# Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum
Gilbert-Varshamov Bound Asymptotically

Jihao Fan, Jun Li, Ya Wang, Yonghui Li, Min-Hsiu Hsieh, and Jiangfeng Du This
work was supported in part by the National Natural Science Foundation of China
(No. 61802175), in part by the National Natural Science Foundation of China
(No. 61872184), in part by the Fundamental Research Funds for the Central
Universities (No. 30921013104), and in part by Future Network Grant of
Provincial Education Board in Jiangsu. _(Corresponding authors: Jun Li; Min-
Hsiu Hsieh.)_ J. Fan is with School of Cyber Science and Engineering, Nanjing
University of Science and Technology, Nanjing 210094, China (e-mail:
jihao.fan@outlook.com).J. Li is with School of Electronic and Optical
Engineering, Nanjing University of Science and Technology, Nanjing 210094,
China (e-mail: jun.li@njust.edu.cn). Y. Wang and J. Du are with CAS Key
Laboratory of Microscale Magnetic Resonance and Department of Modern Physics,
University of Science and Technology of China, Hefei 230026, China (e-mail:
{ywustc, qcmr}@ustc.edu.cn).Y. Li is with School of Electrical and Information
Engineering, the University of Sydney, Sydney, NSW 2006, Australia (e-mail:
yonghui.li@sydney.edu.au).M.H. Hsieh is with Quantum Computing Research
Center, Hon Hai Research Institute, Taipei City 114, Taiwan (e-mail: min-
hsiu.hsieh@foxconn.com).

###### Abstract

In this paper, we utilize a concatenation scheme to construct new families of
quantum error correction codes achieving the quantum Gilbert-Varshamov (GV)
bound asymptotically. We concatenate alternant codes with any linear code
achieving the classical GV bound to construct Calderbank-Shor-Steane (CSS)
codes. We show that the concatenated code can achieve the quantum GV bound
asymptotically and can approach the Hashing bound for asymmetric Pauli
channels. By combing Steane’s enlargement construction of CSS codes, we derive
a family of enlarged stabilizer codes achieving the quantum GV bound for
enlarged CSS codes asymptotically. As applications, we derive two families of
fast encodable and decodable CSS codes with parameters
$\mathscr{Q}_{1}=[[N,\Omega(\sqrt{N}),\Omega(\sqrt{N})]],$ and
$\mathscr{Q}_{2}=[[N,\Omega(N/\log N),\Omega(N/\log N)/\Omega(\log N)]].$ We
show that $\mathscr{Q}_{1}$ can be encoded very efficiently by circuits of
size $O(N)$ and depth $O(\sqrt{N})$. For an input error syndrome,
$\mathscr{Q}_{1}$ can correct any adversarial error of weight up to half the
minimum distance bound in $O(N)$ time. $\mathscr{Q}_{1}$ can also be decoded
in parallel in $O(\sqrt{N})$ time by using $O(\sqrt{N})$ classical processors.
For an input error syndrome, we proved that $\mathscr{Q}_{2}$ can correct a
linear number of ${X}$-errors with high probability and an almost linear
number of ${Z}$-errors in $O(N)$ time. Moreover, $\mathscr{Q}_{2}$ can be
decoded in parallel in $O(\log(N))$ time by using $O(N)$ classical processors.

###### Index Terms:

Stabilizer code, Calderbank-Shor-Steane code, asymmetric quantum code, quantum
Gilbert-Varshamov bound, Pauli Channel

##  I Introduction

Quantum systems are vulnerable to the environment noise induced by
_decoherence_ , which is one of the biggest obstacles in quantum information
processing. Similar to classical systems, one feasible solution is to exploit
quantum error correction codes (QECCs) to encode the primitive quantum
information into a larger quantum state. It is widely believed that QECCs are
necessary in realizing long-term quantum communications and in building fault-
tolerant quantum computers [1, 2]. The construction and design of QECCs with
excellent performance is thus one significant task. QECCs can be constructed
from classical linear codes, e.g., by using the stabilizer formalism [3] or
the Calderbank-Shor-Steane (CSS) construction [4, 5]. But it is not an easy
task to do that straightforwardly since an additional dual-containing
constraint is needed.

How to construct asymptotically good QECCs with positive rates and linear
distances, is one of the key problems in quantum coding theory [5, 6]. The
quantum Gilbert-Varshamov (GV) bound is an important lower bound that promises
the existence of such good quantum codes [5, 7, 8, 9, 10, 11]. Compared with
classical codes, it is usually much more challenging to obtain quantum codes
that can attain the quantum GV bound due to the dual-containing constraint
[3]. Therefore very few types of QECCs can attain the quantum GV bound up to
now. In [5], it was shown that quantum codes derived from dual-containing
codes can attain the quantum GV bound for CSS codes. In [5], random codes were
shown to attain the quantum GV bound for binary stabilizer codes. In [9],
stabilizer codes derived from self-orthogonal quaternary codes were shown to
attain the quantum GV bound for binary stabilizer codes. In [12],
asymptotically good concatenated quantum codes (CQCs) were constructed. In
[13], it was shown that CQCs can attain the quantum GV bound for general
stabilizer codes asymptotically. A finite version of the quantum GV bound was
given in [14]. In [11], quantum GV bounds for asymmetric quantum codes (AQCs)
were given. All the known asymptotic results about quantum GV bound rely on
some random code arguments which mean that there is little structure in them.

It should be noted that, a unique phenomenon in quantum coding theory, called
_error degeneracy_ , exists, which makes degenerate codes more powerful than
nondegenerate ones [15, 16]. In an extreme case, it is possible to construct
asymptotically good quantum codes from asymptotically bad classical codes. We
refer to such codes as _extremely_ degenerate codes. For example, the qLDPC
conjecture asks whether there exist families of asymptotically good quantum
low-density parity-check (qLDPC) codes [17, 18, 19]. Recently, it was shown
that asymptotically good qLDPC codes exist with high probabilities [20, 21]
and the resultant qLDPC codes are extremely degenerate. The previous results
about quantum GV bounds usually rely on asymptotically good classical linear
codes. It was unknown whether there exist extremely degenerate codes derived
from bad classical codes can attain the quantum GV bound.

In [22], an enlargement of the CSS construction was proposed to construct more
efficient stabilizer codes than the standard CSS codes. In [23], the
enlargement was generalized to nonbinary situations. Several asymptotic bounds
for enlarged CSS codes were also given in [22, 23]. Then in [24, 25, 26],
asymptotically good binary quantum algebraic geometry (AG) codes were
constructed by using enlargement of the CSS construction. In [27],
asymptotically good concatenated quantum AG codes were constructed by using
the CSS construction and its enlargement, and these codes can be decoded
efficiently in polynomial time. However, whether binary enlarged CSS codes can
attain the corresponding quantum GV bound is unknown. On the other hand, it
should be noted that quantum AG codes over sufficiently large field size can
even beat the quantum GV bound [28, 29, 30]. However, for codes over small
field size, e.g., binary codes, quantum GV bound is much better than
asymptotic bounds from quantum AG codes.

In addition to the construction of QECCs with good parameters, practical QECCs
need to equip with both efficient encoders and decoders. If the syndrome
decoding is slower than the error accumulations, additional noise will be
introduced during the syndrome decoding [31]. However, error degeneracy does
not simplify the syndrome decoding of QECCs [32]. In contrast, it makes most
decoding methods that are efficient for classical codes fail to fully decode
QECCs, e.g., the belief-propagation (BP) algorithm [33]. How to utilize error
degeneracy to correct more errors is an attractive and crucial issue, and its
importance, in our opinion, is no less than the construction of good QECCs.

On the other hand, the encoding of QECCs is also significant but it has
received much less attention compared to the decoding. Noise not only occurs
in the transmission and decoding, but also in the encoding circuit [6]. Large-
scale encoding circuit and deep encoding depth make the risk of qubits
suffering from faulty gates increase rapidly. Consequently, the qubits might
be corrupted even before the transmission and the computation. The encoding
circuit of stabilizer codes of length $N$ with $g$ stabilizer generators
generally has $O(gN)$ gates and $O(N)$ depth [34]. But in many practical
applications, the encoding circuits need to be with linear size and sublinear
depth [35, 36].

In this work, we combine classical alternant codes with some other linear
codes to construct QECCs achieving the quantum GV bound asymptotically. We use
classical alternant codes as the outer code and use any linear code achieving
the classical GV bound as the inner code to do the concatenation. We call the
constructed codes as partially concatenated CSS (PC-CSS) codes since we only
introduce concatenation in the $X$-stabilizers (or the $Z$-stabilizers). We
show that PC-CSS codes can attain quantum GV bounds for CSS codes and
asymmetric CSS codes asymptotically. Here, our scheme can use any linear code
achieving the classical GV bound to do the concatenation. Then PC-CSS codes
can achieve the quantum GV bound asymptotically. In particular, if we use low-
density parity-check (LDPC) codes as the component codes, we show that error
degeneracy can greatly improve the minimum distance of PC-CSS codes. In such
case, the minimum distance of PC-CSS codes without considering error
degeneracy is constant to the block length. Yet, degenerate PC-CSS codes can
achieve the quantum GV bound whose minimum distance is linear to the block
length. Thus PC-CSS codes are extremely degenerate and error degeneracy has
large advantages in constructing asymptotically good quantum codes. It should
be noted that PC-CSS codes are not qLDPC codes since only $Z$-stabilizers of
PC-CSS codes can satisfy the qLDPC constraints [37]. It is known that
asymptotically good qLDPC codes has been shown to exist with high
probabilities [20, 21]. However, the asymptotic bound of qLDPC codes in [20,
21] is much weaker than quantum GV bound. We show that PC-CSS codes can not
only attain the quantum GV bound but also can approach the capacity of Pauli
channels with large asymmetries.

TABLE I: Summary of Asymptotic Bounds for Different Quantum Codes. Let $p$ be a prime and let $m>0$ be an integer. Denote by $q=p^{m}$. For a stabilizer code with parameters $Q=[[n,k,d]]_{q}$, we denote by $R=k/n$ and $\delta=d/n$. For a nonstabilizer code with parameters $Q=((n,K,d))_{q}$, we denote by $R=(\log_{q}K)/n$ and $\delta=d/n$. For an AQC with parameters $Q=[[n,k,d_{X}/d_{Z}]]$, we denote by $R=k/n$, $\delta_{X}=d_{X}/n$, and $\delta_{Z}=d_{Z}/n$. Nos. | References |  Asymptotic Bounds
---|---|---
1 | [3] | $R\geq 1-2H_{2}(\delta)-O(1)$
2 | [5, 9] | $R\geq 1-2H_{4}(\delta)-O(1)$
3 | [13] |  $R\geq 1-2H_{q^{2}}(\delta)-O(1)$
4 | [24] |  $R=1-\frac{1}{2^{m-1}-1}-\frac{10}{3}m\delta$, $\delta\in[\delta_{m},\delta_{m-1}]$, $m=3,4,5,\ldots,$ $\delta_{2}=1/18$, $\delta_{3}=3/56$, and $\delta_{m}=\frac{3}{5}\frac{2^{m-2}}{(2^{m}-1)((2^{m-1}-1))}$, $m=4,5,6,\ldots$
5 | [25] |  $R=1-\frac{10}{3}m\delta-\frac{2}{2^{m}-1}$, $0<\delta\leq\frac{1}{2m}(\frac{1}{2}-\frac{1}{2^{m}-1})$
6 | [26] |  $R=3t(\delta_{t}-\delta)$, $\delta_{t}=\frac{2}{3}\frac{2^{t}-3}{3(2t+1)(2^{t}-1)}$, $t\geq 3,$ $0<\delta<\delta_{t}$
7 | [12] |  $R\geq\frac{1}{3}(1-4\delta/a)$, $a=H^{-1}_{4}(1/6)$
8 | [27] |  $R\geq\frac{t}{t+1}-\frac{2t}{(t+1)(q^{t}-1)}-2t\delta$, $t>0$ is an integer
9 | [27] |  $R\geq\frac{k_{0}}{n_{0}}(1-2\gamma-\frac{(2q+1)n_{0}}{(q+1)d_{0}}\delta),$ $0\leq\gamma\leq(q^{k_{0}/2}-2)^{-1}$, $[[n_{0},k_{0},d_{0}]]_{q}$ is a quantum code
10 | [28, 29] |  $R\geq 1-2\delta-\frac{2}{\sqrt{q}-1}$, $q$ is a square
11 | [28] |  $R\geq 1-2\delta-\frac{2}{\sqrt{q}-1}+\log_{q}(1+\frac{1}{q^{3}})$, $q$ is a square
12 | [21] |  $0<R<1$, $0<\delta<\frac{1}{2c^{7/2}}$, $c>1$ is an integer
13 | [11] |  $R\geq 1-H_{q}(\delta_{X})-H_{q}(\delta_{Z})-O(1)$, $0\leq\delta_{X}\leq 1-1/q$, $0\leq\delta_{Z}\leq 1-1/q$
14 | [30] |  $R\geq 1-\delta_{X}-\delta_{Z}-\frac{2}{\sqrt{q}-1}+\log_{q}(1+\frac{1}{q^{3}})$, $q$ is a square
15 | Theorem 1 | $R\geq 1-2H_{q}(\delta)$
16 | Corollary 1 |  $R\geq 1-H_{q}(\delta_{X})-H_{q}(\delta_{X})-O(1)$, $0\leq\delta_{X}\leq 1-1/q$, $0\leq\delta_{Z}\leq 1-1/q$
17 | Theorem 3 |  $R\geq 1-H_{q}(\delta)-H_{q}(\frac{q}{q+1}\delta)-O(1)$

As a special case, we construct a family of asymmetric PC-CSS codes with
parameters $\mathscr{Q}=[[N,\Omega(N/n_{0}),d_{Z}\geq\Omega(N/n_{0})/d_{X}\geq
n_{0}]]$ by using expander codes as one of the component codes, where
$n_{0}>1$ and $N>1$ are integers and $n_{0}|N$. Let $n_{0}=\Theta(\sqrt{N})$,
we can derive a family of quantum codes with parameters
$\mathscr{Q}_{1}=[[N,\Omega(\sqrt{N}),\Omega(\sqrt{N})]].$ For an input error
syndrome, $\mathscr{Q}_{1}$ can correct all errors of weight smaller than half
the minimum distance bound in $O(N)$ time. $\mathscr{Q}_{1}$ can also be
decoded in parallel in $O(\sqrt{N})$ time by using $O(\sqrt{N})$ classical
processors. We show that $\mathscr{Q}_{1}$ can be fast encoded by a circuit of
size $O(N)$ and depth $O(\sqrt{N})$. Moreover, $\mathscr{Q}_{1}$ can correct a
random $Z$-error with very high probability provided $p_{z}<25\%$.

If let $n_{0}=\Theta(\log N)$, then we can derive a family of AQCs with
parameters $\mathscr{Q}_{2}=[[N,\Omega(N/\log N),\Omega(N/\log N)/\Omega(\log
N)]].$ For an input error syndrome, $\mathscr{Q}_{2}$ can correct any random
${X}$-error with high probability and correct an almost linear number of
${Z}$-errors in $O(N)$ time. Further, we show that $\mathscr{Q}_{2}$ can be
decoded in parallel in $O(\log(N))$ time by using $O(N)$ classical processors.
While fault-tolerant quantum computation [38] and communication [39, 6] prefer
fast encodale and decodable codings, $\mathscr{Q}_{1}$ and $\mathscr{Q}_{2}$
are thus practical and beneficial for the future use.

We further combine our concatenation scheme with the enlarged CSS construction
in [22, 23]. We show that the enlarged quantum codes can attain the quantum GV
bound for enlarged CSS codes. In Table I, we give a summary of asymptotic
bounds for different quantum codes.

This paper is organized as follows. We begin in Section II with Preliminaries
on classical linear codes and quantum stabilizer codes. In Section III, we
present the partial concatenation construction of CSS codes and show that PC-
CSS codes can attain the quantum GV bound for CSS codes and AQCs
asymptotically. In Section IV, we give a family of fast encodable and
decodable PC-CSS codes. In Section V, we combine our concatenation scheme with
Steane’s enlarged CSS construction. We show that the enlarged quantum codes
can attain the quantum GV bound for enlarged CSS codes asymptotically. The
discussions and conclusions are given in Section VI.

##  II Preliminaries

We shortly introduce some needed knowledge about classical linear codes and
quantum stabilizer codes, for more details, see the literatures, e.g., [40, 5,
3, 41, 42]. In this paper, we usually neglect the index in the code parameters
for binary linear codes and binary stabilizer codes provided there is no
ambiguity.

###  II-1 Classical Alternant Codes

Let $p\geq 2$ be a prime and denote by $GF(p)$ a prime field. Let
$q=p^{m_{0}}$ be a power of $p$, where $m_{0}>0$ is an integer. Let $GF(q)$ be
the Galois field of size $q$ and denote by $GF(q^{m})$ a field extension of
$GF(q)$, where $m>0$ is an integer. Define the trace operation from $GF(q)$ to
$GF(p)$ as $\textrm{Tr}(\alpha)=\sum_{i=0}^{m_{0}-1}\alpha^{p^{i}}$, where
$\alpha\in GF(q)$. We review some known results about classical alternant
codes which will be used in the construction of quantum codes. Let
$\alpha_{1},\alpha_{2},\ldots,\alpha_{n}$ be $n$ distinct elements of
$GF(q^{m})$ and let $v_{1},v_{2},\ldots,v_{n}$ be $n$ nonzero elements of
$GF(q^{m})$, where $2\leq n\leq q^{m}$. Denote by
$\mathbf{a}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ and
$\mathbf{v}=(v_{1},v_{2},\ldots,v_{n})$. Denote the ring of polynomials with
coefficients in finite field $GF(q^{m})$ by $GF(q^{m})[x]$. For any polynomial
$F(x)=\sum_{i=0}^{l}c_{i}x^{i}\in GF(q^{m})[x]$, the degree of $F(x)$ is
denoted by $\deg F(x)=l$. For any $1\leq k\leq n-1$, the generalized Reed-
Solomon (GRS) code $\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})$ is defined by

$\displaystyle\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})\equiv\big{\\{}(v_{1}F(\alpha_{1}),v_{2}F(\alpha_{2}),\ldots,v_{n}F(\alpha_{n}))\
|$ $\displaystyle\ F(x)\in GF(q^{m})[x],\ \deg F(x)<k\big{\\}}.$ (1)

$\nrightarrow$ The parameters of $\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})$ are
given by $[n,k,n-k+1]_{q^{m}},$ and GRS codes are maximum-distance-separable
(MDS) codes which can attain the Singleton bound in [40]. Let $r=n-k$. The
dual of a GRS code is also a GRS code, i.e.,
$\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})^{\bot}=\textrm{GRS}_{r}(\mathbf{a},\mathbf{y})$,
where $\mathbf{y}=(y_{1},y_{2},\ldots,y_{n})$ and $y_{i}\cdot
v_{i}=1/\prod_{j\neq i}(\alpha_{i}-\alpha_{j})$, for $1\leq i\leq n$. The
parity-check matrix of $\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})$ is given by

$H_{\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})}=\left(\begin{array}[]{cccc}y_{1}&y_{2}&\cdots&y_{n}\\\
\alpha_{1}y_{1}&\alpha_{2}y_{2}&\cdots&\alpha_{n}y_{n}\\\
\vdots&\vdots&\vdots&\vdots\\\
\alpha_{1}^{r-1}y_{1}&\alpha_{2}^{r-1}y_{2}&\cdots&\alpha_{n}^{r-1}y_{n}\end{array}\right).$
(2)

For some given GRS code $\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})$ over
$GF(q^{m})$, the alternant code over $GF(q)$ denoted by
$\mathcal{A}_{r}(\mathbf{a},\mathbf{y})$, is defined as the subfield subcode
of $\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})$, i.e.,

$\mathcal{A}_{r}(\mathbf{a},\mathbf{y})=\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})|GF(q).$
(3)

Since $\mathcal{A}_{r}(\mathbf{a},\mathbf{y})$ consists of all codewords of
$\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})$ over $GF(q)$, the parity check
matrix of $\mathcal{A}_{r}(\mathbf{a},\mathbf{y})$ is given by that of
$\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})$ in (2). The integer $r=n-k$ is
called the degree of the alternant code
$\mathcal{A}_{r}(\mathbf{a},\mathbf{y})$. In addition, $\mathbf{a}$ is called
the support of $\mathcal{A}_{r}(\mathbf{a},\mathbf{y})$, and $\mathbf{y}$ is
called a multiplier of $\mathcal{A}_{r}(\mathbf{a},\mathbf{y})$. For the
dimension and minimum distance of alternant codes, there is the following
result:

###### Lemma 1

[40] Let
$\mathscr{C}=\mathcal{A}_{r}(\mathbf{a},\mathbf{y})=[n,k_{\mathscr{C}},d_{\mathscr{C}}]_{q}$
be an alternant code defined in (3). The dimension and minimum distance of
$\mathscr{C}$ can be given by $k_{\mathscr{C}}\geq n-mr$ and
$d_{\mathscr{C}}\geq r+1$, respectively.

Let $u$ be any nonzero vector over $GF(q)$. Let $\mathbf{a}$ be a fixed vector
and let $\mathbf{v}$ be a varying vector. According to [40, Chap. 12, Theorem
3], we know that the number of GRS codes
$\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})$ containing $u$ is at most
$(q^{m}-1)^{k}$. Therefore, the number of alternant codes
$\mathcal{A}_{r}(\mathbf{a},\mathbf{y})=\textrm{GRS}_{k}(\mathbf{a},\mathbf{v})|GF(q)$
containing $u$ is also at most $(q^{m}-1)^{k}$. Here, it should be noted that
the estimate of the number of alternant codes that contain $u$ does not relate
to its Hamming weight.

###### Lemma 2

[40] For any nonzero vector $u\in GF(q)^{n}$, the number of alternant codes
$\mathcal{A}_{r}(\mathbf{a},\mathbf{y})$ in (3) that contain $u$ is at most
$(q^{m}-1)^{n-r}$.

On the other hand, the total number of alternant codes
$\mathcal{A}_{r}(\mathbf{a},\mathbf{y})$ is equal to the number of choices for
different $\mathbf{y}$ (or $\mathbf{v}$), which is given by $(q^{m}-1)^{n}$
[40]. Thus alternant codes form a large family of linear codes and include
many important subclasses, e.g., Goppa codes and Bose-Chaudhuri-Hocquenghem
codes. Further, alternant codes play a significant role in the McEliece
cryptosystem [43]. Moreover, alternant codes can attain the classical GV bound
asymptotically [40]. In the next section, we will show that alternant codes
can be used to construct concatenated CSS codes achieving the quantum GV bound
asymptotically.

For integers $n,\lambda$, denote by
$\textrm{Vol}_{q}(n,\lambda)=\sum_{i=0}^{\lambda}\binom{n}{i}(q-1)^{i}$ the
Hamming ball of radius $\lambda$. Denote by
$H_{q}(x)=x\log_{q}(q-1)-x\log_{q}x-(1-x)\log_{q}(1-x)$ the $q$-ary entropy
function. There exists the following important estimate of the Hamming ball.

###### Lemma 3

[44] Let $q$ be a power of a prime $p\geq 2$. For an integer $n>1$ and
$0\leq\ell\leq 1-1/q$,

$q^{(H_{q}(\ell)-O(1))n}\leq\textrm{Vol}_{q}(n,\ell n)\leq q^{H_{q}(\ell)n}$
(4)

###  II-2 Quantum Stabilizer Codes

Let $\xi=\exp(2\pi i/p)$ be a primitive $p$th root of unity and let $u,v\in
GF(q)$. Denote the unitary operators $X(u)$ and $Z(v)$ on $\mathbb{C}^{q}$ by
$X(u)|a\rangle=|a+u\rangle$ and
$Z(v)|a\rangle=\xi^{\textrm{Tr}(va)}|a\rangle$, respectively. Denote by the
finite group

$G_{n,q}=\left\\{\begin{aligned} \langle
X(\textrm{u}),Z(\textrm{v})|\textrm{u},\textrm{v}\in
GF(q)^{n}\rangle,&&\textrm{if}&\ p\ \textrm{is odd},\\\ \langle
iI,X(\textrm{u}),Z(\textrm{v})|\textrm{u},\textrm{v}\in
GF(q)^{n}\rangle,&&\textrm{if}&\ p\ \textrm{is even}.\end{aligned}\right.$ (5)

A stabilizer code $Q$ is a $q^{k}$-dimensional subspace of the Hilbert space
$\mathbb{H}_{n}=\mathbb{C}^{q^{n}}$, i.e.,

$Q=\\{|\varphi\rangle\in\mathbb{H}_{n}\ |\ E|\varphi\rangle=|\varphi\rangle,\
\forall\ E\in\mathcal{S}\\},$ (6)

where $\mathcal{S}=\langle S_{1},\ldots,S_{n-k}\rangle$ with stabilizer
generators $S_{i}$($1\leq i\leq n-k$) is an Abelian subgroup of $G_{n,q}$.
$\mathcal{S}$ is called the stabilizer group. By measuring the eigenvalues of
the stabilizer generators, the syndrome of errors can be revealed. An error
$E\in G_{n,q}$ is detectable if it anticommutes with some stabilizer generator
$S_{i}(1\leq i\leq n-k)$. A detectable error results in a nonzero syndrome,
otherwise the syndrome is zero and the error is undetectable. The minimum
distance $d$ of code $Q$ is the minimum weight of an undetectable error $E\in
G_{n,q}$, which does not belong to the stabilizer group. We denote by
$Q=[[n,k,d]]_{q}$. The stabilizer code $Q$ is called nondegenerate if the
stabilizer group $\mathcal{S}$ does not contain a nontrivial element whose
weight is less than $d$, otherwise it is degenerate.

The CSS construction in [5, 45] presents a direct way to construct stabilizer
codes from two classical linear codes that satisfy the dual-containing
relationship.

######  Lemma 4 ([5, 45])

Let $C_{1}$ and $C_{2}$ be two $q$-ary linear codes with parameters
$[n,k_{1},d_{1}]_{q}$ and $[n,k_{2},d_{2}]_{q}$, respectively. If
$C_{2}^{\bot}\subseteq C_{1}$, then there exists a stabilizer code $Q$ with
parameters $[[n,k_{1}+k_{2}-n,d_{Q}]]_{q},$ where

$d_{Q}=\min\\{\text{wt}_{H}(C_{1}\backslash
C_{2}^{\bot}),\text{wt}_{H}(C_{2}\backslash C_{1}^{\bot})\\}.$ (7)

If $d_{Q}=\min\\{d_{1},d_{2}\\}$, then $Q$ is nondegenerate, otherwise it is
degenerate.

The enlargement of CSS construction in [22, 23] can lead to more efficient
stabilizer codes than the standard CSS construction.

######  Lemma 5 ([22, 23])

Let $C=[n,k,d]_{q}$ and $D=[n,k^{\prime},d^{\prime}]_{q}$ be two classical
linear codes over $GF(q)$ such that $C^{\bot}\subseteq C\subseteq D$ and
$k^{\prime}>k+1$. Then there exists a stabilizer code with parameters
$Q_{E}=[[n,k+k^{\prime}-n,d_{Q_{E}}]]_{q},$ where
$d_{Q_{E}}=\min\\{d,\lceil\frac{q+1}{q}d^{\prime}\rceil\\}$.

In most quantum channels, the dephasing errors ($Z$-errors) usually happen
much more frequently than the amplitude errors ($X$-errors). Asymmetric
quantum codes (AQCs) are thus proposed to deal with such biased quantum noise
[46, 30, 47]. Moreover, the CSS construction can be used to derive AQCs with
two classical linear codes, one for correcting $Z$-errors and the other for
correcting $X$-errors.

######  Lemma 6 ([30])

Let $C_{X}$ and $C_{Z}$ be two $q$-ary linear codes with parameters
$[n,k_{1},d_{1}]_{q}$ and $[n,k_{2},d_{2}]_{q}$, respectively. If
$C_{Z}^{\bot}\subseteq C_{X}$, then there exists a
${Q_{A}}=[[n,k_{1}+k_{2}-n,d_{Z}/d_{X}]]_{q}$ AQC, where

$\displaystyle d_{Z}$ $\displaystyle=$
$\displaystyle\max\\{\text{wt}_{H}(C_{X}\backslash
C_{Z}^{\bot}),\text{wt}_{H}(C_{Z}\backslash C_{X}^{\bot})\\},$ (8)
$\displaystyle d_{X}$ $\displaystyle=$
$\displaystyle\min\\{\text{wt}_{H}(C_{X}\backslash
C_{Z}^{\bot}),\text{wt}_{H}(C_{Z}\backslash C_{X}^{\bot})\\}.$ (9)

If $\text{wt}_{H}(C_{X}\backslash C_{Z}^{\bot})=d_{1}$ and
$\text{wt}_{H}(C_{Z}\backslash C_{X}^{\bot})=d_{2}$, then $Q_{A}$ is
nondegenerate, otherwise it is degenerate.

##  III Asymptotic Goodness of Partially Concatenated Calderbank-Shor-Steane
Codes

Denote by $C_{1}=[n_{1},k_{1},d_{1}]_{q}$ a linear code with the parity check
and generator matrices given by $H_{1}$ and $G_{1}$, respectively. Let
$C_{2}=[k_{1},k_{2},d_{2}]_{q}$ be another linear code with the parity check
and generator matrices given by $H_{2}$ and $G_{2}$, respectively. We use
$C_{1}$ as the inner code and $C_{2}$ as the outer code, and we concatenate
the generator matrix of $C_{1}$ with the parity-check matrix of $C_{2}$ as
follows:

$\mathscr{H}_{{X}}=H_{2}G_{1}.$ (10)

Denote the null space of $\mathscr{H}_{X}$ by $\mathscr{C}_{X}$ and we have
$\mathscr{C}_{X}=[n_{1},k_{X}]_{q}$, where $k_{X}=n_{1}-k_{1}+k_{2}$. Denote
by

$\mathscr{H}_{{Z}}=H_{1}\ \textrm{and}\ \mathscr{C}_{Z}=C_{1}.$ (11)

It is easy to see that $\mathscr{H}_{{X}}\mathscr{H}_{{Z}}^{T}=0$ and
$\mathscr{C}_{X}^{\bot}\subseteq\mathscr{C}_{Z}$. Thus we can derive a
partially concatenated CSS (PC-CSS) code with parameters
$\mathscr{Q}=[[n_{1},k_{2}]]_{q}$ by using the CSS construction. We show that
PC-CSS codes can attain the quantum GV bound asymptotically.

###### Theorem 1

There exist PC-CSS codes with parameters
$\mathscr{Q}=[[n_{\mathscr{Q}},k_{\mathscr{Q}},d_{\mathscr{Q}}]]_{q}$
achieving the quantum GV bound asymptotically. $\mathscr{Q}$ can achieve a
quantum rate

$\frac{k_{\mathscr{Q}}}{n_{\mathscr{Q}}}\geq
1-2H_{q}(\delta_{\mathscr{Q}})-O(1)$ (12)

for any block length $n_{\mathscr{Q}}\rightarrow\infty$, where
$\delta_{\mathscr{Q}}=d_{\mathscr{Q}}/n_{\mathscr{Q}}$ is the relative minimum
distance.

###### Proof:

Let $C_{1}=[n_{1},k_{1},d_{1}]_{q}$ be any classical linear code. Let
$C_{2}=[k_{1},k_{2},d_{2}]_{q}$ be an alternant code. We construct a pair of
dual-containing codes $\mathscr{C}_{X}^{\bot}\subseteq\mathscr{C}_{Z}$
according to (10) and (11). By using the CSS construction, we can derive a PC-
CSS code $\mathscr{Q}=[[n_{1},k_{2},d_{\mathscr{Q}}]]_{q}$ with

$\displaystyle d_{\mathscr{Q}}$ $\displaystyle=$
$\displaystyle\min\\{\text{wt}_{H}(c)|c\in(\mathscr{C}_{Z}\backslash\mathscr{C}_{X}^{\bot})\cup(\mathscr{C}_{X}\backslash\mathscr{C}_{Z}^{\bot})\\}$
(13) $\displaystyle=$ $\displaystyle\min\\{\omega_{1},\omega_{2}\\},$

where
$\omega_{1}=\min\\{\text{wt}_{H}(c)|c\in\mathscr{C}_{Z}\backslash\mathscr{C}_{X}^{\bot}\\}\geq
d_{1}$, and
$\omega_{2}=\min\\{\text{wt}_{H}(c)|c\in\mathscr{C}_{X}\backslash\mathscr{C}_{Z}^{\bot}\\}$.
Since $\mathscr{C}_{Z}=C_{1}$ can be chosen arbitrarily, we let $C_{1}$ be an
asymptotically good linear code which can attain the classical GV bound, i.e.,

$\frac{k_{1}}{n_{1}}\geq 1-H_{q}(\delta_{1})-O(1),$ (14)

where $\delta_{1}=d_{1}/n_{1}$ is the relative minimum distance. Next we need
to determine $\omega_{2}$. Since we know nothing about the distance of the
dual code $C_{1}^{\bot}$, there maybe some (very) low weight vectors coming
from $C_{1}^{\bot}$. If we do not consider error degeneracy, the minimum
distance $\omega_{2}$ is unknown and maybe very small. Fortunately, vectors
come from $C_{1}^{\bot}$ do not affect the computation of $\omega_{2}$ because
they are degenerate. For an integer $1<\lambda_{X}<n_{1}$, let $\nu\in
GF(q)^{n_{1}}$ be any nonzero vector with Hamming weight less than
$\lambda_{{X}}$. Denote by

$\textrm{S}_{\nu}\equiv G_{1}\nu^{T}.$ (15)

If $\textrm{S}_{\nu}=0$, then $\nu$ must belong to $C_{1}^{\bot}$ and $\nu$ is
degenerate when we compute $\omega_{2}$. Therefore we only need to consider
the case which makes $\textrm{S}_{\nu}\neq 0$. Let $1\leq f_{X}\leq k_{1}$ and
denote by $k_{f_{{X}}}=k_{1}-(k_{1}-f_{{X}})/m$, where $m\geq 2$ is an integer
and $m|(n_{1}-f_{{X}})$.

For some given $\mathbf{a}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{k_{1}})\in
GF(q^{m})^{k_{1}}$ and varying $\mathbf{v}=(v_{1},v_{2},\ldots,v_{n})\in
GF(q^{m})^{k_{1}}$, let
$\textrm{GRS}_{k_{f_{{X}}}}(\mathbf{a},\mathbf{v})=[k_{1},{k_{f_{{X}}}},k_{1}-{k_{f_{{X}}}}+1]_{q^{m}}$
be a GRS code over $GF(q^{m})$. Let

$C_{2}\equiv\mathcal{A}_{k_{1}-k_{f_{X}}}(\textbf{a},\textbf{y})=\textrm{GRS}_{k_{f_{{X}}}}(\mathbf{a},\mathbf{v})|GF(q)$
(16)

be an alternant code, where $\mathbf{y}=(y_{1},y_{2},\ldots,y_{k_{1}})$ and
$y_{i}\cdot v_{i}=1/\prod_{j\neq i}(\alpha_{i}-\alpha_{j})$, for $1\leq
i\leq{k_{1}}$. By Lemma 1, the dimension of $C_{2}$ satisfies

$k_{2}\geq k_{1}-m(k_{1}-k_{f_{{X}}})=f_{{X}}.$ (17)

By Lemma 2, we know that the number of alternant codes that contain
$\textrm{S}_{\nu}$ does not relate to the Hamming weight of $\textrm{S}_{\nu}$
and is at most

$(q^{m}-1)^{k_{1}-(k_{1}-f_{{X}})/m}.$ (18)

On the other hand, the total number of alternant codes $C_{2}$ is equal to the
number of the choice of $\mathbf{v}$, and it is equal to $(q^{m}-1)^{k_{1}}$
[40]. According to the counting argument in [40], if

$(q^{m}-1)^{k_{1}-(k_{1}-f_{X})/m}\sum\limits_{j=1}^{\lambda_{X}-1}(q-1)^{j}\binom{n_{1}}{j}<(q^{m}-1)^{k_{1}},$
(19)

then there exists an alternant code such that there does not exist any nonzero
vector $\nu\in\mathscr{C}_{X}\backslash\mathscr{C}_{Z}^{\bot}$ with weight
less than $\lambda_{X}$. According to the estimate of Hamming ball in Lemma 3
and taking the limit as $n_{1}\rightarrow\infty$ in (19), we can derive

$\frac{k_{2}}{n_{1}}\geq\frac{f_{{X}}}{n_{1}}=\frac{k_{1}}{n_{1}}-H_{q}(\frac{\lambda_{{X}}}{n_{1}})-O(1).$
(20)

Combining (13), (14), and (20) above together, we have

$\left\\{\begin{aligned} \frac{k_{1}}{n_{1}}&\geq
1-H_{q}(\frac{d_{1}}{n_{1}})-O(1),\\\
\frac{k_{2}}{n_{1}}&\geq\frac{k_{1}}{n_{1}}-H_{q}(\frac{\lambda_{{X}}}{n_{1}})-O(1),\\\
d_{\mathscr{Q}}&\geq\min\\{d_{1},\lambda_{{X}}\\}.\end{aligned}\right.$ (21)

If we let $\lambda_{{X}}=d_{1}+c$, where $c$ is any constant, then we have

$\frac{k_{\mathscr{Q}}}{n_{1}}=\frac{k_{2}}{n_{1}}\geq
1-2H_{q}(\frac{d_{\mathscr{Q}}}{n_{1}})-O(1).$ (22)

∎

Suppose that we consider an asymmetry between $\omega_{1}$ and $\omega_{2}$ in
the proof of Theorem 1, and we apply the CSS construction for AQCs in Lemma 6
to PC-CSS codes. Then asymmetric PC-CSS codes can attain the quantum GV bound
for AQCs asymptotically.

###### Corollary 1

There exist asymptotically good asymmetric PC-CSS codes
$\mathscr{A}=[[n_{\mathscr{A}},k_{\mathscr{A}},d_{Z}/d_{X}]]_{q}$ such that

$\frac{k_{\mathscr{A}}}{n_{\mathscr{A}}}\geq
1-H_{q}(\delta_{X})-H_{q}(\delta_{Z})-O(1)$ (23)

for any block length $n_{\mathscr{A}}\rightarrow\infty$, where
$\delta_{X}=d_{X}/n_{\mathscr{A}}$ and $\delta_{Z}=d_{Z}/n_{\mathscr{A}}$ are
the relative distances, and $0\leq\delta_{X}\leq\delta_{Z}\leq 1-1/q$.

It is known that there exist families of asymptotically good LDPC codes which
can achieve the classical GV bound. But the dual of LDPC codes has a low-
density generator matrix and thus the minimum distance is constant. If we use
LDPC codes as the inner code $C_{1}$, then the minimum distance of the PC-CSS
code without considering error degeneracy is constant to the block length.
While considering error degeneracy, the low weight codewords in the dual of
LDPC codes do not affect the counting argument in the proof of Theorem 1.
Therefore, the PC-CSS code is extremely degenerate and the minimum distance is
greatly improved from a constant to be linear with the block length. By
applying the Evra-Kaufman-Zémor distance balancing construction [17], we show
that our scheme can also lead to qLDPC codes wth non-vanishing rates and
minimum distance growing with the square root of the block length [48].

In Corollary 1, PC-CSS codes are shown to attain the quantum GV bound for AQCs
[11]. Moreover, we show that asymmetric PC-CSS codes can approach the capacity
of asymmetric Pauli channels as the channel asymmetry goes to large. Suppose
that we transmit quantum information over the Pauli channel:

$\varrho\mapsto
p_{I}\varrho+p_{X}{X}\varrho{X}+p_{Y}{Y}\varrho{Y}+p_{Z}{Z}\varrho{Z}$ (24)

for an input state $\varrho$, where $\\{I,{X},{Y},{Z}\\}$ are the Pauli
operators, $0\leq p_{I},p_{Z},p_{Y},p_{Z}\leq 1$, and
$p_{I}+p_{X}+p_{Y}+p_{Z}=1$. Denote the total error probability by
$\emph{{p}}=p_{X}+p_{Y}+p_{Z}$. Denote by $\zeta=(p_{Z}+p_{Y})/(p_{X}+p_{Y})$
the asymmetry for the probabilities of ${Z}$-errors and ${X}$-errors. Usually,
we let $p_{X}=p_{Y}$, then we have $p_{X}=\emph{{p}}/(2\zeta+1)$ and
$p_{Z}=\emph{{p}}(2\zeta-1)/(2\zeta+1)$.

In Corollary 1, the inner code $C_{1}$ can be chosen arbitrarily, we let it be
an LDPC code, which is used to correct $Z$-errors. We use alternant codes to
correct $X$-errors. It is known that alternant codes can be decoded up to the
GV bound under the bounded minimum distance decoder [40]. LDPC codes can
approach the Shannon capacity of the binary symmetric channel under the BP
decoding [49, 50]. Then the rate $R_{\mathscr{Q}}$ of PC-CSS codes under the
asymmetric Pauli channel can approach

$1-H_{2}(4p_{X})-H_{2}(p_{Z}+p_{Y})-O(1).$ (25)

In Fig. 1, we compare the the limit of PC-CSS codes in (25) with the Hashing
bound given by $R=1-H_{2}(\emph{{p}})$ over Pauli channels with an asymmetry
$\zeta=(p_{Z}+p_{Y})/(p_{X}+p_{Y})$. If $\zeta=1$, then the channel is
symmetric with $p_{X}=p_{Y}=p_{Z}=\emph{{p}}/3$. As the asymmetry $\zeta$
grows, the gap between PC-CSS codes and the Hashing bound becomes increasingly
smaller. In Fig. 1, when the asymmetry $\zeta=10^{2}$ and $\zeta=10^{3}$, the
code rate gaps are less than $3\times 10^{-2}$ and $4\times 10^{-3}$,
respectively. Therefore, PC-CSS codes can approach the Hashing bound of Pauli
channels as the asymmetry grows. It should be noted that, although our codes
are highly degenerate, they can not attain the bound in [15] for very noisy
channels .

Figure 1: The comparison of the asymptotic bound of asymmetric PC-CSS codes
(solid lines) and the Hashing bound (dashed lines) with different channel
asymmetries $\zeta=1,100,$ and $1000$. The horizontal axis is the total error
probability in the Pauli channel and the vertical axis is the quantum code
rate.

The PC-CSS codes in Theorem 1 can also be used to prove the security of the
famous BB84 quantum key distribution (QKD) protocol directly [51]. By a simple
calculation, we can obtain a total qubit error rate (QBER) less than $7.56\%$
for the security of BB84 over Pauli channels. Although the rate is lower than
the bound of $11\%$ in [51], our codes are not randomly generated and the LDPC
component codes can be efficiently decoded. Since classical LDPC codes can
approach the Shannon bound by using efficient BP decoders, PC-CSS codes are
particularly applicable to efficient and practical QKD. Furthermore, if we
consider asymmetric errors in Pauli channels, we can further improve the total
QBER for the security of BB84. For example, if we admit an asymmetry
$\zeta=100$ between ${Z}$-errors and ${X}$-errors, then the total QBER is less
than $35.56\%$. Our codes are competitive with quantum Polar codes in [52, 53]
that can achieve the coherent information. But Ref. [52] needs some preshared
entanglement assistance.

##  IV A Family of Fast Encodable and Decodable PC-CSS Codes

In practice, we need quantum codes to be efficiently encoded and decoded. Let
the error syndrome as the input, we show that PC-CSS codes can be fast
decoded. Moreover, we also show that PC-CSS codes can be fast encoded.

TABLE II: Parameters and the syndrome decoding complexity of several efficiently decodable quantum codes. References | Parameters |  Weight of $X$-Stabilizers |  Weight of $Z$-Stabilizers |  Syndrome Decoding Complexity |  Parallel Syndrome Decoding Complexity
---|---|---|---|---|---
[54] | $[[N,\Omega(N),\Omega(\sqrt{N})]]$ | $O(1)$ | $O(1)$ | $O(N)$ | $\backslash$
[27] | $[[N,\Omega(N),\Omega({N})]]$ | $O(N)$ | $O(N)$ | Polynomial | $\backslash$
[55] | $[[N,2,\Omega(\sqrt{N})]]$ | $O(1)$ | $O(1)$ | $O(N^{3})$ | $\backslash$
[56] | $[[N,2,\Omega(\sqrt{N})]]$ | $O(1)$ | $O(1)$ | $O(N\log\sqrt{N})$ | $O(\log\sqrt{N})$
[17] | $[[N,\Omega(\sqrt{N/\log N}),\Omega(\sqrt{N\log N})]]$ | $O(1)$ | $O(1)$ | Polynomial | $\backslash$
Corollary 2 | $[[N,\Omega(\sqrt{N}),\Omega(\sqrt{N})]]$ | $O(\sqrt{N})$ | $O(1)$ | $O({N})$ | $O(\sqrt{N})$
Corollary 3 |  $[[N,\Omega(N/\log N),d_{Z}/d_{X}]]$ $d_{Z}=\Omega(N/\log N),$ $d_{X}=\Omega(\log N)$ | $O(\log N)$ | $O(1)$ | $O(N)$ | $O(\log{N})$

###### Theorem 2

Let $n_{0}>1$ and $N>1$ be integers and let $n_{0}|N$. There exists a family
of asymmetric PC-CSS codes with parameters
$\mathscr{A}=[[N,\Omega(N/n_{0}),d_{Z}\geq\Omega(N/n_{0})/d_{X}\geq n_{0}]]$.
The syndrome decoding of $\mathscr{A}$ can be done in $O(N)$ time. Further,
the parallel syndrome decoding of $\mathscr{A}$ can be done in
$O(\log(N/n_{0})+n_{0})$ time by using $O(N/n_{0}+N/n_{0}\cdot\log(N/n_{0}))$
classical processors. Moreover, $\mathscr{A}$ can be encoded by a circuit of
size $O((N/n_{0})^{2}+N)$ and depth $O(N/n_{0}+n_{0})$.

###### Proof:

Let $C_{0}=[n_{0},1,n_{0}]$ be a binary repetition code whose parity check
matrix and generator matrix are given by:

$H_{0}=\left(\begin{array}[]{ccccc}1&0&\cdots&0&1\\\ 0&1&\cdots&0&1\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\cdots&1&1\end{array}\right),\
\textrm{and}\ G_{0}=(\begin{array}[]{ccccc}1&1&\cdots&1\end{array}),$ (26)

respectively. Let $C_{1}=[N,N/n_{0},n_{0}]$ be a linear code with a parity
check matrix $H_{1}=\mathbf{I}\otimes H_{0}$ and a generator matrix
$G_{1}=\mathbf{I}\otimes G_{0}$, where $\mathbf{I}$ is an identity matrix of
size $N/n_{0}$. Let $C_{2}=[N_{2}=N/n_{0},k_{2},d_{2}]$ be an asymptotically
good expander code in [57] such that $k_{2}=\gamma_{2}N_{2}$, and
$d_{2}=\delta_{2}N_{2}$, where $0<\gamma_{2},\delta_{2}<1$. Let the parity
check and generator matrices of $C_{2}$ be $H_{2}$ and $G_{2}$, respectively.
Then we can construct an asymmetric PC-CSS code
$\mathscr{A}=[[N,k_{2},d_{Z}/d_{X}]]$ with

$\mathscr{H}_{X}=H_{2}G_{1},\ \textrm{and}\ \mathscr{H}_{Z}=H_{1}.$ (27)

It is not difficult to verify that $d_{X}\geq d_{2}$ and $d_{Z}\geq n_{0}$.

We consider the syndrome decoding of $\mathscr{A}$. For an $X$-error $e_{X}$,
the error syndrome is given by

$\textrm{S}_{X}\equiv\mathscr{H}_{X}e_{X}^{T}=H_{2}\textrm{S}_{X}^{(1)},$ (28)

where we denote by $\textrm{S}_{X}^{(1)}=G_{1}e_{X}^{T}$. Suppose that
$\textrm{wt}_{H}(e_{X})\leq\lfloor(d_{2}-1)/2\rfloor$, then we must have
$\textrm{wt}_{H}(\textrm{S}_{X}^{(1)})\leq\lfloor(d_{2}-1)/2\rfloor$. Thus we
can decode
$\textrm{S}_{X}^{(1)}=(\textrm{S}_{X_{1}}^{(1)},\ldots,\textrm{S}_{X_{N_{2}}}^{(1)})$
by using the expander code $C_{2}$ in $O(N_{2})$ time [57]. Further, $C_{2}$
can also be decoded in parallel in $O(\log(N_{2}))$ time by using
$O(N_{2}\cdot\log(N_{2}))$ classical processors [57]. Let
$\widetilde{e}_{X}=(\textrm{S}_{X_{1}}^{(1)},0,\ldots,\textrm{S}_{X_{N_{2}}}^{(1)},\ldots,0)$
be the decoded $X$-error, where the $0$s correspond to the redundant qubit
positions in $G_{1}$. Then we have $G_{1}(e_{X}+\widetilde{e}_{X})^{T}=0$ and
$\mathscr{Q}$ is degenerate with respect to $e_{X}+\widetilde{e}_{X}$.

Next, we use $C_{1}$ to correct any $Z$-error $e_{Z}$ such that
$\textrm{wt}_{H}(e_{Z})\leq\lfloor(n_{0}-1)/2\rfloor$. We divide the parity
check matrix $H_{1}$ into $n_{0}$ sub-blocks by columns, and each sub-block
corresponds to a diagonal block $H_{0}$ in $H_{1}$. Divide $e_{Z}$ into
$N_{2}$ sub-blocks, i.e., $e_{Z}=(e_{Z_{1}},\ldots,e_{Z_{N_{2}}})$. Then there
are at most $\lfloor(n_{0}-1)/2\rfloor$ erroneous sub-blocks in $e_{Z}$ and
each erroneous sub-block has at most $\lfloor(n_{0}-1)/2\rfloor$ errors. For
each erroneous sub-block, we use the one-step majority-logic (OSMLG) decoding
method [58] to check the error qubit that corresponds to the last column in
$H_{0}$. Let
$\widehat{e}_{Z}=(\widehat{e}_{Z_{1}},\ldots,\widehat{e}_{Z_{n_{0}}})$ be any
sub-block in $\\{e_{Z_{i}}|1\leq i\leq N_{2}\\}$. Then we can derive the
following syndrome for $\widehat{e}_{Z}$:

$S_{\widehat{e}_{Z}}=\widehat{e}_{Z}H_{0}^{T}=(\widehat{e}_{Z_{1}},\ldots,\widehat{e}_{Z_{n_{0}-1}})+\widehat{e}_{Z_{n_{0}}}E_{0},$
(29)

where $E_{0}=(1,\ldots,1)$ is an all-ones vector of length $n_{0}-1$. If
$\widehat{e}_{Z_{n_{0}}}=0$, then there are at most
$\lfloor(n_{0}-1)/2\rfloor$ ones in the vector
$(\widehat{e}_{Z_{1}},\ldots,\widehat{e}_{Z_{n_{0}-1}})$. Thus the Hamming
weight of the syndrome $S_{\widehat{e}_{Z}}$ is at most
$\lfloor(n_{0}-1)/2\rfloor$. While if $\widehat{e}_{Z_{n_{0}}}=1$, then there
are at most $\lfloor(n_{0}-1)/2\rfloor-1$ ones in the vector
$(\widehat{e}_{Z_{1}},\ldots,\widehat{e}_{Z_{n_{0}-1}})$. Then the Hamming
weight of the syndrome $S_{\widehat{e}_{Z}}$ is at least
$n_{0}-\lfloor(n_{0}-1)/2\rfloor=\lfloor(n_{0}-1)/2\rfloor+1$. Therefore we
can use the OSMLG decoding to determine $\widehat{e}_{Z_{n_{0}}}$, and then
$(\widehat{e}_{Z_{1}},\ldots,\widehat{e}_{Z_{n_{0}-1}})$ can be derived from
(35) directly.

The running time for deriving
$\widehat{e}_{Z}=(\widehat{e}_{Z_{1}},\ldots,\widehat{e}_{Z_{n_{0}}})$ is
determined by the computation of the Hamming weight of the syndrome
$S_{\widehat{e}_{Z}}$. The computation time complexity is $O(n_{0})$. Thus the
running time for decoding all the erroneous sub-blocks is $O(N)$ since there
are $N/n_{0}$ sub-blocks. Here, the decoding of the $n_{0}$ sub-blocks can
also be carried out in parallel by using $N/n_{0}$ classical processors. Thus
the parallel syndrome decoding of the $Z$-error $e_{Z}$ can be done in
$O(n_{0})$ time by using $N/n_{0}$ classical processors. Overall, the syndrome
decoding of $\mathscr{A}$ can be done in $O(N/n_{0}+N)$ time or can be done in
parallel in $O(\log(N/n_{0})+n_{0})$ time by using
$O(N/n_{0}\cdot\log(N/n_{0})+N/n_{0})$ classical processors.

Next, we consider the encoding of $\mathscr{A}$ and we use the standard
encoding method of CSS codes in [34] to do that. We can encode the pure state
as follows:

$|u+\mathscr{C}_{Z}^{\bot}\rangle\equiv\frac{1}{\sqrt{|\mathscr{C}_{Z}^{\bot}|}}\sum_{v\in\mathscr{C}_{Z}^{\bot}}|u+v\rangle,$
(30)

where $u\in\mathscr{C}_{X}\backslash\mathscr{C}_{Z}^{\bot}$. It is not
difficult to verify that the generator matrix of $\mathscr{C}_{X}$ is given by

$\mathscr{G}_{X}=\left(\begin{array}[]{cccc}H_{1}\\\ \mathbf{O}\
G_{2}\end{array}\right),$ (31)

where $\mathbf{O}$ is a zero matrix of size $k_{2}\times(N-N/n_{0})$, and
$G_{2}=[I\ P_{2}]$ is in the standard form. Denote by
$P_{2}=[\textbf{P}_{1},\ldots,\textbf{P}_{r_{2}}]$, where each
$\textbf{P}_{i}(1\leq i\leq r_{2})$ is a column of $P_{2}$ and
$r_{2}=N/n_{0}-k_{2}$. Then the encoding circuit of the PC-CSS code
$\mathscr{A}$ is given in Fig. LABEL:EncodingCircuitBacon. In Stage I of the
encoding circuit, the number of gates is determined by the density of $P_{2}$
and it is at most $O((N/n_{0})^{2})$. It is easy to see that the depth of
Stage I is $k_{2}$. While in Stage II, the number of C-NOT gates is exactly
$N-N/n_{0}$ and the depth is $n_{0}$. Therefore the size of the encoding
circuit is $O((N/n_{0})^{2}+N)$ and the depth is
$k_{2}+n_{0}=O(N/n_{0}+n_{0})$. ∎

Notice that the last column in $H_{0}$ in the proof of Theorem 2 is quite
dense and then $H_{1}$ is also dense. However, we can transform $H_{0}$ into a
sparse matrix by multiplying a series of elementary matrices in the left.
Therefore the ${Z}$-stabilizer generators of $\mathscr{A}$ satisfy the qLDPC
constraint [37] . In the fault tolerant settings, we require quantum codes to
correct a linear number of random errors. We consider the correction of random
errors by using PC-CSS codes. We assume that the noise model used is the
independent error model in [59, 38, 60]. Denote by $p_{z}$ the probability of
each $Z$-error. It is shown that the PC-CSS code $\mathscr{A}$ in Theorem 2
can correct any adversarial $Z$-error $e_{Z}$ of weight less than half the $Z$
distance bound. We count all the uncorrectable errors of weight larger than
half the minimum distance bound. It is easy to see that if there is at least
one sub-block that has $Z$-errors of weight larger than
$d_{0}=\lfloor(n_{0}-1)/2\rfloor$, then $e_{Z}$ is undetectable. All the
uncorrectable $Z$-errors are counted as follows in equations (32) - (34).

$\displaystyle\mathcal{P}_{Z}$ $\displaystyle\leq$ $\displaystyle
C_{N_{2}}^{1}C_{n_{0}}^{d_{0}+1}(p_{z}^{d_{0}+1}(1-p_{z})^{N-d_{0}-1}+C_{N-d_{0}-1}^{1}p_{z}^{d_{0}+2}(1-p_{z})^{N-d_{0}-2}+\cdots+C_{N-d_{0}-1}^{N-d_{0}-1}p_{z}^{N})$
(32) $\displaystyle=$ $\displaystyle
C_{N_{2}}^{1}C_{n_{0}}^{d_{0}+1}p_{z}^{d_{0}+1}\sum_{i=0}^{N-d_{0}-1}C_{N-d_{0}-1}^{i}p_{z}^{i}(1-p_{z})^{N-d_{0}-1-i}$
(33) $\displaystyle\leq$ $\displaystyle N_{2}2^{n_{0}-1}p_{z}^{d_{0}+1},$ (34)

We denote by $\mathcal{P}_{Z}$ the probability of uncorrectable $Z$-errors and
denote by $N_{2}=N/n_{0}$. Let $c>0$ be any constant. If we let
$n_{0}=\Omega(\log N)$ and let $p_{z}<1/4^{c+1}$, then $N^{c}\mathcal{P}_{Z}$
vanishes as $N\rightarrow\infty$. If we let $n_{0}=\Theta(N^{c_{0}})$, where
$0<c_{0}<1$ is constant, then $N^{c}\mathcal{P}_{Z}$ vanishes when
$p_{z}<25\%$ as $N\rightarrow\infty$. Therefore the PC-CSS code $\mathscr{A}$
in Theorem 2 can correct a linear number of random $Z$-errors with high
probability in $O(n_{0})$ time as long as $n_{0}=\Omega(\log N)$. However, for
the correction of random ${Z}$-errors, we cannot get a non vanishing
threshold.

If we let $n_{0}=\Theta(\sqrt{N})$ or $n_{0}=\Theta(\log N)$, we can derive
the following two families of fast encodable and decodable quantum codes.

###### Corollary 2

There exists a family of quantum codes with parameters
$\mathscr{Q}_{1}=[[N,\Omega(\sqrt{N}),\Omega(\sqrt{N})]].$ For an input error
syndrome, $\mathscr{Q}_{1}$ can correct all errors of weight smaller than half
the minimum distance bound in $O({N})$ time. $\mathscr{Q}_{1}$ can also be
decoded in parallel in $O(\sqrt{N})$ time by using $O(\sqrt{N})$ classical
processors. We show that $\mathscr{Q}_{1}$ can be encoded efficiently by a
circuit of size $O(N)$ and depth $O(\sqrt{N})$. Moreover, $\mathscr{Q}_{1}$
can correct a random $Z$-error with very high probability provided
$p_{z}<25\%$.

###### Corollary 3

There exists a family of quantum codes with parameters
$\mathscr{Q}_{2}=[[N,\Omega(N/\log N),\Omega(N/\log N)/\Omega(\log N)]].$ For
an input error syndrome, $\mathscr{Q}_{2}$ can correct an almost linear number
of ${Z}$-errors and can also correct a linear number of ${X}$-errors with high
probability in $O(N)$ time. Further, $\mathscr{Q}_{2}$ can be decoded in
parallel in $O(\log(N))$ time by using $O(N)$ classical processors.

In [27], families of concatenated quantum AG codes were constructed and
decoded efficiently in polynomial time, while our codes in Theorem 2 can be
decoded in linear time. Therefore our codes are more efficient than
concatenated quantum AG codes in [27]. In particular, the parallel syndrome
decoding of Corollary 2 can be carried out in sublinear time, and that of
Corollary 3 can be carried out in logarithmic time. In Table II, we compare
the parameters and decoding complexity of several efficiently decodable
quantum codes.

##  V Enlargement of PC-CSS Codes Achieving the Quantum GV Bound for Enlarged
CSS Codes Asymptotically

In [22, 23], an enlargement construction of CSS codes was proposed by
enlarging the dual-containing codes. Further, several upper and lower bounds
for the enlargement of CSS codes were given in [22, 23]. Among them, whether
enlargement of CSS codes can attain the quantum GV bound for enlarged CSS
codes is unknown. In this work, we show that enlarged PC-CSS codes can attain
the quantum GV bound asymptotically.

###### Theorem 3

There exist a family of enlarged stabilizer codes with parameters
$\mathscr{Q}=[[n_{\mathscr{Q}},k_{\mathscr{Q}},d_{\mathscr{Q}}]]_{q}$
achieving the quantum GV bound for enlarged CSS codes asymptotically.
$\mathscr{Q}$ can achieve a quantum rate

$\frac{k_{\mathscr{Q}}}{n_{\mathscr{Q}}}\geq
1-H_{q}(\delta_{\mathscr{Q}})-H_{q}(\frac{q}{q+1}\delta_{\mathscr{Q}})-O(1)$
(35)

for any block length $n_{\mathscr{Q}}\rightarrow\infty$, where
$\delta_{\mathscr{Q}}=d_{\mathscr{Q}}/n_{\mathscr{Q}}$ is the relative minimum
distance.

###### Proof:

Suppose that $C_{1}=[n_{1},k_{1},d_{1}]_{q}$ is a dual-containing code, i.e.,
$C_{1}^{\bot}\subseteq C_{1}$. Denote the parity check and generator matrices
of $C_{1}$ by $H_{1}$ and $G_{1}$, respectively. We begin with a family of
dual-containing codes. It is known that there exists asymptotically good dual-
containing codes $C_{1}^{\bot}\subseteq C_{1}$ achieving the quantum GV bound
[61, 9], i.e.,

$\frac{k_{1}}{n_{1}}\geq 1-H_{q}(\delta_{1})-O(1),\ n_{1}\rightarrow\infty,$
(36)

where $\delta_{1}=d_{1}/n_{1}$ is the relative distance of $C_{1}$. Let
$r_{1}=n_{1}-k_{1}$ and let $C_{2}=[r_{1},k_{2},d_{2}]_{q}$ be an alternant
code. Denote the parity check and generator matrices of $C_{2}$ by $H_{2}$ and
$G_{2}$, respectively. We construct a linear code
$C_{3}=[n_{1},k_{1}+k_{2},d_{3}]_{q}$ with a parity check matrix given by

$H_{3}=H_{2}H_{1}.$ (37)

Then there is $C_{1}^{\bot}\subseteq C_{1}\subseteq C_{3}$. According to the
enlarged CSS construction in Lemma 5, we can derive a stabilizer code with
parameters
$\mathscr{Q}=[[n_{1},k_{\mathscr{Q}}=2k_{1}+k_{2}-n_{1},d_{\mathscr{Q}}=\min\\{d_{1},\omega_{2}\\}]]_{q},$
where $\omega_{2}=\lceil\frac{q+1}{q}d_{3}\rceil$. We need to determine
$\omega_{2}$ and we chose $C_{2}=[r_{1},k_{2},d_{2}]_{q}$ as an alternant
code.

Let $1\leq f_{A}\leq r_{1}$ and denote by $k_{f_{A}}=r_{1}-(r_{1}-f_{A})/m$,
where $m\geq 2$ is an integer and $m|(n_{1}-f_{A})$. For some given
$\mathbf{b}=(\beta_{1},\beta_{2},\ldots,\beta_{r_{1}})\in GF(q^{m})^{r_{1}}$
and varying $\mathbf{u}=(u_{1},u_{2},\ldots,u_{r_{1}})\in GF(q^{m})^{r_{1}}$,
let
$\textrm{GRS}_{k_{f_{A}}}(\mathbf{b},\mathbf{u})=[r_{1},{k_{f_{A}}},r_{1}-{k_{f_{A}}}+1]_{q^{m}}$
be a GRS code over $GF(q^{m})$. Let

$C_{2}\equiv\mathcal{A}_{r_{1}-k_{f_{A}}}(\textbf{b},\textbf{z})=\textrm{GRS}_{k_{f_{A}}}(\mathbf{b},\mathbf{u})|GF(q)$
(38)

be an alternant code, where $\mathbf{z}=(z_{1},z_{2},\ldots,z_{r_{1}})$ and
$z_{i}\cdot u_{i}=1/\prod_{j\neq i}(\beta_{i}-\beta_{j})$, for $1\leq i\leq
r_{1}$. By Lemma 1, the dimension of $C_{2}$ satisfies

$k_{2}\geq r_{1}-m(r_{1}-k_{f_{A}})=f_{A}.$ (39)

Let $1<\lambda_{A}<n_{1}$ be an integer. Let $\upsilon\in GF(q)^{n_{1}}$ be
any nonzero vector with Hamming weight less than $\lambda_{A}$. Denote by
$\textrm{S}_{\upsilon}=H_{1}\upsilon^{T}$. If $\lambda_{A}<d_{1}$, then we
must have $\textrm{S}_{\upsilon}\neq 0$. If $\upsilon\in C_{3}$, then we have
$H_{2}\textrm{S}_{\upsilon}=H_{2}H_{1}\upsilon^{T}=0$. By Lemma 2, we know
that the number of alternant codes that contain the nonzero vector
$\textrm{S}_{\upsilon}$ does not relate to the Hamming weight of
$\textrm{S}_{\upsilon}$ and is at most

$(q^{m}-1)^{r_{1}-(r_{1}-f_{A})/m}.$ (40)

On the other hand, the total number of alternant codes $C_{2}$ is equal to the
number of the choice of $\mathbf{u}$, and it is equal to $(q^{m}-1)^{r_{1}}$.
According to the counting argument in [40], if

$(q^{m}-1)^{r_{1}-(r_{1}-f_{A})/m}\sum\limits_{j=1}^{\lambda_{A}-1}(q-1)^{j}\binom{n_{1}}{j}<(q^{m}-1)^{r_{1}},$
(41)

then there exists an alternant code such that there does not exist any nonzero
vector $\upsilon\in C_{3}$ with weight less than $\lambda_{A}$. Therefore we
have $d_{3}\geq\lambda_{A}$. According to the estimate of Hamming ball in
Lemma 3 and taking the limit as $n_{1}\rightarrow\infty$ in (41), we can
derive

$\frac{k_{2}}{n_{1}}\geq\frac{f_{A}}{n_{1}}=\frac{r_{1}}{n_{1}}-H_{q}(\frac{\lambda_{A}}{n_{1}})-O(1).$
(42)

Combining (36), (42), and the parameters of $\mathscr{Q}$ above together, we
have

$\left\\{\begin{aligned} \frac{k_{1}}{n_{1}}&\geq
1-H_{q}(\frac{d_{1}}{n_{1}})-O(1),\\\
\frac{k_{2}}{n_{1}}&\geq\frac{r_{1}}{n_{1}}-H_{q}(\frac{\lambda_{A}}{n_{1}})-O(1),\\\
d_{\mathscr{Q}}&\geq\min\\{d_{1},\lceil\frac{q+1}{q}\lambda_{A}\rceil\\},\\\
d_{1}&>\lambda_{A}.\end{aligned}\right.$ (43)

Then there is

$\left\\{\begin{aligned} \frac{k_{\mathscr{Q}}}{n_{1}}&\geq
1-H_{q}(\frac{d_{1}}{n_{1}})-H_{q}(\frac{\lambda_{A}}{n_{1}})-O(1),\\\
d_{\mathscr{Q}}&\geq\min\\{d_{1},\lceil\frac{q+1}{q}\lambda_{A}\rceil\\},\\\
d_{1}&>\lambda_{A}.\end{aligned}\right.$ (44)

If we let $d_{1}=\lceil\frac{q+1}{q}\lambda_{A}\rceil$, then we have

$\frac{k_{\mathscr{Q}}}{n_{1}}\geq
1-H_{q}(\frac{d_{\mathscr{Q}}}{n_{1}})-H_{q}(\frac{q}{q+1}\cdot\frac{d_{\mathscr{Q}}}{n_{1}})-O(1).$
(45)

∎

##  VI Discussions and Conclusions

Constructing asymptotically good QECCs is one central problem in quantum
coding theory. In this paper, we have proposed a partial concatenation
construction of CSS codes by using alternant codes as the outer code and any
linear code achieving the GV bound as the inner code. We show that PC-CSS
codes can attain the quantum GV bound for CSS codes and AQCs asymptotically.
Further, we show that PC-CSS codes can approach the Hashing bound for
asymmetric Pauli channels with a large asymmetry. In addition, PC-CSS codes
can be used to prove the security of the BB84 QKD protocol. Moreover, the
concatenation scheme can asymptotically attain the quantum GV bound for
enlarged CSS codes by using Steane’s enlargement construction of CSS codes.

We derive a family of PC-CSS codes that can be encoded by a circuit of linear
size and sublinear depth, and the syndrome decoding of PC-CSS codes can be
done in linear time. Moreover, we show that the parallel syndrome decoding of
the family of PC-CSS codes can be done in sublinear time. We derive a family
of PC-CSS codes that can correct a linear number of ${X}$-errors with high
probability and an almost linear number of ${Z}$-errors in linear time. The
parallel syndrome decoding of the family of PC-CSS codes can be done in
logarithmic time. One interesting future work is how to compute the threshold
of PC-CSS codes for correcting random quantum errors.

## Acknowledgment

The authors would like to thank the anonymous referees and the Associate
Editor for their valuable comments and helpful suggestions.

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Jihao Fan received the B.S. degree in Mathematics and Applied Mathematics from
Lanzhou University, Lanzhou, China, in 2009, and the Ph.D. degree in Computer
Software and Theory from Southeast University, Nanjing, China, in 2016. He is
currently an Associate Professor with the Nanjing University of Science and
Technology, Nanjing, China. His research interests include classical and
quantum coding theory, information theory, and machine learning.
---
Jun Li (M’09-SM’16) received Ph. D degree in Electronic Engineering from
Shanghai Jiao Tong University, Shanghai, P. R. China in 2009. From January
2009 to June 2009, he worked in the Department of Research and Innovation,
Alcatel Lucent Shanghai Bell as a Research Scientist. From June 2009 to April
2012, he was a Postdoctoral Fellow at the School of Electrical Engineering and
Telecommunications, the University of New South Wales, Australia. From April
2012 to June 2015, he was a Research Fellow at the School of Electrical
Engineering, the University of Sydney, Australia. From June 2015 to now, he is
a Professor at the School of Electronic and Optical Engineering, Nanjing
University of Science and Technology, Nanjing, China. He was a visiting
professor at Princeton University from 2018 to 2019. His research interests
include network information theory, game theory, distributed intelligence,
multiple agent reinforcement learning, and their applications in ultra-dense
wireless networks, mobile edge computing, network privacy and security, and
industrial Internet of things. He has co-authored more than 200 papers in IEEE
journals and conferences, and holds 1 US patents and more than 10 Chinese
patents in these areas. He is serving as an editor of IEEE Transactions on
Wireless Communication and TPC member for several flagship IEEE conferences.
---
Ya Wang received the Ph.D. degree in Physics from the University of Science
and Technology of China (USTC) in 2012. From August 2012 to July 2016, he was
a Postdoctoral Fellow at Stuttgart University. From July 2016 to March 2018,
he was a Research Fellow at the University of Science and Technology of China.
From April 2018 to now, he has been a professor with the School of Physical
Sciences of USTC. His research interests include spin-based quantum devices,
spin quantum control, and their applications in quantum science. He currently
focuses on diamond quantum device engineering and applications in quantum
technologies.
---
Yonghui Li (M’04-SM’09-F’19) received his PhD degree in November 2002 from
Beijing University of Aeronautics and Astronautics. Since 2003, he has been
with the Centre of Excellence in Telecommunications, the University of Sydney,
Australia. He is now a Professor and Director of Wireless Engineering
Laboratory in School of Electrical and Information Engineering, University of
Sydney. He is the recipient of the Australian Queen Elizabeth II Fellowship in
2008 and the Australian Future Fellowship in 2012. He is a Fellow of IEEE. His
current research interests are in the area of wireless communications, with a
particular focus on MIMO, millimeter wave communications, machine to machine
communications, coding techniques and cooperative communications. He holds a
number of patents granted and pending in these fields. He is now an editor for
IEEE transactions on communications, IEEE transactions on vehicular
technology. He also served as the guest editor for several IEEE journals, such
as IEEE JSAC, IEEE Communications Magazine, IEEE IoT journal, IEEE Access. He
received the best paper awards from IEEE International Conference on
Communications (ICC) 2014, IEEE PIRMC 2017 and IEEE Wireless Days Conferences
(WD) 2014.
---
Min-Hsiu Hsieh received the B.S. and M.S. degrees in electrical engineering
from National Taiwan University in 1999 and 2001, respectively, and the Ph.D.
degree in electrical engineering from the University of Southern California at
Los Angeles, Los Angeles, CA, USA, in 2008. From 2008 to 2010, he was a
Researcher with the ERATO-SORST Quantum Computation and Information Project,
Japan Science and Technology Agency, Tokyo, Japan. From 2010 to 2012, he was a
Post-Doctoral Researcher with the Statistical Laboratory, the Centre for
Mathematical Sciences, The University of Cambridge, U.K. From 2012 to 2020, he
was an Australian Research Council (ARC) Future Fellow and an Associate
Professor with the Centre for Quantum Software and Information, Faculty of
Engineering and Information Technology, University of Technology Sydney,
Ultimo, NSW, Australia. He is currently the Director of the Hon Hai (Foxconn)
Quantum Computing Center. His scientific research interests include quantum
information, quantum learning, and quantum computation.
---
Jiangfeng Du received the Ph.D. degree in Physics from the University of
Science and Technology of China (USTC) in 2000. Since 2004, he has been a
professor with the School of Physical Sciences of USTC. He is an expert in the
area of spin quantum physics and its applications. His research interests are
interdisciplinary. He currently focuses on quantum computation, quantum
simulation and quantum sensing, and the development of new technologies for
medical applications.
---


11institutetext: Instituto de Astrofisíca de Canarias, 38200 La
Laguna,Tenerife, Canary Islands, Spain 22institutetext: Departamento de
Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain

# Analysis of Galactic molecular cloud polarization maps: a review of the
methods

Frédérick Poidevin 1122<EMAIL_ADDRESS>
###### Abstract

The Davis-Chandrasekhar-Fermi (DCF) method using the Angular Dispersion
Function (ADF), the Histogram of Relative Orientations (HROs) and the
Polarization-Intensity Gradient Relation (P-IGR) are the most common tools
used to analyse maps of linearly polarized emission by thermal dust grains at
submilliter wavelengths in molecular clouds and star-forming regions. A short
review of these methods is given. The combination of these methods will
provide valuable tools to shed light on the impact of the magnetic fields on
the formation and evolution of subparsec scale hub-filaments that will be
mapped with the NIKA2 camera and future experiments.

##  1 Introduction

This article gives a short review of the main methods used to analyse linear
polarization maps. Given the more and more prolific literature on the subject,
not all the articles using each technique are cited but the generic articles
introducing the conceptual ideas behind each procedure are used as references.
Additional methods employing wavelet transform analysis technics and Bayesian
inference statistical tools currently in development are not discussed here,
and we refer the interested reader to the method investigated by
robitaille2019  and to the IMAGINE consortium project imagine2018 ,
respectively. The review focuses on the analysis of maps obtained from the
observation of linearly polarized thermal dust emission at submillimeter
(submm) wavelengths toward Galactic Molecular Clouds (MCs) and protostellar
cores, but the same methods can be used on maps obtained from simulations.
Independently of the dust grain alignment mechanism that is considered (see
and2015  for a review on this subject), the main accepted current picture is
that of dust grains aligned perpendicular to the local magnetic field
direction pervading the Interstellar Medium (ISM). Each polarization pseudo-
vector displayed in one pixel of a polarization map is therefore an average
measurement of the weighted contribution by all dust grains along a given
Line-Of-Sight (LOS) in a direction perpendicular to the average magnetic field
on the Plane-Of-Sky (POS). From the measurements of the Stokes parameters $I$,
$Q$ and $U$ the total fraction of polarization
($p=\frac{\sqrt{(Q^{2}+U^{2})}}{I}$) is often represented by the length of the
pseudo-vector and the polarization angle (P.A.; $\theta=\frac{1}{2}\times\rm
arctan(\frac{U}{Q})$) by its orientation with respect to a given reference
frame. Other representations of $p$ exists in the literature and maps showing
only drapery patterns of the POS magnetic field lines, or of the P.A.s, are
more and more common. An example of submm polarization is shown in figure 1
(left panel).

Figure 1: Left: Figure 1 from koch2012  showing the Submillimeter Array (SMA)
870 $\mu$m polarization map of the Collapsing core W51 e2 obtained by tang2009
. The thick red segments show the magnetic field orientation after rotation by
90∘ of the polarization pseudo-vectors (see Introduction) where polarization
was detected such that $p/\sigma_{p}>3$. The degree of polarization is not
represented in this map. The blue pseudo-vectors show the gradient directions
(see Section 3 and Section 4) of the dust emission continuum mapped at 1.3 mm
by lai2001  with the Berkeley-Illinois-Maryland Association (BIMA). Right:
Figure 2 from poidevin2013  showing the ADF of the SCUBA JCMT DR21 850 $\mu$m
polarization maps (see Section 2).

##  2 Davis-Chandrasekhar and Fermi (DCF) method and Angular Dispersion
function (ADF)

The DCF method was introduced in the middle of the twentieth century by
davis1951  and cf1953 . It was first designed to get estimates of the POS
magnetic field strength, $B_{pos}$, assuming a turbulent diffuse ISM. It was
applied on polarimetry data obtained on star fields by hiltner1951  at visible
wavelengths. In this regime, the polarization is produced by dichroic
absorption of starlight by dust grain layers pervading the diffuse ISM, in a
direction perpendicular to the one produced in emission at submm wavelengths
by identical polarizing dust grains (see planck_ir_xxi ). Inferred from the
data is a large-scale uniform magnetic field along the Galactic Plane (GP).
The fluctuations around the mean of the distribution of the polarization
pseudo-vectors are assumed by davis1951  and cf1953  to be produced by Alfvén
waves that are coupled to the gas such that there is equipartition between
kinetic and perturbed magnetic energies. With this model $B_{pos}$ is a
function of the ISM gas density $\rho$, of the gas velocity dispersion
$\sigma_{v}$, and of the polarization angle dispersion $\sigma_{\theta}$, such
that: $B_{pos}=Q\sqrt{4\pi\rho}\frac{\sigma_{v}}{\sigma_{\theta}}$, where $Q$
is a factor of proportionality. With this method, cf1953  calculated a diffuse
ISM $B_{pos}$ estimate of a few $\mu$G consistent with those obtained with
other independent methods. Later on, in the 1990s, when polarimetry detectors
at submm wavelengths started to be sensitive enough, the DCF method was used
on submm maps of molecular clouds, meaning transposed to spatial scales 1000
to 10000 times smaller than the GP scale in regions of density two orders or
more magnitudes that of the diffuse ISM density. The first comprehensive study
was done by gonatas1990  from 10 measurements obtained at 100$\,\mu$m with the
Kuiper Airborne Observatory (KAO) in the Orion Nebula, leading to $B_{pos}$
estimates of a few mG. The DCF method has been tested numerically by
ostriker2001  and dfg2008  and some refinements to the calculations of
$B_{pos}$ have been proposed. A review on the values of $Q$ has been given by
poidevin2013 .

Improvements to the Method: Further major improvements to the method were
designed by hildebrand2009 , houde2009  and houde2011 . The Angular Dispersion
Function (ADF) was introduced by hildebrand2009  to avoid inaccurate estimates
of magnetohydrodynamic or turbulent dispersion, as well as to avoid inaccurate
estimates of $B_{pos}$, due to the large-scale, non turbulent field structure.
The ADF is expressed by
$<\Delta\Phi^{2}(l)>^{1/2}\equiv\left\\{\frac{1}{N(l)}\Sigma_{i=1}^{N(l)}[\Phi(x)-\Phi(x+l)]^{2}\right\\}^{1/2}$,
where $\Phi(x)$ is the angle asssociated to the projected POS magnetic field
vector $B(x)$ at position $x$ in a map. The difference in angle between two
points is obtained by $\Delta\Phi(l)\equiv\Phi(x)-\Phi(x+l)$, and is
calculated between the $N(l)$ pairs of vectors separated by displacement, or
lag, $l$. $<...>$ denotes an average and $l=|l|$. The square of the ADF, a
second order structure function, is also often used (e.g. dfg2008 ). One
example of ADF is shown in Figure 1 (right panel), where the angular
dispersion $b$ is the intercept of the fit at $l=0$. Correlations in
polarization angles at lags $l$ smaller than the telescope beam
($1.22\lambda/D$) or than the turbulent correlation length ($\delta$) have to
be avoided. The fit is ideally applied on the set of points calculated by
taking into account the measurement uncertainties and such that the lag
distance $l$ is smaller that the typical length scale $d$ for variations in
the large-scale magnetic fields. Once $b$ is estimated from a map this method
also provides an estimate of the turbulent to large-scale magnetic field
strength ratio such that
$\frac{<B_{t}^{2}>^{1/2}}{B_{pos}}=\frac{b}{\sqrt[]{2-b^{2}}}$, and the POS
strength of the large-scale component is estimated by
$B_{pos}\simeq\sqrt[]{8\pi\rho}\frac{\sigma_{v}}{b}$. The method to take into
account the effect of the signal integration through the thickness of the
clouds as well as across the area sustended by the telescope is fully
incorporated by houde2009 . The authors also show how to evaluate the
turbulent magnetic field correlation length scale from polarization maps
obtained with sufficiently high resolution and high enough spatial sampling
rate. Further examples are given and discussed by houde2011  as well as the
application of the technique to interferometry measurements.

Results: The DCF method has been applied to data from many Galactic MCs
obtained on sky patches, including Gould belt MCs (e.g. coude2019 ) and some
of the closests low, intermediate or high star-forming regions. Estimates of
$B_{pos}$ lie typically in the range of a few $\mu$G to a few mG. In OMC-1 the
turbulent correlation length is estimated to $\delta\approx$ 10 mpc (e.g.
houde2009 ). Independently of the uncertainties coming from the propagation of
the errors, the estimates of $B_{pos}$ and $\delta$ will rely on the choice
(or availabilty) of the gas tracers and of the value of $Q$ used for the
calculation of $B_{pos}$ (e.g. see discussion in pattle2017 ). In principle
accurate and reliable results can be obtained with polarization data of
sufficient spatial resolution and high enough spatial sampling rate (houde2009
). The DCF method has been applied to a large fraction of the sky by
planck_ir_xix  and on the full sky by planck2018_xii . Using the Planck
Release 3 planck2018_xii  have obtained the following relation between the
ADF, $S$, and the fraction of polarization, $p$, as a function of the map
resolution, $w$:
$<S_{p}>=\frac{0^{\circ}.31}{p}\left(\frac{w}{160^{{}^{\prime}}}\right)$. The
results are displayed in Figure 2 (top-left panel), and show that down to a
resolution of $10^{\prime}$ the systematic decrease in $p$ with $N_{\rm H}$ is
determined mostly by the magnetic-field structure. At a lower resolution of
$2.5^{\prime}$, using the $500\mu$m BLASTPol polarization map of Vela C,
fissel2016  discuss the dependence of $p$ on the dust temperature and on
$N_{\rm H}$ and show that $p\varpropto N_{\rm H}^{-0.45}S^{-0.60}$, suggesting
that dust grain alignment properties may also contribute to the decrease of
$p$ in some conditions.

Figure 2:  Top-Left: Figure 11 from planck2018_xii  showing the $S\times p$
relation as a function of column density $N_{\rm H}$ where $w$ is the map
resolution parameter (see Section 2). Bottom-Left: Figure 6 from fissel2019
showing the PRS $Z_{x}$ corrected for oversampling obtained with several
molecular tracers (see Section 3). Right: Figure 3 from koch2012  showing the
90∘ rotated P.A. $\alpha$, and the angle associated to the gradient
orientation $\psi$, used to retrieve information on the magnetic field
strength and significance (see Section 4).

##  3 Histogram of Relative Orientations (HROs) between ISM tracer structures
and magnetic field structures

This method has been designed by soler2013  and subsequently applied to many
molecular cloud regions (e.g. soler2017 , planck_ir_xxxv ). The concept of the
method relies on the calculation of the gradient of the column density
structure provided by a given material tracer (e.g. $N_{\rm H}$) in the pixels
of a polarization map. As an illustration the blue pseudo-vectors displayed in
Figure 1 (left panel), show the gradient orientations obtained from a dust
emission continuum map. Once the gradients are quantified their relative
orientation with the POS magnetic field orientations inferred from the P.A.s
can be estimated and the HROs can be analysed. Several methods exist to
calculate the gradients in a map as a function of the morphology structure of
the object one is interested to isolate (see e.g. alina2019  and references
therein). HROs can be built by considering the relative orientation of P.A.s
with respect to the average orientation of a large-scale structure identified
in a map (e.g. alina2019 ). More frequently, HROs are calculated by
considering the relative orientation angle $\phi$ between the POS magnetic
field $<\widehat{B}_{pos}>$ and a line tangent to the local iso-contour (see
soler2013 , soler2017 , fissel2019 ) which is equivalent to the angle between
the polarization direction $\widehat{E}$ and the intensity gradient
$\bigtriangledown I$: $\phi=\rm arctan(|\bigtriangledown
I\times\widehat{E}|,\bigtriangledown I.\widehat{E})$. Statistical measures of
HROs have been quantified using the histogram shape parameter (e.g. soler2013
, soler2017 ), $\zeta=\frac{A_{0}-A_{90}}{A_{0}+A_{90}}$, where $A_{0}$ is a
measure of the total number of points in the quartile
$0^{\circ}<\phi<22^{\circ}.5$, and $A_{90}$ a measure of the same quantity in
the quartile $67^{\circ}.5<\phi<90^{\circ}$. Smoother and more accurate
definitions of $\zeta$ have been investigated by jow2018 . For this Rayleigh
statistics $Z$ is used to test whether a given set ${\theta_{i}}$ of $n$
independent angles are uniformly distributed within the range$[0,2\pi]$. Using
the relation $\theta=2\phi$, where $\phi$ is the relative orientation angle
discussed above, jow2018  showed that the Projected Rayleigh Statistic (PRS)
of $Z$: $Z_{x}=\Sigma_{i}^{n_{\rm ind}}\rm cos\theta{{}_{i}}/\sqrt{n_{\rm
ind}/2}$, where $n_{\rm ind}$ is the number of independent data samples in the
map, can be used to test for a preference for perpendicular or parallel
alignment between the magnetic field orientations and the iso-contours. The
variables $\zeta$ or $Z_{x}$ are often calculated on samples of data lying in
different ranges of intensity of the material tracer map, and used to explore
variations of the relative orientations between the magnetic field structures
and the ISM morphology structures as a function of the column density or
number density parameters. Figure 2 (bottom-left panel) shows the PRS obtained
by fissel2019  using several molecular tracers. For each tracer, $Z_{x}>0(<0)$
indicates the $I$ structure preferentially aligns parallel (perpendicular) to
$<\widehat{B}_{pos}>$.

Results: The main trend found from the analysis of HROs derived with lines
tangent to local iso-contours is that the POS magnetic field orientations
change from mostly parallel (or not clearly defined) to perpendicular with
respect to the molecular cloud structures probed with a given material tracer
(soler2017 , planck_ir_xxxv , fissel2019 ). This translates by a change of
orientation from lower to higher column densities $N_{\rm H}$. In Vela C the
transition is estimated to occur at molecular hydrogen number density of
approximately $n_{\rm H}\approx 10^{3}\rm cm^{-3}$ (fissel2019 ). This local
iso-contours approach has been tested by simulations (e.g. soler2013 ). In
their study alina2019  first produce a component separation analysis of the
magnetic fields in the diffuse ISM and in higher column density regions, and
use image analysis technics to extract filaments and clumps embedded in
different background column densities, i.e. showing density contrast varying
with their environment. Their analysis of the HROs obtained between filaments,
embedded clumps and internal and background magnetic field orientations lead
to a more complex picture than in other studies. Overall their results support
the possibility of magnetic fields strong enough to influence the formation of
molecular clouds and also of their embedded clumps.

##  4 Polarization-Intensity Gradient Relation (P-IGR)

This method has been designed by koch2012  to study star-forming regions. In
the case of negligible viscosity and infinite conductivity (ideal MHD case)
the force equation is given by the following expression
$\rho\left(\frac{\partial}{\partial
t}+\mathbf{v}.\bigtriangledown\right)\mathbf{v}=-\bigtriangledown\left(P+\frac{B^{2}}{8\pi}\right)-\rho\bigtriangledown\phi+\frac{1}{4\pi}(\mathbf{B}.\bigtriangledown)\mathbf{B},$
where $\rho$ and v are the dust density and velocity, respectively. B is the
magnetic field, $P$ is the hydrostatic dust pressure, $\phi$ is the
gravitational potential resulting from the total mass contained in the region
of interest and, $\bigtriangledown$ denotes the gradient. The left-hand side
term in the equation represents the resulting motion of the dust produced by
the right-hand side terms which are the gradients of the hydrostatic pressure
terms of the gas, the magnetic field, and the gravitational potential together
as well as the magnetic field tension term (last term). The force equation can
be transformed and under several assumptions the magnetic field strength can
in principle be derived geometrically at each position of a polarization map
and expressed by: $B=\sqrt{\frac{\rm sin(\psi)}{\rm
sin(\alpha)}(\bigtriangledown P+\rho\bigtriangledown\phi)4\pi R}$, where
$\alpha$ = P.A.-90∘, and $\psi$ is the angle associated to the gradient
orientation. Figure 2 (right panel), illustrates the terms displayed in this
equation. The red and blue pseudo-vectors displayed in the figure can be
compared to those displayed in figure 1 (left panel) where the gradients in
intensity are estimated assuming central symmetry towards the brightest pixel
in the map, but the method can be generalized to arbitrary cloud shapes. An
important outcome of the method is given by the magnetic field significance:
$\Sigma_{B}\equiv\left(\frac{\rm sin(\psi)}{\rm
sin(\alpha)}\right)_{local}=\left(\frac{F_{B}}{\mid
F_{G}+F_{P}\mid}\right)_{local}$, which gives a direct physical meaning to the
factor $\frac{\rm sin(\psi)}{\rm sin(\alpha)}$ derived geometrically from a
map.

Results: The method provides a quantification of the local significance of the
magnetic field force compared to the other forces in a model-independent way.
In W51 e2 it allows derivations of the azimuthally averaged radial profile
$B(r)\sim r^{-1/2}$ (koch2012 ). The potential of the method is explored in
additional works (e.g. koch2018  and references therein). In their study,
tang2019  propose that in G34 the varying relative importance between magnetic
field, gravity, and turbulence from large-to-small-scale drives and explains
the different fragmentation types seen at subparsec scales (no fragmentation,
aligned fragmentation and clustered fragmentation).

##  5 Conclusions and perspectives

The methods discussed above are complementary to each other. If submm
polarization maps are available at different resolutions they are suited to
explore the role of magnetic fields, turbulence and gravity on different
spatial scales. Their combination starts to give insights on the interplay
between magnetic fields, gravity and turbulence (e.g. tang2019 ) and they are
promising tools to shed light on the physics of hubs-filaments (e.g. andre2019
) detected at subparsec scales in molecular clouds. In addition to these
methods, getting estimates of the mean inclination with respect to the LOS of
the large-scale ordered magnetic field considered as the main factor
regulating the mean level of polarization in a map can help guiding the
overall interpretation. In this regard, a method first proposed by
poidevin2013  has been further explored by other authors (e.g. chen2019 ).
Multi-wavelengths submm polarimetry of a region can give insights about
polarization dust grain properties and also help to put constraints on the
interpretation of the maps. On this matter we refer the reader to and2015 ,
vaillancourt2012 , shariff2019  and references therein.

Acknowledgments: FP acknowledges support from the Spanish Ministerio de
Economia y Competitividad (MINECO) under grant numbers ESP2015-65597-C4-4-R
and ESP2017-86852-C4-2-R.

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