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)
		$\displaystyle=$	$\displaystyle\min\{\omega_{1},\omega_{2}\},$		(13)

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 .

Refer to caption — 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.