Fidelity Lower Bounds for Stabilizer and CSS Quantum Codes

TL;DR

This paper provides bounds on the fidelity of stabilizer and CSS quantum codes, analyzing their performance and asymptotic behavior to guide code design for desired fidelity levels.

Contribution

It introduces new lower bounds on fidelity using quantum enumerators, compares stabilizer and CSS codes, and explores asymptotic performance limits.

Findings

Stabilizer codes outperform CSS codes in fidelity and rate in symmetric depolarizing channels.

Expurgated ensembles improve fidelity bounds over basic ensembles.

Asymptotic analysis shows CSS codes have fundamental performance limitations compared to stabilizer codes.

Abstract

In this paper we estimate the fidelity of stabilizer and CSS codes. First, we derive a lower bound on the fidelity of a stabilizer code via its quantum enumerator. Next, we find the average quantum enumerators of the ensembles of finite length stabilizer and CSS codes. We use the average quantum enumerators for obtaining lower bounds on the average fidelity of these ensembles. We further improve the fidelity bounds by estimating the quantum enumerators of expurgated ensembles of stabilizer and CSS codes. Finally, we derive fidelity bounds in the asymptotic regime when the code length tends to infinity. These results tell us which code rate we can afford for achieving a target fidelity with codes of a given length. The results also show that in symmetric depolarizing channel a typical stabilizer code has better performance, in terms of fidelity and code rate, compared with a typical CSS codes, and that balanced CSS codes significantly outperform other CSS codes. Asymptotic results demonstrate that CSS codes have a fundamental performance loss compared to stabilizer codes.

Equations

$$\mathcal{M}(\rho )=\sum _{k\in \mathcal{K}}{M}_k\rho M_k^{†},$$

$$F(Q,\mathcal{M},\mathcal{R})=\underset{|\psi ⟩\in Q}{\min }⟨\psi |\mathcal{R}\mathcal{M}(|\psi ⟩⟨\psi |)|\psi ⟩.$$

$$\sigma_{x}=\left[\begin{array}[]{rr}0&1\\ 1&0\end{array}\right],\sigma_{z}=\left[\begin{array}[]{rr}1&0\\ 0&-1\end{array}\right],\sigma_{y}=\left[\begin{array}[]{rr}0&i\\ -i&0\end{array}\right].$$

$$E_{\mathbf{x}}=e_1\otimes \dots \otimes e_n,$$

$$e_{j}=\left\{\begin{array}[]{ll}I_{2},&x_{j}=0,\\ \sigma_{x},&x_{j}=1,\\ \sigma_{z},&x_{j}=\omega,\\ \sigma_{y},&x_{j}=\omega^{2}.\end{array}\right.$$

$$\text{mbox}\mathrm{wt}(\mathbf{x})=|\{x_j\ne 0,j=1,\dots ,n\}|.$$

$$\mathcal{M}\sim \{{\left(\frac{p}{3}\right)}^{\text{mbox}\mathrm{wt}(\mathbf{x})}(1-p{)}^{n-\text{mbox}\mathrm{wt}(\mathbf{x})}{E}_{\mathbf{x}}:\mathbf{x}\in {\mathbb{F}}_4^n\}.$$

$$\mathrm{Pr}(e_j=I_2)=1-p,$$

$$\mathrm{Pr}(e_j={\sigma }_x)=\mathrm{Pr}(e_j={\sigma }_z)=\mathrm{Pr}(e_j={\sigma }_y)=\frac{p}{3}.$$

$$\mathrm{Pr}(E_{\mathbf{x}})={\left(\frac{p}{3}\right)}^{\text{mbox}\mathrm{wt}(\mathbf{x})}(1-p{)}^{n-\text{mbox}\mathrm{wt}(\mathbf{x})}.$$

$$A_j(C)=|\{\mathbf{c}\in C:\text{mbox}\mathrm{wt}(\mathbf{c})=j\}|,j=0,\dots ,n.$$

$$C^{\perp }=$$

$$\text{mbox}forall{\mathbf{c}}^{\prime }=(c_1^{\prime },\dots ,c_n^{\prime })\in C\},$$

$$\mathbf{c}\cdot {\mathbf{c}}^{\prime }=c_1{c}_1^{\prime }+\dots +c_n{c}_n^{\prime }$$

$$\overline{0}=0,\overline{1}=1,\overline{\omega }={\omega }^2,\overline{{\omega }^2}=\omega .$$

$$C^{\perp }=\{\mathbf{c}:\mathbf{c}\ast {\mathbf{c}}^{\prime }=0\text{mbox}forall{\mathbf{c}}^{\prime }\in C\},$$

$$\mathbf{c}\ast {\mathbf{c}}^{\prime }=\text{mbox}{\mathrm{Tr}}_{{\mathbb{F}}_2}^{{\mathbb{F}}_4}(c_1{\overline{c}}_1^{\prime }+\dots +c_n{\overline{c}}_n^{\prime })$$

$$A_j^{\perp }=\frac{1}{|C|}\sum _{i=0}^n{A}_i{\mathcal{K}}_j(i),j=0,\dots ,n,$$

$${\mathcal{K}}_j(i)=\sum _{t=0}^j(-q{)}^t(q-1{)}^{j-t}\left(\genfrac{}{}{0ex}{}{n-t}{j-t}\right)\left(\genfrac{}{}{0ex}{}{i}{t}\right)$$

$$A^{\perp }(x,y)=\sum _{j=0}^n{A}_j^{\perp }{x}^{n-j}{y}^j\text{mbox}andA(x,y)=\sum _{j=0}^n{A}_j{x}^{n-j}{y}^j.$$

$$A(x,y)=\frac{1}{|C^{\perp }|}{A}^{\perp }(x+(q-1)y,x-y).$$

$$B_j^{\perp }(Q)$$

$$B_j(Q)$$

$$R(C^{\perp })=\frac{1}{n}{\log }_4|C^{\perp }|\text{mbox}andR(C)=\frac{1}{n}{\log }_4|C|$$

$$R(C^{\perp })=\frac{1-R}{2}\text{mbox}andR(C)=\frac{R+1}{2}.$$

$$B_j(Q)=A_j(C),B_j^{\perp }(Q)=A_j(C^{\perp }).$$

$$B_j⩾B_j^{\perp },B_0^{\perp }=B_0=1.$$

$$\dim (Q)=2^k=\frac{1}{2^n}\sum _{j=0}^n{B}_j=2^n\cdot \frac{1}{{\sum }_{j=0}^n{B}_j^{\perp }}.$$

$$B_r=\frac{\dim (Q)}{2^n}\sum _{j=0}^n{B}_j^{\perp }{\mathcal{K}}_r(j),$$

$$B(x,y)=\frac{\dim (Q)}{2^n}{B}^{\perp }(x+3y,x-y),$$

$$B^{\perp }(x,y)$$

Full text

These results tell us which code rate we can afford for achieving a target fidelity with codes of a given length. The results also show that in symmetric depolarizing channel a typical stabilizer code has better performance, in terms of fidelity and code rate, compared with a typical CSS codes, and that balanced CSS codes significantly outperform other CSS codes. Asymptotic results demonstrate that CSS codes have a fundamental performance loss compared to stabilizer codes.

I Introduction

In recent years quantum error correcting codes were subject of intensive studies as they allow protection of quantum information from decoherence during quantum computations. The main focus of these studies was on various constructions of quantum codes, such as block codes, convolutional code, LDPC quantum codes and others, and their combinatorial, geometrical and topological properties, such as the minimum distance and others (see, for example, numerous publications in IEEE Trans. on Information Theory, Physical Review A, Physical Review Letters, International Journal of Quantum Information).

At the same time it is not so much known about the fidelity that one can hope to achieve using good quantum codes. A number of lower bounds on the fidelity $F$ of quantum codes in the asymptotic regime, as the code length tends to infinity, were derived in [hamada2001],[hamada2002],[barg]. These bounds were derived in terms of reliability functions (error exponents).

In this article we consider the problem from a different angle. First, we would like to derive lower bounds on the fidelity $F$ as a function of the code length. While a reliability function tells us what kind of fidelity we may expect in the asymptotic regime, it does not give a good estimate on the fidelity for quantum codes of short and moderate code length, like $50$ or $1000$ qubits.

Second, we would like to estimate the fidelity for several ensembles of quantum codes. In particular, we are interested in the ensembles of stabilizer codes, linear stabilizer codes, and CSS codes with different choices of parameters $k_{1}$ and $k_{2}$. CSS codes form an important subfamily of stabilizer codes. They are attractive for numerous applications, such as error correction in quantum memory, quantum fault-tolerant computations, quantum cryptography, and others. Therefore estimates on their fidelity can be very important for proper use of CSS codes in these applications.

Third, we are interested in analysis of performance of CSS codes in the asymptotic regime, as the code length tends to infinity. In particular, we would like to understand what is the fundamental performance loss of CSS codes compared with the performance of unrestricted stabilizer codes as the code length tends to infinity.

The paper is organized as follows. In Section II-A we remind the main definitions of quantum depolarizing channel. In Section II-B we remind the definitions of classical and quantum enumerators and their main properties, which will be used later in the paper. In Section III we derive a lower bound on the fidelity $F$ of quantum stabilizer code $Q$ as a function of its quantum enumerators. Further we derive two lower bounds on the average fidelity of an ensemble of quantum codes and apply these bounds to the ensemble of stabilizer codes of given length and code rate. In Section LABEL:sec:F_CSS_codes we find the average quantum enumerators of the ensemble of CSS codes and apply them for obtaining lower bounds on the average fidelity of this ensemble. In Section LABEL:sec:ErrExpon we investigate the behavior of $F$ for stabilizer and CSS codes in the asymptotic regime as the code length tends to infinity.

II Preliminaries

II-A Quantum Depolarizing Channel

An $((n,K))$ binary quantum code $Q$ is a $K$-dimensional linear subspace of the complex space $\mathbb{C}^{2^{n}}$.

A quantum channel is a trace-preserving completely positive linear map ${\cal M}$. Any such map has an operator sum representation

[TABLE]

for some operators $M_{k}$. Here $\rho$ is a density operator on $\mathbb{C}^{2^{n}}$ and ${\cal K}$ is a set of indices. We will write ${\cal M}\sim\{M_{k}\}_{k\in K}$.

Decoding, or state-recovery operator, ${\cal R}$ associated with $Q$ is another trace-preserving completely positive linear map. The minimum fidelity of $Q$ is defined by

[TABLE]

In what follows it will be more convenient for us to use the bound (8) (see Section III) instead of working with the definition (1) itself. More details on trace-preserving completely positive linear maps and fidelity of quantum codes can found in [hamada2001],[hamada2002], and references within.

The quantum symmetric depolarizing channel is defined with the help of Pauli operators

[TABLE]

Denote by $0,1,\omega,\omega^{2}$ the elements of the Galois field $\mathbb{F}_{4}$. Let us associate with a vector ${\bf x}=(x_{1},\ldots,x_{n})\in\mathbb{F}_{4}^{n}$ the linear operator

[TABLE]

where

[TABLE]

The operators $E_{\bf x}$ are called error operators. The Hamming weight of ${\bf x}$ is defined in the standard way by

[TABLE]

The quantum symmetric depolarizing channel is the channel with the trace-preserving completely positive linear map

[TABLE]

The quantity $p,0\leqslant p\leqslant 1$, is channel error probability.

Equivalently the quantum symmetric depolarizing channel can be defined as a channel in which the $j$-th qubit is effected by $e_{j}\in\{I_{2},\sigma_{x},\sigma_{y},\sigma_{z}\}$ with probabilities

[TABLE]

[TABLE]

Thus a quantum code state $|\psi\rangle\in Q$ is effected by the error operator $E_{\bf x}$ with probability

[TABLE]

II-B Quantum Enumerators

Important parameter of any classical linear code is its weight enumerator, or its weight distribution. The weight enumerator of a linear $[n,k]$ code $C$ over $\mathbb{F}_{q}$ is defined as the set of numbers:

[TABLE]

The Euclidian dual code of $C$ is defined by

[TABLE]

where

[TABLE]

is the Euclidian inner product.

We say that a code $C$ over $\mathbb{F}_{4}$ is additive if ${\bf c}+{\bf c}^{\prime}\in C$ for any ${\bf c},{\bf c}^{\prime}\in C$. The conjugate elements of $\mathbb{F}_{4}$ are defined by

[TABLE]

The Hermitian dual code of an additive code $C$ is defined by

[TABLE]

where

[TABLE]

is the trace inner product. Here $\mbox{{\rm Tr}\,}^{\mathbb{F}_{4}}_{{\mathbb{F}}_{2}}$ is the trace operator from $\mathbb{F}_{4}$ into $\mathbb{F}_{2}$. For codes over fields $\mathbb{F}_{q},q>4,$ their Hermitian dual codes are defined in [AshKnill].

In what follows we will use the same notation $C^{\perp}$ for both Euclidean and Hermitian dual codes. The meaning will be clear from the context.

If $C^{\perp}$ is Euclidean or Hermitian dual of $C$ over $\mathbb{F}_{q}$ then its weight enumerator is connected to the weight enumerator of $C$ via the MacWilliams identities:

[TABLE]

where

[TABLE]

are Krawtchouk polynomials. Often it is more convenient to formulate the MacWilliams identities in the following polynomial form. Let

[TABLE]

Then

[TABLE]

In [shor] P. Shor and R. Laflamme generalized the notion of weight enumerators for the case of quantum codes as follows. A quantum code $Q$ is a linear subspace of $\mathbb{C}^{2^{n}}$ and therefore there exists the orthogonal projector $P$ on $Q$. The code $Q$ has two quantum enumerators $B_{j}$ and $B_{j}^{\perp}$ defined by

[TABLE]

In this paper we are interested only in quantum binary stabilizer codes. A binary $[[n,k]]$ quantum stabilizer code $Q$ is a $2^{k}$-dimensional linear subspace of ${\mathbb{C}}^{2^{n}}$ that is defined by a classical additive code $C$ of length $n$ and size $2^{n+k}$ over $\mathbb{F}_{4}$. The code $C$ has the property that its Hermitian dual $C^{\perp}$ is a subset of $C$, that is $C^{\perp}\subseteq C$. See [gott],[stean],[cald] for the exact definition of stabilizer codes. If $C$ is a linear code over $\mathbb{F}_{4}$ then the corresponding quantum code $Q$ is called linear stabilizer code.

Denote by

[TABLE]

the code rates of $C^{\perp}$ and $C$ respectively. These codes rates are connected to the code rate $R={k\over n}$ of $Q$ by

[TABLE]

The quantum enumerators of a stabilizer code $Q$ are equal to the enumerators of $C$ and $C^{\perp}$, that is

[TABLE]

The quantum enumerators of a stabilizer code have a number of useful properties. In particular,

The enumerators $B_{j}$ and $B_{j}^{\perp}$ are nonnegative integers and

[TABLE] 2. 2.

If $t$ is the smallest integer such that $B_{t}>B_{t}^{\perp}$ then the minimum distance of $Q$ is $d(Q)=t$. 3. 3.

The sum of $B_{j}$ defines the size of $Q$:

[TABLE] 4. 4.

Similar to the classical case quantum enumerators are connected to each other via the MacWilliams identities

[TABLE]

where ${\cal K}_{r}(j)$ is the quaternary Krawtchouk polynomial defined in (3). In the polynomial form the quantum MacWilliams identities have the form

[TABLE]

where

[TABLE]

III Bound on Fidelity via Quantum Enumerators

Correctable and uncorrectable errors of a classical linear code in $q$-ary symmetric channel can be characterized with the help of its standard array, see for example [blahut, Ch.3.3]. For a quantum stabilizer code $Q$ associated with classical code $C$ one can generalize the standard array as it is shown in Fig.1.

Remind that a coset of a linear code $C$ generated by a vector ${\bf v}$ is the set

[TABLE]

The space $\mathbb{F}_{4}^{n}$ can be partitioned into the cosets of $C$. Each coset of $C$, say $C+{\bf v}_{l}$,${\bf v}_{l}\in\mathbb{F}_{4}^{n}$, can be further partitioned into cosets of $C^{\perp}$ for appropriately chosen coset leaders ${\bf a}_{0},\ldots,{\bf a}_{2^{2k}-1}\in C\backslash C^{\perp}$. The cosets of $C^{\perp}$ can be permuted inside the coset $C+{\bf v}_{l}$. For example, we can assign ${\bf v}_{l}^{\prime}={\bf v}_{l}+{\bf a}_{j}$, and use ${\bf v}_{l}^{\prime}$ instead of ${\bf v}_{l}$. After this permutation the leading (the most left) coset of $C^{\perp}$ in the $l$-th row of the array will be $C^{\perp}+{\bf v}_{l}+{\bf a}_{j}$.

Any vector ${\bf x}\in\mathbb{F}_{4}^{n}$ appears in the standard array one and only one time. The error operators $E_{{\bf x}}$ that correspond to the vectors ${\bf x}$ from the leading (the most left) cosets $C^{\perp}+{\bf v}_{l},l=0,\ldots,2^{n-k}-1$ (here we assume ${\bf v}_{0}=(0,\ldots,0)$) form the set $J$ of correctable error operators (see [gott],[cald] and references within).

In [hamada2001],[hamada2002] M. Hamada proved, using a result from [Preskil], the following lower bound on minimum average fidelity of $Q$:

[TABLE]

As we discussed above any coset $C^{\perp}+{\bf v}_{l}+{\bf a}_{j},j=0,\ldots,2^{2k}-1,$ (here ${\bf a}_{0}=(0,\ldots,0)$) can be used as the leading coset in the $l$-th row of the standard array. The optimal choice $j^{*}$, which maximizes $F(Q)$, is defined by

[TABLE]

The value $j^{*}$ is not fixed and depends on the channel error probability $p$. So for different values of $p$ we may have different $j^{*}$s. For deriving a lower bound on $F(Q)$ we may choose any coset $C^{\perp}+{\bf v}_{l}+{\bf a}_{j}$ as the leading one in the $l$-th row of the standard array. Our goal, of course, is to choose it such that to get large $F(Q)$. We will use the ”classical” approach. Let ${\bf w}\in C+{\bf v}_{l}$ be a minimum weight vector in the $l$-th row of the standard array, that is

[TABLE]

Then we choose as a leading coset the one that contains ${\bf w}$, that is we chose $j$ such that

[TABLE]

Examples show that when $p$ is not too large this is a good and, in fact, most likely the optimal choice of the leading coset.

In [poltyrev] G. Poltyrev derived an upper bound on the probability of decoding error of classical linear code via its weight enumerator $A_{j}$. The following theorem generalizes this bound for quantum stabilizer codes.

Theorem 1

Let $Q$ be a stabilizer code of length $n$ with quantum enumerators $B_{j}^{\perp}$ and $B_{j}$. Then

[TABLE]

where

[TABLE]

and

[TABLE]

Proof:

Let us consider an error vector

[TABLE]

According to our choice of the leading cosets in the standard array, the vector ${\bf x}$ may belong to $\mathbb{F}_{4}^{n}\setminus J$ only if $d({\bf x},{\bf c})\leqslant d({\bf x},{\bf 0})$ for some ${\bf c}=(c_{1},\ldots,c_{n})\in C\backslash C^{\perp}$. Let $\mbox{{\rm wt}\,}({\bf c})=w$ and denote

[TABLE]

Then

[TABLE]

Using the above notations we get

[TABLE]

Using (12) and (13) we get that the number of vectors ${\bf x}$ for which $d({\bf x},{\bf c})\leqslant d({\bf x},{\bf 0})=m$ is equal to $G(m,w)$.

Using union bound we can upper bound the number of error vectors ${\bf x}\in\mathbb{F}_{4}^{n}\setminus J,\mbox{{\rm wt}\,}({\bf x})=m$, by $N(m)$. Taking into account that the total number of error vectors ${\bf x},\mbox{{\rm wt}\,}({\bf x})=m$, is ${n\choose m}3^{m}$ we obtain (9). ∎

IV Bounds on the Average Fidelity of an Ensemble of Quantum Codes

In this section we derive bounds on the average fidelity over the ensemble of stabilizer codes. These bounds can be considered as achievability bounds in the sense that they prove existence of codes whose fidelity is at least as good as the bounds.

IV-A Bound on the Average Fidelity

Denote by ${\cal Q}_{stab}$ the ensemble of all $[[n,k]]$ stabilizer codes, that is

[TABLE]

Let ${\cal Q}$ be an arbitrary sub-ensemble of ${\cal Q}_{stab}$. For instance ${\cal Q}$ can be ${\cal Q}_{stab}$ itself, or it can be the ensemble of all $(n,k_{1},k_{2})$ CSS codes (see Section LABEL:sec:F_CSS_codes). The average enumerators of ${\cal Q}$ are defined as

[TABLE]

Similar to the classical case [polyanskiy][Theorem 5], using the average enumerators $\bar{B}_{j}^{\perp}$ and $\bar{B}_{j}$ in Theorem 1, we can obtain an upper bound on the average value of $1-F$ over the ensemble ${\cal Q}$. We formulate it as a Corollary of Theorem 1

Corollary 2

[TABLE]

where

[TABLE]