Quantum Data-Syndrome Codes

TL;DR

This paper introduces quantum data-syndrome (DS) codes that correct errors in both data qubits and syndrome bits, providing theoretical bounds, properties, and constructions to enhance quantum error correction reliability.

Contribution

It develops the theory of quantum DS codes, including bounds, properties, and new constructions based on classical codes, improving error correction in quantum computing.

Findings

Quantum DS codes can correct both data and syndrome errors.

Random DS codes achieve the Gilbert-Varshamov bound.

Constructed DS codes include CSS-type codes like Steane and Golay codes.

Abstract

Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of error information--error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose the idea of quantum data-syndrome (DS) codes that are capable of correcting both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes, including split weight enumerators, generalized MacWilliams identities, and linear programming bounds. In particular, we derive Singleton and Hamming-type upper bounds on degenerate quantum DS codes. Then we study random DS codes and show that random DS codes with a relatively small additional syndrome measurements achieve the Gilbert-Varshamov bound of stabilizer codes. Constructions of quantum DS codes are also discussed. A family of quantum DS codes is based on classical linear block codes, called syndrome measurement codes, so that syndrome bits are encoded in additional redundant stabilizer measurements. Another family of quantum DS codes is CSS-type quantum DS codes based on classical cyclic codes, and this includes the Steane code and the quantum Golay code.

Tables

Table 1. TABLE I: The distance of H ~ ~ 𝐻 \tilde{H} corresponding to different number of rows.

$r$ 0 1 2 3 4 5 6 7 8 9 $d$ 3 4 4 4 5 5 5 6 6 7

Equations

$$\displaystyle\langle g,h\rangle_{{{\cal G}}_{n}}=\begin{cases}0,&\mbox{if $gh=hg$};\\ 1,&\mbox{otherwise}.\end{cases}$$

$$\mathbf{x}\ast \mathbf{y}=\text{mbox}{\mathrm{Tr}}_{{\mathbb{F}}_2}^{{\mathbb{F}}_4}\left(\sum _{i=0}^n{x}_i{\overline{y}}_i\right),$$

$$\displaystyle H=\left[\begin{array}[]{c}{\bf g}_{1}\\ \vdots\\ {\bf g}_{n-k}\end{array}\right],$$

$$\hat{H}=[H{I}_m],$$

$$D$$

$$G_C=\left[I_{m}A\right],$$

$${\mathbf{f}}_j=a_{1,j}{\mathbf{g}}_1+\cdots +a_{m,j}{\mathbf{g}}_m,\text{mbox}forj=1,\dots ,r.$$

$$H_{\mathrm{DS}}=\left[\begin{array}[]{ccc}H&I_{m}&0\\ 0&A^{T}&I_{r}\end{array}\right].$$

$$C_{\mathrm{DS}}$$

$$C_{\mathrm{DS}}^{\perp }$$

$$\mathbf{x}\star \mathbf{y}=\text{mbox}{\mathrm{Tr}}_{{\mathbb{F}}_2}^{{\mathbb{F}}_4}\left(\sum _{i=1}^n{x}_i{\overline{y}}_i\right)+\sum _{j=1}^{m+r}{x}_{n+j}{y}_{n+j}.$$

$$G_{C_{\mathrm{DS}}^{\perp}}=\left[\begin{array}[]{lll}G&{\bf 0}&{\bf 0}\\ H_{1}&{\bf 0}&{\bf 0}\\ H_{2}&I_{n-k}&A\end{array}\right],\mbox{ where }G=\left[\begin{array}[]{c}{\bf g}_{1}\\ \vdots\\ {\bf g}_{n}\end{array}\right],H_{1}=\left[\begin{array}[]{c}{\bf h}_{n-k+1}\\ \vdots\\ {\bf h}_{n}\end{array}\right],H_{2}=\left[\begin{array}[]{c}{\bf h}_{1}\\ \vdots\\ {\bf h}_{n-k}\end{array}\right],$$

$$p_{err}({\mathbf{g}}_j)=\mathrm{Pr}({\hat{s}}_j\ne s_j)$$

$$\mathbf{x}=(s_1,\dots ,s_m,z_1,\dots ,z_r)$$

$$\hat{\mathbf{x}}=({\hat{s}}_1,\dots ,{\hat{s}}_m,{\hat{z}}_1,\dots ,{\hat{z}}_r).$$

$$P_{se}=\mathrm{Pr}((s_1,\dots ,s_m)\ne ({\tilde{s}}_1,\dots ,{\tilde{s}}_m)),$$

$$P_{SBER}=\frac{1}{m}\sum _{j=1}^m\mathrm{Pr}({\tilde{s}}_j\ne s_j).$$

$$G_{C}=\left[\begin{array}[]{l}100001111111000\\ 010010011110110\\ 001101100110111\end{array}\right].$$

$$C_{\mathrm{DS}}^{\perp }\setminus \{(\mathbf{g},0):\mathbf{g}\in C\}.$$

$$B_{i,j}=B_{i,j}(C_{\mathrm{DS}})=|\{$$

$$B_{i,j}^{\perp }=B_{i,j}(C_{\mathrm{DS}}^{\perp })=|\{$$

$$B_{i,0}^{\perp }=\sum _{j=0}^{m+r}{B}_{i,j},\text{ for }i=1,\dots ,d-1.$$

$$B_{i,0}^{\perp }⩾$$

$$B_{0,0}=B_{0,0}^{\perp }=1\text{mbox}and$$

$$B_{d(0),0}^{\perp }>\sum _{j=1}^{m+r}{B}_{d(0),j},\text{mbox}and{B}_{d(i),i}^{\perp }>0,\text{mbox}fori=1,\dots ,m+r.$$

$$d=\underset{0⩽i⩽m+r}{\min }d(i)+i.$$

$$K_i(x;n,q)=\sum _{j=0}^i(-1{)}^j(q-1{)}^{i-j}\left(\genfrac{}{}{0ex}{}{x}{j}\right)\left(\genfrac{}{}{0ex}{}{n-x}{i-j}\right).$$

$$f(x,y)=\sum _{i=0}^{d_x}\sum _{j=0}^{d_y}{f}_{i,j}{K}_i(x;n,q_1)K_j(y;m+r,q_2),$$

$$f_{i,j}=\frac{1}{q_1^n{q}_2^{m+r}}\sum _{x=0}^n\sum _{y=0}^{m+r}f(x,y)K_x(i;n,q_1)K_y(j;m+r,q_2).$$

$$B_{x,y}=\frac{1}{4^n}\sum _{i=0}^n\sum _{j=0}^{m+r}{B}_{i,j}^{\perp }{K}_x(i;n,4)K_y(j;m+r,2).$$

$$\text{mbox}nonnegativeintergers{B}_{i,j},B_{i,j}^{\perp }$$

$$(\text{ref}eq:stabilize{r}_{n}umber),(\text{ref}eq:Bperp>B),(\text{ref}eq:Bi0=0),\text{mbox}and(\text{ref}eq:MacW).$$

Full text

I Introduction

Quantum error-correcting codes provide a method of actively protecting quantum information [1]. In a quantum error-correcting code, quantum information is stored in the joint $+1$ eigenspace of a set of Pauli operators, called stabilizers. To perform quantum error correction, we have to learn knowledge of errors, the error syndromes (in bits), through quantum measurements. More precisely, the error syndrome are given by measuring a generating set of the stabilizers. Realistically, the quantum gates used to perform quantum error correction are themselves faulty and thus the measurement outcomes for the error syndrome can be wrong due to faulty measurements or newly introduced errors from faulty gates.

In this article, we are interested in eliminating the effect of faulty syndrome measurement. Typically, this can be done with the syndromes being measured repeatedly in the case of Shor’s syndrome extraction [2]. In other words, more redundant measurements than necessary are required to determine a reliable error syndrome. Other protocols have also been proposed to handle measurement errors for color codes and other topological codes s [3, 5, 4], following Bombin’s seminal work on the so-called single-shot fault-tolerant quantum error correction on color codes [3]. Very recently Campbel has proposed a theory for this one-shot error correction [6]. Herein we consider a general scheme of quantum stabilizer codes that are capable of correcting data qubit errors and syndrome bit errors simultaneously with the help of additional stabilizer measurements. These codes are called quantum data-syndrome (DS) codes [7, 8]. This idea is also independently studied by Fujiwara in [9]. Constructions and simulations of LDPC and Convolutional DS codes have been studied in [7, 10].

In this paper, we give a comprehensive study of quantum DS codes, which completes our previous work [7, 8]. We first define quantum DS codes. In addition to quantum data errors, a measurement outcome suffers a measurement error depending on the weight of the measured stabilizer. One may use redundant stabilizer measurements to decode the error syndrome first and then do quantum error correction. The overall procedure can be seen as decoding a larger code that is a concatenation of a binary code (for syndrome bits) and a quaternary code (for qubits). Thus we call such scheme a quantum DS code. Given an $[[n,k,d]]$ stabilizer code that encodes $k$ information qubits in $n$ physical qubits with minimum distance $d$, we denote the corresponding DS code with additiona $r$ redundant stabilizer measurements by the parameters $[[n,k,d:r]]$. To get familiar with how quantum DS codes work, we introduce a family of quantum DS codes such that $r$ additional stabilizer measurements are based on classical error-correcting codes. The idea of repeated syndrome measurements is similar to using a classical repetition code. We generalize this approach by introducing the idea of syndrome measurement (SM) codes based on classical linear block codes. Examples show that syndrome decoding can be improved using SM codes than repeted syndrome measurements.

Then we define notions of minimum distance and split weight enumerators. Naturally, the generalized MacWilliams identities hold for DS codes, which lead to the linear programming bounds on the minimum distance of small DS codes. Armed with the MacWilliams identities for DS codes, we further analyze algebraic linear programming bounds for DS codes, generalizing the approach proposed in [11]. In the case of $r=0$, we derive Singleton and Hamming type upper bounds on the code size of degenerate quantum DS codes. Especially, we demonstrate that the Hamming bounds for nondegnerate codes and for degenerate codes will merge for sufficiently large $n$.

Next we study the properties of random quantum DS codes for the case of $0\leq r\leq n-k$. We will show that the minimum distance of random DS codes with a relatively small $r\leq n-k$ achieves the Gilbert-Varshamov bound of stabilizer codes. Along the way average weight enumerators are also derived, which may be of independent interest, since in classical coding theory, those numerators also lead to upper bound on the probability of decoding error.

Finally, we consider CSS-type ([12, 13]) quantum DS codes and provide DS code constructions from CSS-type quantum cyclic codes such that their minimum distances are as high as their underlying stabilizer codes. The quantum Golay code and Steane code, which are shown to have fault-tolerant error thresholds [14, 15], are included as two examples.

II Preliminaries

In this paper we focus on two-dimenstional quantum systems: qubits. An $n$-qubit state space is a $2^{n}$-dimensional complex Hilbert space $\mathbb{C}^{2^{n}}$ and a pure quantum state is a unit vector in the state space. A basis of the linear operators on the $n$-qubit state space is the $n$-fold Pauli operators $\{M_{1}\otimes\cdots\otimes M_{n}:M_{j}\in\{I,X,Y,Z\}\},$ where $I=\begin{bmatrix}1&0\\ 0&1\end{bmatrix},X=\begin{bmatrix}0&1\\ 1&0\end{bmatrix},Z=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix},Y=iXZ$ are the Pauli matrices. The $n$-fold Pauli group ${\cal G}_{n}$ is the set of $n$-fold Pauli operators with possible phases $\pm 1,\pm i$. Note that Pauli operators either commute or anticommute with each other. We can define an inner product in ${{\cal G}}_{n}$: for $g,h\in{\cal G}_{n}$,

[TABLE]

Often it is convenient to consider the quantum coding problem via codes over the Galios filed of four elements ${\mathbb{F}}_{4}=\{0,1,\omega,{\omega^{2}}\}$ [16]. We can define a homomorphism $\tau$ on ${\cal G}_{1}$ that maps $I,X,Z,Y$ to $0,1,\omega,\omega^{2}$, respectively, regardless of a possible phase $\pm 1,\pm i$ in front of a Pauli matrix, and this homomorphism extends to an $n$-fold Pauli operator naturally. For example, $\tau(\pm iX\otimes Y\otimes Z\otimes I\otimes I)=1\omega^{2}\omega 00$. Throughout the text for an $n$-fold Pauli operator $g$, we denote by ${\bf g}\in{\mathbb{F}}_{4}^{n}$ the corresponding vector and vice versa (up to an appropriate phase). Likewise, we define a trace inner product on $\mathbb{F}_{4}^{n}$: for ${\bf x}=(x_{1},x_{2},\dots,x_{n}),{\bf y}=(y_{1},y_{2},\dots,y_{n})\in\mathbb{F}_{4}^{n}$,

[TABLE]

where $\overline{y}_{i}$ denotes the conjugation of $y_{i}$ in ${\mathbb{F}}_{4}=\{0,1,\omega,{\omega^{2}}\}$ with $\bar{0}=0,\bar{1}=1,\bar{\omega}=\omega^{2},$ and $\bar{\omega^{2}}=\omega$. It can be checked that $\langle g,h\rangle_{{{\cal G}}_{n}}={\bf g}*{\bf h}$ for $g,h\in{\cal G}_{n}$ and ${\bf g}=\tau(g),{\bf h}=\tau(h)$.

Suppose ${\cal S}=\langle g_{1},\dots,g_{n-k}\rangle$ is an Abelian subgroup of ${{\cal G}}_{n}$, where $g_{j}$ are independent generators of ${\cal S}$, such that the minus identity $-I^{\otimes n}\notin{\cal S}$. Then ${\cal S}$ defines a quantum stabilizer code $Q=\{|\psi\rangle\in\mathbb{C}^{2^{n}}:g|\psi\rangle=|\psi\rangle,\forall g\in{\cal S}\}$ of dimension $2^{k}$. The vectors $|\psi\rangle\in{\cal C}({\cal S})$ are called the codewords of ${\cal C}({\cal S})$ and the operators $g\in{\cal S}$ are called the stabilizers of ${\cal C}({\cal S})$. By the quantum error correction conditions [17, 18], it suffices to consider error correction on a discrete set of error operators. Thus we only treat errors that are Pauli operators in this paper.

In the following we will use the corresponding codes over $\mathbb{F}_{4}$ to discuss the quantum error correction procedure of $Q$. Define a check matrix

[TABLE]

where ${\bf g}_{1},\ldots,{\bf g}_{n-k}\in{\mathbb{F}}_{4}^{n}$ are the corresponding vectors of $g_{1},g_{2},\dots,g_{n-k}\in{\cal G}_{n}$. Let $C$ be the classical $[n,n-k]$ code over ${\mathbb{F}}_{4}$ generated by the rows of $H$, and $C^{\perp}$ be its dual with respect to the trace inner product $*$ defined in (2). We have $C\subseteq C^{\perp}$ since ${\bf g}_{i}*{\bf g}_{j}=0$ for all $i,j$. Suppose a quantum codeword $|\psi\rangle\in Q$ is corrupted by a Pauli error $e\in{\cal G}_{n}$ and let ${\bf e}=\tau(e)\in{\mathbb{F}}_{4}^{n}$ be the corresponding vector. Then the syndrome of $e$ is ${\bf s}=(s_{1},\dots,s_{n-k})\in{\mathbb{F}}_{2}^{n-k}$, where $s_{j}={\bf g}_{j}*{\bf e}.$ More explicitly, the syndrome ${\bf s}$ has the commutation relations between the Pauli error $e$ and the stabilizers $g_{1},g_{2},\dots,g_{n-k}$ and it can be obtained by measuring the observables $g_{j}$’s on $e|\psi\rangle$.

III Quantum Data-Syndrome Codes

Using a quantum stabilizer code with a corresponding check matrix $H$, quantum error correction can be done according to the error syndrome ${\bf s}\in{\mathbb{F}}_{2}^{n-k}$. However, the syndrome ${\bf s}$ itself could be measured with an error. So instead of the true vector ${\bf s}\in{\mathbb{F}}_{2}^{n-k}$, we may get, after measurement, a vector $\widehat{{\bf s}}={\bf s}+{\bf z},$ for syndrome error ${\bf z}\in{\mathbb{F}}_{2}^{n-k}$. In other words, syndrome bits could be flipped. Herein we discuss stabilizer codes that are capable of correcting both data errors and syndrome errors. To shorten notation, we will use $m\triangleq n-k$ in the following.

The central idea is that the error syndrome of a measurement error on the $i$th syndrome bit is the vector $(0\cdots 0\underbrace{1}_{i}0\cdots 0)$. Thus we define a new parity-check matrix

[TABLE]

where $I_{m}$ is considered as a matrix over ${\mathbb{F}}_{2}$. Define codes

[TABLE]

where $D^{\perp}$ is the dual code of $D$ with respect to the inner product: ${\bf x}\star{\bf y}=\mbox{{\rm Tr}\,}^{{\mathbb{F}}_{4}}_{{\mathbb{F}}_{2}}\left(\sum_{i=0}^{n}x_{i}\overline{y}_{i}\right)+\sum_{j=0}^{m}x_{n+j}y_{n+j},$ for ${\bf x},{\bf y}\in{\mathbb{F}}_{4}^{n}\times{\mathbb{F}}_{2}^{m}.$ Therefore, a quantum stabilizer code is inherently capable of handling both data and syndrome errors. Fujiwara, [9], noticed that the error-correcting capabilities of $D^{\perp}$ depend on the choice of generators in (7). Choosing generators properly, we can get a code capable of correcting simultaneously multiple data and syndrome errors.

In addition, the error-correcting capabilities of $D^{\perp}$ can be further inhanced. The standard approach to reduce the probability of syndrome measurement error is repeated syndrome measurement. That is, we repeat the syndrome measurement several times and take a majority vote. This is the same idea as in classical repetition codes. We propose a generalization of this idea by measuring additional stabilizers according to more powerful linear classical codes.

Let $C$ be an $[m+r,m]$ linear binary code with a generator matrix in the systematic form

[TABLE]

where $A=[a_{i,j}]$ is an $m\times r$ binary matrix. We define a new set of $r$ stabilizers ${\bf f}_{j}$ by

[TABLE]

These ${\bf f}_{j}$ belong to the stabilizer group ${\cal S}$, and can be measured without disturbing the underlying quantum codewords. For this reason, we call $C$ the syndrome measurement (SM) code. Let H^{\prime T}=\left[\begin{array}[]{ccc}{\bf f}_{1}^{T}&\cdots&{\bf f}_{r}^{T}\end{array}\right]. With additional $r$ stabilizers being measured, it is equivalent to considering the code defined by the following parity-check matrix \left[\begin{array}[]{ccc}H&I_{m}&0\\ H^{\prime}&0&I_{r}\end{array}\right]. This parity-check matrix can be transformed into the form

[TABLE]

We will say that (10) defines a quantum data-syndrome code $Q_{\mathrm{DS}}$. It is convenient to define codes

[TABLE]

where $C_{\mathrm{DS}}^{\perp}$ is the dual code of $C_{\mathrm{DS}}$ with respect to the inner product:

[TABLE]

Slightly abusing terminology, we will call $C_{\mathrm{DS}}^{\perp}$ also a data-syndrome code. We will say that $Q_{\mathrm{DS}}$ (or $C_{\mathrm{DS}}^{\perp}$) has length $n$, dimension $k$, and size $2^{k}$. Note that a DS code of length $n$ and dimension $k$ encodes $k$ (information) qubits into $n$ (code) qubits.

It is easy to see that $|C_{\mathrm{DS}}|=2^{n-k+r}$ and $|C_{\mathrm{DS}}^{\perp}|=2^{2n+n-k+r}/|C_{\mathrm{DS}}|=2^{2n}$, so the size of $C_{\mathrm{DS}}^{\perp}$ does not depend on a choice of $k$ and $r$. For a given matrix (10), we always can find vectors ${\bf g}_{n-k+1},\ldots,{\bf g}_{n}$ and ${\bf h}_{1},\ldots,{\bf h}_{n}$ over $\mathbb{F}_{4}$ such that ${\bf g}_{i}*{\bf g}_{j}=0$, ${\bf g}_{i}*{\bf h}_{j}=0$ for $i\neq j$ and ${\bf g}_{i}*{\bf h}_{i}=1$. These vectors allow us to write down a generator matrix of $C_{\mathrm{DS}}^{\perp}$ in the following explicit form

[TABLE]

and ${\bf 0}$s are all zero matrices of appropriate sizes.

We will say that a code defined by (10) is an $[[n,k:r]]$ DS-code. If the minimum distance $d$ (defined in Section V) is known we will say that it is an $[[n,k,d:r]$ code.

IV Syndrome measurement codes

In this section, we consider some examples of stabilizer codes for which it is easy to find a efficient SM code that beat the repetitive syndrome measurement approach.

Suppose we are using an $[[n,k]]$ stabilizer code defined by a stabilizer group $\mathcal{S}$ with generators ${\bf g}_{1},\ldots,{\bf g}_{n-k}$. Let ${\bf s}=(s_{1},\ldots,s_{n-k})$ be the correct syndrome measurement outcomes. Let $\hat{{\bf s}}=(\hat{s}_{1},\ldots,\hat{s}_{n-k})$ be the (imperfect) syndrome bits output by Shor’s syndrome extraction. Assume that the probability that an $X$ or $Z$ measurement error occurs with probability $p_{m}$. Then the probability of incorrect measurement outcome on ${\bf g}_{j}$ is

[TABLE]

Now suppose that an $[m+r,m]$ SM code $C$ is used. Denote by $s_{j}$ and $z_{j}$ the results of correct measurement of ${\bf g}_{j}$ and ${\bf f}_{j}$, respectively. It is not difficult to see that

[TABLE]

is a valid codeword of $C$. After the measurement of ${\bf g}_{1},\ldots,{\bf g}_{m}$ and ${\bf f}_{1},\ldots,{\bf f}_{r}$, we obtain a vector

[TABLE]

We can correct quantum and syndrome errors simultaneously by decoding vector $(\underbrace{0,\ldots,0}_{n\mbox{\small times}},\widehat{{\bf x}})$ using a decoder of $C_{\mathrm{DS}}^{\perp}$. Alternatively we can first correct syndrome errors by decoding vector $\widehat{{\bf x}}$ using a decoder of the SM code $C$, and next correct quantum errors. The latter approach is typically simpler, though its performance is always suboptimal. In this section we consider this type of decoding.

Applying a decoding algorithm of $C$ to $\widehat{{\bf x}}$, we obtain bits $\tilde{s}_{1},\ldots,\tilde{s}_{m}$. For a given $C$ and its decoding algorithm, we define the syndrome decoding error and average syndrome decoding error, respectively, as

[TABLE]

[TABLE]

The $l$-fold repeated syndrome measurement can be considered as a particular case of an encoded syndrome measurement. It corresponds to the SM code with generator matrix $G=[\underbrace{I_{m}\cdots I_{m}}_{l\mbox{ times}}].$ Choosing a good SM code is not equivalent to finding a good $[m+r,m]$ linear code in the usual sense. This is because for a typical $[m+r,m]$ code with a large minimum distance, the matrix $A$ in (8) will have “heavy” columns. This may result in that $\mbox{{\rm wt}\,}({\bf f}_{j})>>\mbox{{\rm wt}\,}({\bf g}_{l})$ and therefore $p_{err}({\bf f}_{j})>>p_{err}({\bf g}_{i})$, which, in turn, will lead to large $P_{se}$ and $P_{SBER}$.

Below we present several families of high rate quantum codes with the property that all their stabilizers ${\bf g}\in{\cal S}$ have the same or almost the same weight and therefore any good linear codes can be used for robust syndrome measurement.

Let $S_{a}$ be a generator matrix of the $[2^{a}-1,a,2^{a-1}]$ simplex code. Let ${\cal S}_{a}$ be the $[[2^{a}-1,2^{a}-1-2a,3]]$ CSS code defined by the generators \left[\begin{array}[]{cc}S_{a}&0\\ 0&S_{a}\end{array}\right]. Any liner combination of the first (second) $a$ generators is a vector of weight $2^{a-1}$. Thus we can use any good $[a+r,a]$ linear code for syndrome measurement of the first (second) $a$ syndrome bits. For example, consider ${\cal S}_{3}$, which is the $[[7,1,3]]$ Steane code. Let us use as an SM code the $[15,3]$ code $C$ defined by:

[TABLE]

The code $C$ requires $15$ measurements, which is the same as for the 5-fold repeated measurements of 3 bits. The corresponding probabilities $P_{se}$ are shown in Fig.1. One can see that the code $C$ has significantly lower $P_{se}$.

Another important family is the $[[n,n-2a,3]]$ quantum Hamming codes ${\cal H}_{a}$ for $n=(4^{a}-1)/3$ [16, V]. It is not difficult to prove that all generators of ${\cal H}_{a}$ have weight $4^{a-1}$.

In [16, Thm 11] a family of $[[n,n-a-2,3]]$ codes with $n=\sum_{i=1}^{(a-1)/2}2^{2i+1}$ is defined for odd $a$. The generators of these codes can have only weights $2^{a}-2$ and $2^{a}$.

V Mimimum Distance and Split Weight Enumerators

Let ${\bf e}_{\mathrm{DS}}=({\bf g},{\bf 0})\in{\mathbb{F}}_{4}^{n}\times{\mathbb{F}}_{2}^{m+r},$ with ${\bf g}\in C$. Since ${\bf g}\in C$, we have $g\in{\cal S}$ and thus ${\bf e}_{\mathrm{DS}}$ is harmless. If ${\bf e}_{\mathrm{DS}}=({\bf e},{\bf z})\in C_{\mathrm{DS}}^{\perp}\setminus\{({\bf g},{\bf 0}):{\bf g}\in C\}$, then ${\bf e}\not\in C$. Therefore the operator $e$ does not belong to ${\cal S}$ and acts on $Q$ nontrivially. Since $H_{\mathrm{DS}}\star{\bf e}_{\mathrm{DS}}={\bf 0}^{T}$ by definition, we conclude that such ${\bf e}_{\mathrm{DS}}$ is an undetectable and harmful error. Naturally, the weight $\mbox{{\rm wt}\,}({\bf e},{\bf z})$ is defined as the number of its nonzero entries. We define the minimum distance $d$ of $Q_{\mathrm{DS}}$ (equivalently $C^{\perp}_{\mathrm{DS}}$) as the minimum weight of any element in

[TABLE]

It is not difficult to see that $Q_{\mathrm{DS}}$ (or equivalently $C_{\mathrm{DS}}^{\perp}$) can correct any error ${\bf e}_{\mathrm{DS}}=({\bf e},{\bf z})$ (here we do not assume ${\bf e}_{\mathrm{DS}}\in C_{\mathrm{DS}}^{\perp}$) with $\mbox{{\rm wt}\,}({\bf e})=t_{{}_{\mathrm{D}}},\mbox{{\rm wt}\,}({\bf z})=t_{{}_{\mathrm{S}}}$ if $t_{{}_{\mathrm{D}}}+t_{{}_{\mathrm{S}}}<{d\over 2}.$ Apparently the minimum distance of a DS code cannot be greater than that of the underlying stabilizer code.

Define the split weight enumerators of $C_{\mathrm{DS}}$ and $C_{\mathrm{DS}}^{\perp}$ by

[TABLE]

respectively. The minimum distance $d$ of $Q_{\mathrm{DS}}$ implies that

[TABLE]

We will say that $Q_{\mathrm{DS}}$ is a degenerate quantum DS code if there exists $B_{i,j}>0$ for $i<d$. Otherwise, it is a nondegenerate quantum DS code. Clearly, we also have

[TABLE]

For $0\leqslant i\leqslant m+r$, let us define $d(i)$ as the smallest integer such that

[TABLE]

Then the minimum distance of $Q_{\mathrm{DS}}$ is

[TABLE]

Denote the $q$-ary Krawtchouk polynomial of degree $i$ by

[TABLE]

We list the properties of Krawtchouk polynomials needed in this work in Appendix -A. Their proof and other information on these polynomials can be found in [19, 20, 11]. Let $f(x,y)$ be a two-variable polynomial and its maximal degrees of $x$ and $y$ be $d_{x}\leqslant n$ and $d_{y}\leqslant m+r$, respectively. Then the following Krawtchouk expansion of this polynomial holds:

[TABLE]

where

[TABLE]

Proofs of these equalities are straightforward generalizations of the proofs (see [33, Chapter 5]) for equivalent expressions for single variable polynomials.

In what follows we will need the following generalization of MacWilliams identities [19].

Theorem 1**.**

[TABLE]

A proof of this theorem can be found in Appendix -B.

Like [16, 21, 27], for small $n$ one could apply linear programming techniques to obtain upper bounds on the minimum distance of $[n,k:r]]$ DS codes. More explicitly, we have the following linear program: given $n,k,d,$ and $r$,

[TABLE]

If there is no solution to this feasibility problem, it means that no $[[n,k,d:r]]$ DS code exists.

Example 2**.**

Let $n=7,m=6,r=0$, and $d=3$. Using MAPLE, we find out that the liner program has solutions. This means that a $[[7,1,3]]$ code may be capable of fighting a syndrome bit error by measuring only six stabilizer generators. Indeed, it is the case that a $[[7,1,3:6]]$ DS code exists as shown by Fujiwara [9].

VI Upper Bounds on Unrestricted (Degenerate and Non-Degenerate) DS codes

In this Section we propose a general method defined in Theorem 3 for deriving upper bounds on the minimum distance of both non-degenerate and degenerate DS codes. Next we use this method for obtaining several explicit bounds for DS codes with $r=0$. Theorem 3 can be used for deriving bounds in the case of $r>0$, but this will be done in future work.

Let $1\leqslant d_{\mathrm{D}}\leqslant n$ be an integer and ${\cal N}=\{(i,j):0\leqslant i\leqslant n,1\leqslant j\leqslant m+r\}.$ Let also ${\cal A}\subset{\cal N}$ and $\overline{{\cal A}}={\cal N}\setminus{\cal A}$. We would like to upper bound quantum code rate $R=k/n$ under the conditions:

[TABLE]

Theorem 3**.**

Let $f(x,y)$ be an arbitrary polynomial with nonnegative coefficients $f_{i,j}$ that satisfies the conditions:

[TABLE]

For non-degenerate $C^{\perp}_{\mathrm{DS}}$, it must hold that

[TABLE] 2. 2.

For unrestricted $C^{\perp}_{\mathrm{DS}}$, it must hold that

[TABLE]

Proof.

We prove the second claim. Let $M=|C_{\mathrm{DS}}^{\perp}|=2^{2n}$. Using (28), (26), and (21), we get

[TABLE]

From this and (22), we get

[TABLE]

∎

For getting a bound on the size of DS codes with minimum distance $d$, it suffices to choose

[TABLE]

and a polynomial $f(x,y)$ that satisfies constraints (31) and (32). In the following subsections, we discuss two polynomials and their corresponding bounds on DS codes with $r=0$.

On the other hand, we also have upper bounds for a DS code inherited from its underlying stabilizer code. Let $Q_{\mathrm{DS}}$ be a DS code defined by (10) ($r=0$) with minimum distance $d(Q_{\mathrm{DS}})$. Let $C$ be the $[n,n-k]$ code with generator matrix $H$ used in (7) and $C^{\perp}$ be its dual code. Let $Q$ be the $[[n,k]]$ stabilizer code defined by $C$, and let its minimum distance be $d(Q)$. From (12) it is easy to see that vectors of the form $({\bf v},{\bf 0}),\mbox{ where }{\bf v}\in C^{\perp},~{}{\bf 0}=(\underbrace{0,\ldots,0}_{n-k}),$ form a subcode of $C_{\mathrm{DS}}^{\perp}$ and therefore $d(Q_{\mathrm{DS}})\leqslant d(Q)$. Thus any upper bound on degenerate $[[n,k]]$ stabilizer code $Q$ is also an upper bound on the minimum distance of degenerate $[[n,k;0]]$ DS code. The same is true for non-degenerate codes.

VI-A Singleton Bound

As we mentioned in Section III code $C_{\mathrm{DS}}^{\perp}$ has size $2^{2n}$, and if ${\bf v}=(v_{1},\ldots,v_{n},w_{1},\ldots,w_{n-k})\in C_{\mathrm{DS}}^{\perp}$ then $v_{i}\in\mathbb{F}_{4}$ and $w_{i}\in\mathbb{F}_{2}$. This leads to the Singleton bound for nondegenerate DS codes.

Theorem 4**.**

In any nondegenerate $[[n,k,d:0]]$ DS code we have

[TABLE]

The proof of this theorem is a simple generalization of the well known case of codes over $\mathbb{F}_{q}$ [22].

In [11] several upper bounds for degenerate stabilizer codes have been derived. In particular, the Singleton bound $k\leqslant n-2(d(Q)-1)$ has been proven. Thus we conclude that bound (39) also holds for degenerate DS codes.

It is instructive to prove this result using (34). To do this, we first note that if $f(x,y)=0$ for $y\geqslant 1$, then the coefficients $f_{i,j}$ do not depend on $j$. Indeed, let $f(x,y)=g(x)\delta_{y,0}$ and $f(x,0)=g(x)=\sum_{i=0}^{n}g_{i}K_{i}(x;n,4).$ Then, according to (26) and (68), we have

[TABLE]

Theorem 5**.**

(Singleton Bound) For an unrestricted (non-degenerate or degenerate) DS code, we have

[TABLE]

Proof.

We will use the polynomial

[TABLE]

Using (27) and (72), we obtain

[TABLE]

It is easy to see that $f_{i,j}\geqslant 0$ and $f(i,0)=0$ for $i\geqslant d$. Simple computations show that $f(0,0)/f_{0,0}>f(l,0)/f_{l,j}=f(l,0)/f_{l,0}$ for $1\leqslant l\leqslant d-1$. Finally, $f(0,0)/f_{0,0}=4^{n-d+1}2^{m}$. ∎

This approach gives us additional information on DS codes achieving the Singleton bound. If a DS code, say $Q_{\mathrm{DS,MDS}}$, meets the Singleton bound, then in (35) we must have equality. Noticing that $f_{i,j}=0$ (defined in (43)) for $i>n-d+1$, we conclude that $Q_{\mathrm{DS,MDS}}$ must have $B_{i,j}=0$ for $d\leqslant n-d+1$ and $j\geqslant 0$. In (36) we always have equality since $f(x,0)=0$ if $x\geqslant d$. Finally, in order to have

[TABLE]

in (37), code $Q_{\mathrm{DS,MDS}}$ must have $\sum_{j=0}^{m}B_{x,j}=0$ for $1\leqslant x\leqslant d_{\mathrm{D}}-1$. Thus $B_{x,j}=0$ for $1\leqslant x\leqslant n-d_{\mathrm{D}}-1$ and $j\geqslant 0$. This means that any generator ${\bf g}$ of $Q_{\mathrm{DS,MDS}}$ should have large weight, $\mbox{{\rm wt}\,}({\bf g})\geqslant n-d$. Hence such $Q_{\mathrm{DS,MDS}}$ will have large syndrome measurement error.

Extensive research have been conducted on construction of quantum codes meeting the Singleton bound (see for example [23], [24], [25], [26], references within, and numerous other papers on this subject). The above result however shows that such codes most likely will not be useful for practical applications due to their large syndrome measurement error probability.

VI-B Hamming Bound

Let $C^{\perp}_{\mathrm{DS}}$ be a non-degenerate DS code with minimum distance $d=2t+1$. Standard combinatorial arguments (see [9]) lead to that $k\leqslant\tilde{k}$, where $\tilde{k}$ is the largest integer such that

[TABLE]

This is the Hamming Bound for non-degenerate DS codes. Below we show that this bounds also holds for degenerate DS codes if $n$ is sufficiently large.

Let $d_{\mathrm{D}}$ and ${\cal A}$ be defined as in (38).

Lemma 6**.**

For a positive integer $\lambda$, let $f^{(k)}(x,y)$ be the polynomial defined by the coefficients

[TABLE]

Then

[TABLE]

where

[TABLE]

A proof can be found in Appendix -C.

Note that in the above lemma, $\lambda$ is a parameter over which we will optimize our bound. Next we give an important property of the polynomial $f^{(k)}(x,y)$.

Lemma 7**.**

For $x+y\geqslant d$, we have $f^{(k)}(x,y)=0$.

The proof is given in Appendix -D.

Thus $f^{(k)}(x,y)$ satisfies constraints (31) and (32) and hence we can use it for obtaining a bound on the minimum distance of DS codes. Choosing $\lambda=1$, we get a polynomial with $f^{(\tilde{k})}(0,0)/f^{(\tilde{k})}_{0,0}$ equal to the right hand side of (44). Numerical computations show that for large $n$, the first entry in the set defined in (34) dominates. Thus, for large $n$, this polynomial gives the Hamming bound (44) for unrestricted (non-degenerate and degenerate) DS codes. The “disatvantage” of this polynomial is that its coefficients $f^{(k)}_{i,j}$ may agresively decrease with $j$, which for certain parameters makes $\min_{1\leqslant j\leqslant m}f(x,j)$ in (34) being very small, that results in a loose bound.

If we choose $\lambda=t+1$, we get $f^{(k)}(x,y)=f(x)\delta_{0,y}$, where $f(x)$ is the polynomial with $f_{i}=(\sum_{g=0}^{t}K_{g}(i;n,4))^{2}$, that is the polynomial that leads to the Hamming bound for classical codes over ${\mathbb{F}}_{4}$, see [33, Chapter 17]. For this polynomial, the value $f^{({k})}(0,0)/f^{({k})}_{0,0}$ is larger than in the case of $\lambda=1$. However, its advantage is that its coefficients $f^{(k)}_{i,j}$ do not decrease with $j$ (in fact they do not depend on $j$), which often leads to better bound than with $\lambda=1$.

Theorem 8**.**

(Hamming bound for unrestricted DS codes.) For an unrestricted DS code, we have $k\leqslant\bar{k}$, where $\bar{k}$ is the largest integer such that

[TABLE]

For $d=7$, the Hamming bounds (44) and (47) are shown in Fig. 2. For small values of $n$, bound (47) is only marginally weaker than (44), and for $n\geqslant 36$, these bounds coincide. We observed the same behavior for other values of $d$. So we make the following conjecture.

Conjecture 9**.**

For any $d$, there exists $n(d)$ such that for $n\geqslant n(d)$, the Hamming bound (44) holds for unrestricted DS codes.

In [9] Fujiwara obtained a hybrid Hamming bound for nondegenerate DS codes that can correct any $t_{\mathrm{D}}$ data and $t_{\mathrm{S}}$ syndrome errors: $k\leqslant\hat{k}$, where $\hat{k}$ is the largest integer such that

[TABLE]

We can also derive this hybrid bound using Theorem 3 with ${\cal A}=\{(i,j):0\leqslant i\leqslant 2t_{\mathrm{D}}\mbox{ and }1\leqslant j\leqslant 2t_{\mathrm{S}}\}$, and polynomial

[TABLE]

Tedious but straightforward computations show that $f^{(k)}_{i,j}\geqslant 0$, $f^{(k)}(x,y)=0$ if $(x,y)\in\overline{{\cal A}}$, and that $f^{(\hat{k})}(0,0)/f^{(\hat{k})}_{0,0}$ is equal to the right hand side of (48).

Thus we obtained a different proof of (48). We cannot use this polynomial for degenerate DS codes, since for some $i\leqslant d_{\mathrm{D}}-1$, we have $f^{(k)}_{i,j}=0$. Finding good polynomials for deriving hybrid bounds on degenerate DS codes is an open problem.

VI-C Asymptotic Bounds

In this subsection, we consider the asymptotic regime in which both the code length $n$ and the number of information qubits $k$ tend to infinity but the code rate $R=k/n$ remains constant.

It is instructive to consider the Hamming bound (44) for nondegenerate DS codes in this regime. In order of doing this we have to find the leading term of the denominator of (44).

Recall that if $v$ grows linearly with $n$ and $a_{j^{*}}>a_{j},j\not=j^{*}$, then

[TABLE]

Let $H(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)$ be the binary entropy function. Denoting $\xi=j/n,\tau=t/n$, and $\iota=i/n$ and using (49), we get for the second sum of the denominator of (44)

[TABLE]

where $o(1)$ is a function that tends to [math] as $n$ increases and we have used Stirling’s approximation that ${1\over n}\log_{2}{n\choose i}=H(i/n)+o(1)$ [19]. This function achieves its maximum at $\xi=\frac{1}{2}(1-R)$. However, according to the Singleton bound, the relative distance $\delta\triangleq\frac{d}{n}\leqslant\frac{1}{2}(1-R)$ and therefore

[TABLE]

Thus the maximum is achieved at $\xi=\tau-\iota$. Hence for the denominator of (44) we have

[TABLE]

Taking the derivative and finding its roots, we conclude that the maximum is achieved at

[TABLE]

It is not difficult to show that $\iota^{*}$ is always smaller than $\tau$. Thus the exponent of the denominator of (44) is

[TABLE]

The exponents of the left part and the numerator of (44) are $2$ and ${1\over n}\log_{2}4^{n}2^{n-\tilde{k}}=3-R,$ respectively. Combining the above results, we obtain the following theorem.

Theorem 10**.**

For a given relative distance $\delta$, the code rate $R$ cannot exceed the root, say $R_{Ham,nondeg}(\delta)$, of

[TABLE]

In [11] the Hamming and so-called first linear programming (LP1) bounds have been derived in asymptotic form for unrestricted (degenerate and non-degenerate) quantum codes:

[TABLE]

The Hamming bound was obtained by applying the polynomial $f_{Ham}(x)$ defined by its coefficients $f_{i}=K_{\delta/2}(i)^{2}$. LP1 bound was obtained with the help of the polynomial

[TABLE]

where $t={\delta\over 2}n$ and $a$ is a real number located between the first roots $r_{t+1}$ and $r_{t}$ of $K_{t+1}(x)$ and $K_{t}(x)$, and chosen so that $K_{t}(a)/K_{t+1}(a)=-1$.

As we mentioned prior to Section VI-A, any bound on degenerate quantum codes is also a bound on degenerate DS codes with the corresponding $n$ and $k$. Thus bounds (53) and (54) also hold for unrestricted (degenerate or non-degenerate) DS codes. Note that these bounds can be also obtained using Theorem 3 and polynomials $f(x,y)=f_{Ham}(x)\delta_{y,0}\mbox{ and }f(x,y)=f_{LP1}(x)\delta_{y,0}.$ As we showed in (40) the coefficients $f_{i,j}$ of these polynomials do not depend on $j$.

The bounds (52), (41), (53), and (54) for unrestricted DS codes are shown in Fig. 3. One can see that at certain interval the Hamming bound for non-degenerate quantum codes beats all the bounds for degenerate DS codes.

It looks natural to try to improve bounds (53), and (54) by using polynomials $f(x,y)$ whose coefficients $f_{i,j}$ depend on both indices $i$ and $j$. At this moment we did not find such polynomials and leave this as an interesting open problem.

VII Random DS Codes

The enumerators $B_{i,j}^{\perp}$ define the decoding error probability of a DS code in a number of communication/computational scenarios, similar to [28]. Below we study the behavior of $B_{i,j}$ and $B_{i,j}^{\perp}$ of random DS codes. In particular, we are interested in how the normalized minimum distance $d(r)/n$ depends on the ratio $r/n$ when $n\rightarrow\infty$.

We will consider the ensemble ${\cal E}_{n,k,r}$ of $C_{\mathrm{DS}}$ codes defined by matrices of the form (10) with $r\leqslant n-k$ and full rank matrices $A$, i.e., $\mbox{rank}(A)=r$. We will use this ensemble to show that the minimum distance of random DS codes with a relatively small $r\leqslant n-k$ achieves the Gilbert-Varshamov bound of stabilizer codes [30].

Let ${\cal E}_{n,k,r}^{\perp}$ be the ensemble of $C_{\mathrm{DS}}^{\perp}$ codes that are dual to codes from ${\cal E}_{n,k,r}$. Note that $|{\cal E}_{n,k,r}^{\perp}|=|{\cal E}_{n,k,r}|$. Define the average enumerators (weight distribution) of codes from ${\cal E}_{n,k,r}$ and ${\cal E}_{n,k,r}^{\perp}$, respectively, by

[TABLE]

where $B_{i,j}(C)$ and $B_{ij}(C^{\perp})$ are defined in (18) and (19). The following theorem finds these weight distributions explicitly.

Theorem 11**.**

For $1\leqslant i\leqslant n\mbox{ and }1\leqslant j\leqslant m+r$, we have

[TABLE]

A combinatorial proof of this result can be found in Appendix -E.

Let us now consider the asymptotic case when the code length $n\rightarrow\infty$. Again, denote $\iota=i/n,~{}\xi=j/n,\mbox{ and }\rho=r/n.$ For a DS code with $B_{i,j}$ and $B_{i,j}^{\perp}$, we define $b_{\iota,\xi}={1\over n}\log_{2}{B}_{\lfloor\iota n\rfloor,\lfloor\xi n\rfloor},b_{\iota,\xi}^{\perp}={1\over n}\log_{2}{B}_{\lfloor\iota n\rfloor,\lfloor\xi n\rfloor}^{\perp},\mbox{ and }\delta=d/n,$ where the minimum distance $d$ is defined by (23) and (24). Denote by $d_{Q}$ the minimum distance of a generic quantum code. In [30], it was shown that there are quantum codes, and quantum stabilizer codes in particular, whose normalized minimum distance $\delta_{Q}=d_{Q}/n$ is at least as large as the quantum Gilbert-Varshamov (GV) bound $\delta_{GV}(R)$. This bound is defined by the equation

[TABLE]

In the next theorem, we prove that there exist DS codes whose weight distributions $B_{i,j}$ and $B_{i,j}^{\perp}$ are upper bounded by the analytical expressions presented in the theorem for all $i$ and $j$, and present a GV bound $\delta_{DS,GV}$ for such codes.

Theorem 12**.**

For $r\leqslant m$, there exist DS codes with rate $R$ and $b_{\iota,\xi}\leqslant\overline{b}_{\iota,\xi}$ and $b_{\iota,\xi}^{\perp}\leqslant\overline{b}_{\iota,\xi}^{\perp}$ and

[TABLE]

and the normolized minimum distance $\delta_{\mathrm{DS}}\geqslant\delta_{\mathrm{DS,GV}}(R,\rho)$, where

[TABLE]

A proof can be found in Appendix -F.

It is instructive to compare the bounds $\delta_{DS,GV}(R,\rho)$ and $\delta_{GV}(R)$. In the left part of Fig. 4, we plot these bounds for the case $\rho=0$. One can see that $\delta_{DS,GV}(R,0)<\delta_{GV}(R)$, especially for low rate quantum codes. This means that DS codes with $\rho=0$ have inferior performance compared to stabilizer codes (in which only qubits are valnurable to errors). However, we can improve DS codes by taking nonzero $\rho$. It is not difficult to see that $\delta_{DS,GV}(R,\rho)$ grows with $\rho$. So, for each $R$ we can choose $\rho^{*}(R)$ so that $\delta_{DS,GV}(R,\rho^{*}(R))=\delta_{GV}(R)$. It happened that $\rho^{*}(R)<1-R$ for any $R$ (that is the corresponding $r^{*}(R)=\rho^{*}(R)n<n-k$, what we assumed for ensemble ${\cal E}_{n,k,r}$). In the right part of Fig. 4 we plot the normalized length of syndrome $\mu=m/n=1-R$ for stabilizer codes and $\mu+\rho^{*}(R)=1-R+\rho^{*}(R)$ for DS codes. One can observe that $\rho^{*}(R)$ is not very large even for low rate quantum codes. This means that relatively small number of additional generator measurements are needed for achieving the quantum GV bound by DS codes.

VIII CSS-type Quantum DS Codes

In this section we discuss CSS-type DS codes with $r\geq 0$. Suppose that H_{\mathrm{CSS}}=\left(\begin{array}[]{c|c}{H}&0\\ 0&{H}\end{array}\right) defines an $[[n,k,d]]$ CSS code, where $H$ is a binary $(\frac{n-k}{2})\times n$ matrix and $HH^{T}={\bf 0}$. Let

[TABLE]

where H^{\prime T}=\left(\begin{array}[]{cccc}H^{T}&{\bf f}_{1}^{T}&\cdots&{\bf f}_{r/2}^{T}\end{array}\right), and vectors ${\bf f}_{j}$ are obtained as linear combinations of rows of $H$. The matrix $\tilde{H}=[H^{\prime}\ {I_{(m+r)/2}}]$ defines a classical $[n^{\prime},k^{\prime},d^{\prime}]$ code. The minimum distance of the corresponding DS code is $d^{\prime}\leq d$ and therefore we obtain an $[[n,k,d^{\prime}:r]]$ quantum DS code. Below we discuss how to extend $H$ to $H^{\prime}$ so that the minimum distance of the DS code would not decrease and remain equal to $d$.

For a vector ${\bf y}\in\mathbb{F}_{2}^{n+(m+r)/2}$ we define the extended syndrome as ${\bf s}=(s_{1},\ldots,s_{(m+r)/2})=H^{\prime}{\bf y}$. One can see that these syndromes belong to the column space of $H^{\prime}$. Hence if any nonzero vector ${\bf w}$ from the column space of $H^{\prime}$ has weight $\mbox{{\rm wt}\,}({\bf w})\geqslant d$, then for any two extended syndromes, say ${\bf s}$ and ${\bf s}^{\prime}$, we have $\mbox{{\rm dist}\,}({\bf s},{\bf s}^{\prime})\geqslant d$ and hence the DS code can correct any $\lfloor\frac{d-1}{2}\rfloor$ syndrome bit errors. If the CSS code defined by $H_{\mathrm{CSS}}$ also has minimum distance $d$ or larger then the DS code can correct any combination of qubit and syndrome errors whose total number does not exceed $\lfloor\frac{d-1}{2}\rfloor$. This leads us to the following result.

Theorem 13**.**

If there exists an $[n,k,d]$ classical dual-containing cyclic code $C$ with $2k>n$, then there exists an $[[n,2k-n,d:r]]$ quantum DS code with $r\leq 2k$.

Proof.

Suppose that $H$ is an $n\times(n-k)$ parity-check matrix of $C$. Since $C$ contains its dual code $C^{\perp}$, we have that $HH^{T}=0$. Hence $H$ can be used for construction of a CSS code according to $H_{\mathrm{CSS}}$. Let ${\bf c}=(c_{0}\ c_{1}\ \cdots\ c_{n-1})\in C^{\perp}$. Since $C^{\perp}$ is also cyclic, any cyclic shift of ${\bf c}$ is also a codeword of $C^{\perp}$. Hence $n-k$ cyclic shifts of ${\bf c}$ can be used to construct $H$ and additional $k$ cyclic shifts can be used to construct $H^{\prime}$. Thus, we can construct $H^{\prime}$ as follows

[TABLE]

Clearly, the column space and the row spaces of $H^{\prime}$ are the same and they generate code $C^{\perp}$. Since $C^{\perp}\subseteq C$ we have $d(C^{\perp})\geqslant d$. Now from the arguments preceding this theorem, it follows that $H^{\prime}$ defines an $[[n,2k-n,d:2k]]$ DS code. ∎

To demonstrate an application of the above theorem, we consider quadratic-residue (QR) codes. QR codes are cyclic codes and they are dual-containing for certain parameters [19, 31], and therefore can be used for construction of CSS quantum codes. In particular, they lead to $[[p,1,d]]$ CSS codes with $d^{2}-d+1\geq p$ for $p=8j-1$.

Theorem 14**.**

For $p=8j-1$, there exist $[[p,1,d]]$ CSS codes with $d^{2}-d+1\geq p$. Moreover, there are $[[p,1,d:r]]$ quantum DS codes with $r\leq p+1$.

Example 15**.**

Consider the QR code with $p=23$. Suppose $H^{\prime}$ is cyclicly generated by $11+r$ cyclic shifts of a code vector of the dual code. Table I provides the distances of the corresponding DS codes with different values of $r$. Table I shows that there exists $[[23,1,7:18]]$ quantum DS code and therefore we need only $18$ additional redundant stabilizers, instead of $24$ by Theorem 13.

The family of quantum QR codes in Theorem 14, which includes the Steane code and the quantum Golay code, are important in the theory of fault-tolerant quantum computation. In particular they are used for finding error thresholds [14]. Here we have shown that these codes also induce nontrivial quantum DS codes.

ACKNOWLEDGEMENT

CYL was was financially supported from the Young Scholar Fellowship Program by Ministry of Science and Technology (MOST) in Taiwan, under Grant MOST107-2636-E-009-005.

-A Properties of Krawchuk polynomial

The following equalities holds (see [19, Chapter 5])

[TABLE]

In [33, eq. A.19] and [11, Lemma 2], it is shown that

[TABLE]

where $\beta(u,a,b)$ and $\alpha(q,g,h,w)$ are defined in (46) and (45) respectively.

Lemma 16**.**

[TABLE]

Proof.

The generating function of the binary Krawtchouk polynomials (see [19, Sec. 5.7]) is $(1+x)^{n-j}(1-x)^{j}=\sum_{u=0}^{n}K_{u}(j;n,2)x^{u}.$ Using this equauion, we obuain $\sum_{j=0}^{m}{m\choose j}(1+x)^{n-j}(1-x)^{j}=\sum_{u=0}^{n}x^{u}\sum_{j=0}^{m}{m\choose j}K_{u}(j;n,2).$ At the same time

[TABLE]

Comparing these two expressions, we finish the proof. ∎

-B Proof of Theorem 1

Proof.

We can use the techniques in [32] as follows. We define a Fourier transform operator with respect to the inner product (11) and find a MacWilliams identity that relates the two split weight enumerators. Then Theorem 1 follows directly.

-C Proof of Lemma 6

Using (73) and (74), we obtain

[TABLE]

Now, using (26), we obtain

[TABLE]

where in the last step we used the orthogonality property of Krawtchouk polynomials (70). ∎

-D Proof of Lemma 7

A binomial coefficient ${i\choose j}$ is assumed to be zero if:

1. $i<j$, 2) $j<0$, or 3) $j$ is not an integer. The polynomial $f^{(k)}(x,y)$ is a sum of non negative terms:

[TABLE]

A particular term is not zero if all the five binomial coefficients are not zeros. Each of those binomial coefficients is not zero if and only if neither of the above conditions hold, e.g, the first binomial coefficient is not zero if and only if $a+b\geqslant y,~{}m-y\geqslant(a+b-y)/2,\mbox{ and }(a+b-y)/2\mbox{ is an integer}.$ In the following discussion, we drop condition 3 since it is not needed for our purpose. From the above arguments it follows that $f^{(k)}(x,y)>0$ for $x+y\geqslant 2t+1$ if only if there is a solution to the system of linear inequalities $A(a,b,w,g,h,x,y)^{T}\leqslant{\bf b},$ with

[TABLE]

Conducting the Fourier–Motzkin elimination [34] in the order of $h,y,g,a,b,w,x$ (any other order can also be used, but this one requires computations that are not too long), we come to the incompatible condition $0\leqslant-1/2$. This completes the proof. ∎

-E Proof of Theorem 11

We would like to analyze the weight distribution of a random DS code from ${\cal E}_{n,k,r}$ with a generator matrix of the form \left[\begin{array}[]{ccc}H&I_{m}&0\\ 0&A&I_{r}\end{array}\right], where $m=n-k$. Let ${\cal E}_{n,m}$ be the set of DS codes with a generator matrix of the form $[H\ I_{m}]$, and ${\cal F}_{m,r}$ be the set of binary codes with a generator matrix of the form $[A\ I_{r}]$, where $A$ has rank $r$. A code from ${\cal E}_{n,k,r}$ can be considered as a combination of codes from ${\cal E}_{n,m}$ and ${\cal F}_{m,r}$.

Lemma 17**.**

The size of the ensemble ${\cal E}_{n,m}$ is

[TABLE]

and any vector ${\bf w}=({\bf a},{\bf b})$ with ${\bf a}\in{\mathbb{F}}_{4}^{n}\setminus{\bf 0}$ and ${\bf b}\in{\mathbb{F}}_{2}^{m}\setminus{\bf 0}$ is contained in

[TABLE]

codes from ${\cal E}_{n,m}$.

Proof.

It is proved that the number of $[n,m]$ additive self-orthogonal codes over $\mathbb{F}_{4}$ is $S\triangleq\prod_{u=0}^{m-1}{(2^{2(n-u)}-1)\over 2^{u+1}-1}$ [28]. For any $[n,m]$ additive self-orthogonal code, we can choose $m$ generators (rows of matrix $H$) in $T\triangleq\prod_{u=0}^{m-1}(2^{m}-2^{u})$ ways. Hence, using any $[n,m]$ self-orthogonal code, we can form $T$ different matrices $[H\ I_{m}]$. Thus, $|{\cal E}_{n,m}|=ST$.

It is shown that any nonzero vector ${\bf a}\in{\mathbb{F}}_{4}^{n}\setminus{\bf 0}$ is contained in $P=\prod_{u=1}^{m-1}{2^{2(n-u)}-1\over 2^{u}-1}$   $[n,m]$ self-orthogonal codes [28]. We can use any of those codes for building a code from ${\cal E}_{n,m}$ with vector $({\bf a},{\bf b})$ as its first basis vector. The other $(m-1)$ basis vectors can be chosen in $R=\prod_{u=1}^{m-1}(2^{m}-2^{u})$ ways. Hence any $({\bf a},{\bf b})$ is contained in $PR$ codes from ${\cal E}_{n,m}$. ∎

Lemma 18**.**

The size of the ensemble ${\cal F}_{m,r}$ is

[TABLE]

where $\left[{m\atop r}\right]=\frac{(2^{m}-1)(2^{m-1}-1)\cdots(2^{m-r+1}-1)}{(2^{r}-1)(2^{r-1}-1)\cdots(2-1)}$ is the binary Gaussian binomial coefficient, and any vector $({\bf b},{\bf c})$ for ${\bf b}\in{\mathbb{F}}_{2}^{m}\setminus{\bf 0}$ and ${\bf c}\in{\mathbb{F}}_{2}^{r}\setminus{\bf 0}$ is contained in

[TABLE]

codes from ${\cal F}_{m,r}$.

The proof of this lemma is similar to the previous one and is omitted.

Lemma 19**.**

(1) Any vector $({\bf a},{\bf b},{\bf c})$, where ${\bf a}\in{\mathbb{F}}_{4}^{n},\setminus{\bf 0}$, ${\bf b}\in{\mathbb{F}}_{2}^{m},\setminus{\bf 0}$, ${\bf c}\in{\mathbb{F}}_{2}^{r}\setminus{\bf 0}$, is contained in

[TABLE]

codes from ${\cal E}_{n,k,r}$. (2) Any $({\bf a},{\bf b},{\bf 0})$, where ${\bf a}\in{\mathbb{F}}_{4}^{n}\setminus{\bf 0}$, ${\bf b}\in{\mathbb{F}}_{2}^{m}\setminus{\bf 0}$, ${\bf 0}\in{\mathbb{F}}_{2}^{r}$, is contained in

[TABLE]

codes from ${\cal E}_{n,k,r}$. (3) Any $({\bf 0},{\bf b},{\bf c})$, where ${\bf 0}\in{\mathbb{F}}_{4}^{n}$, ${\bf b}\in{\mathbb{F}}_{2}^{m}\setminus{\bf 0}$, ${\bf c}\in{\mathbb{F}}_{2}^{r}\setminus{\bf 0}$, is contained in

[TABLE]

codes from ${\cal E}_{n,k,r}$. (4) Any $({\bf a},{\bf 0},{\bf c})$, where ${\bf a}\in{\mathbb{F}}_{4}^{n},\setminus{\bf 0}$, ${\bf 0}\in{\mathbb{F}}_{2}^{m}$, ${\bf c}\in{\mathbb{F}}_{2}^{r}\setminus{\bf 0}$, is contained in

[TABLE]

codes from ${\cal E}_{n,k,r}$.

Proof.

We prove the first one and the other three follow similarly.

A vector $({\bf a},{\bf b},{\bf c})$ can be obtained only as the sum of a vector $({\bf a},{\bf x},{\bf 0})$ (where ${\bf x}\in{\mathbb{F}}_{2}^{m}\setminus{\bf 0},{\bf x}\not={\bf b},~{}{\bf 0}\in{\mathbb{F}}_{2}^{r}$, and $({\bf a},{\bf x})$ is a code vector of a code from ${\cal E}_{n,m}$) and a code vector $({\bf 0},{\bf b}+{\bf x},{\bf c})$ (where $({\bf b}+{\bf x},{\bf c})$ is a code vector of of a code from ${\cal F}_{m,r}$). Any given $({\bf b}+{\bf x},{\bf c})$ is contained in $\prod_{u=1}^{r-1}(2^{m}-2^{u})$ codes from ${\cal F}_{m,r}$ by Lemma 18. Since $({\bf a},{\bf x})$ is contained in $L$ codes from ${\cal E}_{n,m}$ and vector ${\bf x}$ can be chosen in $2^{m}-2$ ways, we have $({\bf a},{\bf b},{\bf c})$ contained in $L(2^{m}-2)\prod_{u=1}^{r-1}(2^{m}-2^{u})$ codes from ${\cal E}_{n,k,r}$, which gives (80). ∎

In addition, we note that the total number of codes in ${\cal E}_{n,k,r}$ is

[TABLE]

Now we have all the ingredients needed for finding $\overline{B}_{i,j}$. It will be convenient to assume that ${a\choose b}=0$ if $a<b$ or $b<0$. Let consider the case of $i>0$ and $j>0$. Then

[TABLE]

Taking into account that $\sum_{u=0}^{j}{m\choose u}{r\choose j-u}={m+r\choose j}$, after some computations, we obtain (56). Equation (55) is obtained in a similar way.

To derive $\overline{B}_{i,j}^{\perp}$ we use MacWilliams identities (28), which also hold for average weight enumerators $\overline{B}_{i,j}$ and $\overline{B}_{i,j}^{\perp}$. Changing the role of codes $C$ and $C^{\perp}$, we get, similar to (28):

[TABLE]

Using (69), (71), and (75), after long manipulations, we obtain (57), (58), and (59).

∎

-F Proof of Theorem 12

According to Markov’s inequality for a given pair $i$ and $j$, we have

[TABLE]

for any $\epsilon>0$. Applying the union bound, we obtain

[TABLE]

and further $\Pr\left(B_{i,j}(C^{\perp})<((n+1)(m+r+1))^{1+\epsilon}\overline{B}_{i,j}^{\perp}\mbox{ for all }i,j\right)\geqslant 1-{1\over((n+1)(m+r+1))^{\epsilon}}.$ Hence there exists a code $C^{\perp}\in{{\cal E}}_{n,m}^{\perp}$ such that

[TABLE]

Now we consider codes of growing lengths, i.e., $n\rightarrow\infty$. Note that $m/n=(n-k)/n=1-R$. Recall that ${1\over n}\log_{2}{n\choose i}=H(i/n)+o(1)$ [19]. So the three terms of the last factor of (58) are that

[TABLE]

Simple analysis shows that for $\rho\leqslant 1-R$ (which is the same as $r\leqslant n-k$), we have that (86) is always larger than (87) and (88). Hence

[TABLE]

The equation (61) is obtained in a similar way.

Let us have $C^{\perp}$ that satisfies (85). Since $C^{\perp}$ is linear, all ${B}_{i,j}(C^{\perp})$ are integers. Hence if $\iota^{*}$ and $\xi^{*}$ are such that $\overline{b}_{\iota,\xi}\leqslant 0$ for $\iota\leqslant\iota^{*}$ and $\xi\leqslant\xi^{*}$, then $B_{i,j}(C^{\perp})=0$ for $1\leqslant i\leqslant(\iota^{*}-\epsilon)n,~{}1\leqslant j\leqslant(\xi^{*}-\omega)n$ for any $\epsilon,\omega>0$ and sufficiently large $n$. It is not difficult to see that if $\iota\leqslant\delta_{GV}(R)$, then $\overline{b}^{\perp}_{\iota,0}\leqslant 0$. Similarly, if $\xi(\iota)=H^{-1}\left(1-H\left({\iota+\iota\log_{2}(3)\over 1-R+\rho}\right)\right)$, then $\overline{b}^{\perp}_{\iota,\xi(\iota)}=0$. Thus $B_{i,0}(C^{\perp})=0$ for all $i\leqslant(\delta_{GV}(R)-\epsilon)n$ and $B_{i,j}(C^{\perp})=0$ if $i+j\leqslant(\iota+\xi(\iota)-\omega)n$. Hence (64) follows. ∎