Two Gilbert-Varshamov Type Existential Bounds for Asymmetric Quantum Error-Correcting Codes

TL;DR

This paper presents two new Gilbert-Varshamov type bounds that establish the existence of asymmetric quantum error-correcting codes, advancing the theoretical understanding of quantum code construction.

Contribution

It introduces two novel bounds specifically tailored for asymmetric quantum codes, filling a gap in quantum coding theory.

Findings

Derived two Gilbert-Varshamov type bounds for asymmetric quantum codes

Provided theoretical existence criteria for such codes

Enhanced the understanding of quantum code parameters

Abstract

In this note we report two versions of Gilbert-Varshamov type existential bounds for asymmetric quantum error-correcting codes.

Equations

$$♯B_{n,x}(\vec{e})$$

$$♯B_{n,z}(\vec{e})$$

$$\frac{{\sum }_{\vec{0}\ne \vec{e}\in {\mathbf{F}}_q^n}♯B_{n,x}(\vec{e})}{♯B_n}=q^{k_1}-q^{k_2}.$$

$$\sum _{\vec{0}\ne \vec{e}\in {\mathbf{F}}_q^n}♯B_{n,x}(\vec{e})=(q^n-1)♯B_{n,x}(\vec{e}).$$

$$♯B_{n,x}(\vec{e})=\frac{q^{k_1}-q^{k_2}}{q^n-1}♯B_n.$$

$$♯B_{n,x}({\vec{e}}_1)$$

$$\frac{q^{k_1}-q^{k_2}}{q^n-1}\sum _{i=1}^{d_x-1}\left(\genfrac{}{}{0ex}{}{n}{i}\right)(q-1{)}^i+\frac{q^{n-k_2}-q^{n-k_1}}{q^n-1}\sum _{i=1}^{d_z-1}(\genfrac{}{}{0ex}{}{n}{i})(q-1{)}^i<1,$$

$$\sum _{i=1}^{\lfloor n\delta \rfloor }\left(\genfrac{}{}{0ex}{}{n}{i}\right)(q-1{)}^i\le q^{n{h}_q(\delta )},$$

$$h_q({\delta }_x)$$

$$h_q({\delta }_z)$$

$$0$$

$$n{h}_q({\delta }_x)<n-n{R}_1$$

$$n{h}_q({\delta }_z)<n{R}_2$$

$$\sum _{i=1}^{d_x-1}\left(\genfrac{}{}{0ex}{}{n}{i}\right)(q-1{)}^i\times \sum _{i=1}^{d_z-1}(\genfrac{}{}{0ex}{}{n}{i})(q-1{)}^i.$$

$$\frac{1-q^{-2k}}{1-q^{-2n}}\cdot \frac{1}{q^{n-k}}\sum _{i=1}^{d_x-1}\left(\genfrac{}{}{0ex}{}{n}{i}\right)(q-1{)}^i\times \sum _{i=1}^{d_z-1}(\genfrac{}{}{0ex}{}{n}{i})(q-1{)}^i<1$$

$$h_q({\delta }_x)+h_q({\delta }_z)<1-R\le 1,$$

Full text

††journal: Quantum Information Processing

1 Introduction

Quantum error-correcting codes (QECC) are important for construction of quantum computers, as the fault-tolerant quantum computation is based on QECC chuangnielsen . There are two kinds of errors in quantum information, one is called a bit error and the other is called a phase error. Steane steane96aqc first studied the asymmetry between probabilities of the bit and the phase errors, and he also considered QECC for asymmetric quantum errors, which are called asymmetric quantum error-correcting codes (AQECC). Research on AQECC has become very active recently, see galindo16 ; ioffe06 ; steane96aqc and the references therein.

On the other hand, in the study of error-correcting codes, it is important to know the optimal performance of codes. For classical error-correcting codes, the Gilbert-Varshamov (GV) bound macwilliams77 is a sufficient condition for existence of codes whose parameters satisfies the GV bound. By the GV bound, one can know that the optimal performance of classical codes is at least as good as the GV bound.

For QECC, Ekert and Macchiavello obtained a GV type existential bound for general QECCs. An important subclass of general QECCs is the stabilizer codes calderbank97 ; calderbank98 ; gottesman96 , as they enable efficient encoding and decoding. Calderbank et al. calderbank97 obtained a GV type existential bound for the stabilizer QECCs. After that, Feng and Ma feng04 and Jin and Xing jin11 obtained improved versions of GV type bounds for the stabilizer QECCs.

The Calderbank-Shor-Steane (CSS) QECCs calderbank96 ; steane96 are an important subclass of the stabilizer QECCs, as the CSS codes enable more efficient implementation of the fault-tolerant quantum computation than the stabilizer codes.

Those existential bounds calderbank97 ; calderbank96 ; ekert96 ; feng04 ; jin11 did not consider the asymmetric quantum errors, while the asymmetry in quantum errors is important in practice sarvepalli09 . As far as the author know, nobody has reported existential bounds for the stabilizer or the CSS QECC for asymmetric quantum errors. In this note we report such ones. Our proof arguments are similar to ones in calderbank97 ; calderbank96 .

2 A GV type existential bound

for the CSS codes

An $[[n,k,d_{x},d_{z}]]_{q}$ QECC encodes $k$ $q$-ary qudits into $n$ $q$-ary qudits and detects up to $d_{x}$ bit errors and up to $d_{z}$ phase errors. It is known ashikhmin00 ; calderbank98 that a nested classical code $C_{2}\subset C_{1}\subset\mathbf{F}_{q}^{n}$ with dimensions $k_{2}$ and $k_{1}$ can construct an $[[n,\dim C_{1}-\dim C_{2}]]_{q}$ CSS code, where $\mathbf{F}_{q}$ is a finite field with $q$ elements. A quantum error can be expressed as a pair $(\vec{e}_{x}$, $\vec{e}_{z})$, where $\vec{e}_{x}\in\mathbf{F}_{q}^{n}$ corresponds to the bit error component of a quantum error and $\vec{e}_{x}\in\mathbf{F}_{q}^{n}$ does to the phase error component.

Let $\mathrm{GL}_{n}(\mathbf{F}_{q})$ be the group of $n\times n$ invertible matrices over $\mathbf{F}_{q}$. Let $B_{n}=\{(C_{1}$, $C_{2})\mid$ $C_{2}\subset C_{1}\subset\mathbf{F}_{q}^{n}$, $\dim C_{1}=k_{1}$, $\dim C_{2}=k_{2}\}$. For a nonzero vector $\vec{e}\in\mathbf{F}_{q}^{n}$, let $B_{n,x}(\vec{e})$ (resp. $B_{n,z}(\vec{e})$) be the set of nested code pairs that cannot detect $\vec{e}$ as a bit error (resp. a phase error), that is, $B_{n,x}(\vec{e})=\{(C_{1},C_{2})\in B_{n}\mid\vec{e}\in C_{1}\setminus C_{2}\}$ (resp. $B_{n,z}(\vec{e})=\{(C_{1},C_{2})\in B_{n}\mid\vec{e}\in C_{2}^{\perp}\setminus C_{1}^{\perp}\}$), where $C_{1}^{\perp}$ is the dual code of $C_{1}$ with respect to the standard inner product.

Lemma 1

For nonzero $\vec{e}$, we have

[TABLE]

Proof

As each pair $C_{2}\subset C_{1}$ has $\sharp C_{1}\setminus C_{2}=q^{k_{1}}-q^{k_{2}}$ undetectable errors, we have

[TABLE]

For nonzero $\vec{e}_{1},\vec{e}_{2}\in\mathbf{F}_{q}^{n}$, we claim $\sharp B_{n,x}(\vec{e}_{1})=\sharp B_{n,x}(\vec{e})_{2}$. Assuming the claim, we have

[TABLE]

Combining these two equalities, we have

[TABLE]

We finish the proof by proving the claim. Let $\vec{e}_{1},\vec{e}_{2}$ be nonzero vectors. We have

[TABLE]

where $\tau^{\prime}\in\mathrm{GL}_{n}(\mathbf{F}_{q})$ such that $\tau^{\prime}\vec{e}_{1}=\vec{e}_{2}$.

For phase errors, we can make a similar argument with $C_{2}^{\perp}\supset C_{1}^{\perp}$. ∎

Theorem 2.1

Let $n$, $k_{1}$, $k_{2}$, $d_{x}$ and $d_{z}$ be positive integers such that

[TABLE]

then an $[[n,k_{1}-k_{2},d_{x},d_{z}]]_{q}$ CSS QECC exists.

Proof

Recall that each quantum error can be expressed by its bit error component $\vec{e}_{x}\in\mathbf{F}_{q}^{n}$ and its phase error component $\vec{e}_{z}\in\mathbf{F}_{q}^{n}$. The bit error component $\vec{e}_{x}$ cannot be detected by codes in $B_{n,x}(\vec{e}_{x})$ and the phase error component $\vec{e}_{z}$ cannot be detected by codes in $B_{n,z}(\vec{e}_{z})$. The detectabilities of the bit errors and the phase errors are independent of each other. Therefore, if Eq. (1) holds then there exists at least one $(C_{1},C_{2})\in B_{n}$ that can detect all the bit errors with weight up to $d_{x}-1$ and all the phase errors with weight up to $d_{z}-1$, which implies it is an $[[n,k_{1}-k_{2},d_{x},d_{z}]]_{q}$ quantum code. ∎

Classical coding theorists often have interest in asymptotic versions of GV type existential bounds macwilliams77 . They are stated in terms of information rate and relative distance of classical error-correcting codes. In the classical error correction, information rate is the ratio of the number of information symbols to the code length, and relative distance is the ratio of the minimum distance to the code length.

We can also derive an asymptotic version of Theorem 2.1. For an $[[n,k,d_{x},d_{z}]]_{q}$ AQECC, we may define the relative distance $\delta_{x}$ for bit errors as $d_{x}/n$, and the relative distance $\delta_{z}$ for bit errors as $d_{z}/n$. The information rate of an $[[n,k]]_{q}$ QECC is defined as $k/n$ chuangnielsen .

Recall macwilliams77 that for $0\leq\delta\leq 1-1/q$ we have

[TABLE]

where $h_{q}(\delta)=\delta\log_{q}(q-1)-\delta\log_{q}\delta-(1-\delta)\log_{q}(1-\delta)$.

Corollary 1

Let $\delta_{x}$ and $\delta_{z}$ be real numbers such that $0\leq\delta_{x}\leq 1-1/q$ and $0\leq\delta_{z}\leq 1-1/q$. If

[TABLE]

then, for sufficiently large $n$, there exists an $[[n,\lfloor nR_{1}\rfloor-\lceil nR_{2}\rceil,\lfloor n\delta_{x}\rfloor,\lfloor n\delta_{z}\rfloor]]_{q}$ CSS QECC exists.

In Corollary 1, $R_{1}$ is the information rate of classical ECC $C_{1}$, and $R_{2}$ is the information rate of classical ECC $C_{2}$. The corresponding quantum CSS code has information rate $R_{1}-R_{2}$, relative distance $\delta_{x}$ for bit errors, and relative distance $\delta_{z}$ for phase errors.

Proof

Assume that Eq. (3) holds. Then for sufficiently large $n$ we have

[TABLE]

Similarly, for sufficiently large $n$ Eq. (4) implies

[TABLE]

Equations (5) and (6) imply that the assumption of Theorem 2.1 becomes true for sufficiently large $n$, which shows Corollary 1. ∎

3 A GV type existential bound

for the stabilizer codes

Let $C\subset\mathbf{F}_{q}^{2n}$ be a $\mathbf{F}_{q}$-linear space of dimension $n-k$ self-orthogonal with respect to the standard symplectic inner product in $\mathbf{F}_{q}^{2n}$. $C$ can be viewed as an $[[n,k]]_{q}$ stabilizer QECC. Let $A_{n}$ be the set of all such $C$’s. A nonzero $\vec{e}\in\mathbf{F}_{q}^{2n}$ can be viewed as a quantum error on $n$ qudits. Let $A_{n}(\vec{e})$ be the set of stabilizer codes that cannot detect $\vec{e}$ as an error, that is, $A_{n}(\vec{e})=\{C\in A_{n}\mid\vec{e}\in C^{\perp\mathrm{s}}\setminus C\}$, where $C^{\perp\mathrm{s}}$ is the dual of $C$ with respect to the symplectic inner product. Then $\sharp A_{n}(\vec{e})\leq\frac{1-q^{-2k}}{1-q^{-2n}}\cdot\frac{1}{q^{n-k}}\sharp A_{n}$ (matsumotouematsu01, , Lemma 9).

Recall that, for $C$ to be $[[n,k,d_{x},d_{z}]]_{q}$, $C$ must be able to detect all $d_{x}$ or less bit errors and all $d_{z}$ or less phase errors. The number of such errors is

[TABLE]

By the same argument as matsumotouematsu01 [Remark 10] (or as the last section), we have the following theorem:

Theorem 3.1

Let $n$, $k_{1}$, $k_{2}$, $d_{x}$ and $d_{z}$ be positive integers such that

[TABLE]

then there exists an $[[n,k,d_{x},d_{z}]]_{q}$ stabilizer QECC. ∎

By almost the same argument as Corollary 1 we can derive the following asymptotic version of Theorem 3.1.

Corollary 2

Let $\delta_{x}$ and $\delta_{z}$ be real numbers such that $0\leq\delta_{x}\leq 1-1/q$ and $0\leq\delta_{z}\leq 1-1/q$. If

[TABLE]

then, for sufficiently large $n$, there exists an $[[n,\lfloor nR\rfloor,\lfloor n\delta_{x}\rfloor,\lfloor n\delta_{z}\rfloor]]_{q}$ stabilizer QECC. ∎

The quantum statilizer code in Corollary 2 has information rate $R$, relative distance $\delta_{x}$ for bit errors, and relative distance $\delta_{z}$ for phase errors.

By the relation between the CSS and the stabilizer QECCs calderbank98 , we see that the assumption in Corollary 1 is less demanding than that in Corollary 2 for the same $n$, $R=R_{1}-R_{2}$, $\delta_{x}$ and $\delta_{z}$, which means that Corollary 2 is a stronger existential bound than Corollary 1.

Remark 1

Theorems 2.1 and 3.1, and Corollaries 1 and 2 do not admit direct comparisons against previously known GV type bounds even when $d_{x}=d_{z}$. The reason is as follows: For a binary QECC to be $[[n,k,2,2]]_{2}$, it must detect at least $n^{2}$ different errors. On the other hand, for a binary $[[n,k]]_{2}$ QECC to detect all single symmetric errors, it only has to detect $3n$ errors, which is generally much fewer than $n^{2}$. The above example shows that the number of asymmetric quantum errors is much different from that of corresponding symmetric quantum errors, even if we assume the same number of bit errors and phase errors in asymmetric quantum errors.

In addition, the famous $[[5,1,3]]_{2}$ binary stabilizer code in calderbank98 ; gottesman96 can detect up to four bit errors if there is no phase error, and can detect up to four phase errors if there is no bit error. Thus it is simultaneously both $[[5,1,1,5]]_{2}$ AQECC and $[[5,1,5,1]]_{2}$ AQECC. This phenomenon makes the direct comparison even more difficult.

Acknowledgements.

The author would like to thank an anonymous reviewer for careful reading and the helpful report that improved this note, and Prof. Fernando Hernando for drawing his attention to the asymmetric quantum error correction.