] P. Schindler, J. T. Barreiro, T. Monz, V. Nebendahl, D. Nigg, M. Chwalla, M. Hennrich, R. Blatt, Experimental repetitive quantum error correction, Science 332 (6033) (2011) 1059–1061.
  * [16] J. R. Wootton, D. Loss, Repetition code of 15 qubits, Physical Review A 97 (5) (2018) 052313.
  * [17] P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, et al., Quantum error correction of a qubit encoded in grid states of an oscillator, Nature 584 (7821) (2020) 368–372.
  * [18] E. Dennis, A. Kitaev, A. Landahl, J. Preskill, Topological quantum memory, Journal of Mathematical Physics 43 (9) (2002) 4452–4505.
  * [19] C. Castelnovo, C. Chamon, Topological order in a three-dimensional toric code at finite temperature, Physical Review B 78 (15) (2008) 155120.
  * [20] S. Bravyi, J. Haah, Analytic and numerical demonstration of quantum self-correction in the 3d cubic code, arXiv preprint arXiv:1112.3252 (2011).
  * [21] J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, B. J. Brown, The xzzx surface code, Nature communications 12 (1) (2021) 1–12.
  * [22] C. Chamon, Quantum glassiness in strongly correlated clean systems: An example of topological overprotection, Physical review letters 94 (4) (2005) 040402.
  * [23] S. Bravyi, B. Leemhuis, B. M. Terhal, Topological order in an exactly solvable 3d spin model, Annals of Physics 326 (4) (2011) 839–866.
  * [24] W. Shirley, X. Liu, A. Dua, Emergent fermionic gauge theory and foliated fracton order in the chamon model, Physical Review B 107 (3) (2023) 035136.
  * [25] M. A. Nielsen, I. Chuang, Quantum computation and quantum information (2002).
  * [26] E. Knill, R. Laflamme, L. Viola, Theory of quantum error correction for general noise, Physical Review Letters 84 (11) (2000) 2525.