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IEEE Access

The Gilbert-Varshamov Bound for Stabilizer Codes Over $\mathbb{Z}_m$

Nianqi Tang, Zhuo Li, Lijuan Xing, Ming Zhang

Published Jan 1, 2018• 2 citations
Publisher

Abstract

Quantum codes over finite rings have received a great deal of attention in recent years. Compared with quantum codes over finite fields, a notable advantage of quantum codes over finite rings is that they can adapt to quantum physical systems of arbitrary order. Moreover, operations are much easier to execute in finite rings than they are in fields. The modulo m residue class ring ℤmis the most common finite ring. This paper investigates stabilizer codes over ℤmand presents the Gilbert-Varshamov (GV) bound. The GV bound shows that surprisingly good quantum codes exist over ℤm, which makes quantum coding feasible for arbitrary quantum physical system. We also provide an enhanced version of the GV bound for non-degenerate stabilizer codes over ℤm. The enhanced GV bound has an asymptotical form and ensures the existence of asymptotically good stabilizer codes over ℤm. Finally, these two bounds are well suited for computer searching.

DOI
10.1109/access.2018.2865918

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The Gilbert-Varshamov Bound for Stabilizer Codes Over Zm\mathbb{Z}_mZm​
Quantum codes over finite rings have received a great deal of attention in recent years. Compared with quantum codes over finite fields, a notable advantage of quantum codes over finite rings is that they can adapt to quantum physical systems of arbitrary order. Moreover, operations are much easier to execute in finite rings than they are in fields. The modulo m residue class ring ℤmis the most common finite ring. This paper investigates stabilizer codes over ℤmand presents the Gilbert-Varshamov (GV) bound. The GV bound shows that surprisingly good quantum codes exist over ℤm, which makes quantum coding feasible for arbitrary quantum physical system. We also provide an enhanced version of the GV bound for non-degenerate stabilizer codes over ℤm. The enhanced GV bound has an asymptotical form and ensures the existence of asymptotically good stabilizer codes over ℤm. Finally, these two bounds are well suited for computer searching.
We show that for any R,ε>0R,ε>0R,ε>0 and any prime p≥2p\geq 2p≥2, there exists an infinite family of ppp-ary quantum low-density parity-check (QLDPC) codes, rate RRR, checks of weight Oε(1)O_ε(1)Oε​(1), and normalized distance at least δGV(p,R)−εδ_{\mathrm{GV}}(p,R)-εδGV​(p,R)−ε. Here, δGV(p,R)δ_{\mathrm{GV}}(p,R)δGV​(p,R) denotes the quantum Gilbert-Varshamov (GV) bound for ppp-ary stabilizer codes. In fact, we construct an ensemble of such QLDPC codes, such that a random code from this ensemble is close to the GV bound with high probability. Moreover, this ensemble matches the performance of random stabilizer codes on several quantum channels. Specifically, it approaches the quantum capacity of the erasure channel and the hashing bound for memoryless Pauli channels, including the depolarizing channel. A significant challenge in working with QLDPC codes is that they are necessarily \emph{degenerate}, i.e., contain many low-weight stabilizers. A key contribution of our work is a construction of QLDPC codes with quantitative control on their degeneracy. These codes are obtained by combining known constructions of asymptotically good QLDPC codes with the expander-based distance amplification procedure of Alon, Edmonds, and Luby [FOCS'95]. Our random ensemble is constructed by starting with these low-degeneracy QLDPC codes near the quantum Singleton bound and concatenating each coordinate with a random inner code. This can be viewed as a quantum analogue of Thommesen's construction, and as an LDPC version of a result of Ouyang, with an appropriately designed outer code.
In this short letter, our main objective is to construct good quantum stabilizer codes over small fields, which have a more potential possibility to be employed in actual quantum processors. We firstly consider a suitable class of two-generators quasi-cyclic (QC) codes and determine their parameters. Then by virtue of their symplectic dual algebraic structure, we derive a sufficient condition to be symplectic dual-containing of these codes. Subsequently, an explicit construction of stabilizer codes from quasi-cyclic codes is presented. As for computational results, we provide many stabilizer codes with good parameters over binary, ternary and quaternary fields that all can not be deduced by the quantum Gilbert-Varshamov (GV) bound. In the binary case, our several codes strictly improve the lower bounds on the minimum distance in current records. In the ternary and quaternary occasions, numerous codes fill some gaps or have better parameters than the current results.
The Gilbert--Varshamov (GV) bound is a central benchmark in coding theory, establishing existential guarantees for error-correcting codes and serving as a baseline for both Hamming and quantum fault-tolerant information processing. Despite decades of effort, improving the GV bound is notoriously difficult, and known improvements often rely on technically heavy arguments and do not extend naturally to the quantum setting due to additional self-orthogonality constraints. In this work we develop a concise probabilistic method that yields an improvement over the classical GV bound for qqq-ary linear codes. For relative distance δ=d/n<1−1/qδ=d/n<1-1/qδ=d/n<1−1/q, we show that an [n,k,d]q[n,k,d]_q[n,k,d]q​ linear code exists whenever qk−1q−1  <  cδn qnVolq(n,d−1)\frac{q^{k}-1}{q-1}\;<\;\frac{c_δ\sqrt{n}\, q^{n}}{\mathrm{Vol}_q(n,d-1)}q−1qk−1​<Volq​(n,d−1)cδ​n​qn​, for positive constant cδc_δcδ​ depending only on δδδ, where Volq(n,d−1)\mathrm{Vol}_q(n,d-1)Volq​(n,d−1) denotes the volume of a qqq-ary Hamming ball. We further adapt this approach to the quantum setting by analyzing symplectic self-orthogonal structures. For δ<1−1/q2δ<1-1/q^2δ<1−1/q2, we obtain an improved quantum GV bound: there exists a qqq-ary quantum code [[n, n−k, d]][[n,\,n-k,\,d]][[n,n−k,d]] provided that q2n−k−1q−1<cδn⋅q2n∑i=0d−1(ni)(q2−1)i\frac{q^{2n-k}-1}{q-1}<\frac{c_δ\sqrt{n}\cdot q^{2n}}{\sum_{i=0}^{d-1}\binom{n}{i}(q^2-1)^i}q−1q2n−k−1​<∑i=0d−1​(in​)(q2−1)icδ​n​⋅q2n​. In particular, our result improves the standard quantum GV bound by an Ω(n)Ω(\sqrt{n})Ω(n​) multiplicative factor.
Guruswami et al. showed that the list-decodability of random linear codes is as good as that of general random codes. In this paper, we further strengthen the result by showing that the list-decodability of random Euclidean self-orthogonal codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. In particular, we show that, for any fixed finite field Fq, error fraction δ ∈ (0,1 - 1/q) satisfying 1 - Hq(δ) ≤ 1/2, and small ε > 0, with high probability a random Euclidean self-orthogonal code over Fqof rate 1 - Hq(δ) - ε is (δ, O(1/ε))-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding symplectic dual-containing codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.
We consider design of the quantum stabilizer codes via a two-step, low-complexity approach based on the framework of codeword-stabilized (CWS) codes. In this framework, each quantum CWS code can be specified by a graph and a binary code. For codes that can be obtained from a given graph, we give several upper bounds on the distance of a generic (additive or nonadditive) CWS code, and the lower Gilbert-Varshamov bound for the existence of additive CWS codes. We also consider additive cyclic CWS codes and show that these codes correspond to a previously unexplored class of single-generator cyclic stabilizer codes. We present several families of simple stabilizer codes with relatively good parameters.