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IEEE Transactions on Information Theory

A Finite Gilbert–Varshamov Bound for Pure Stabilizer Quantum Codes

Kai Feng, Zhi Ma

Published Nov 30, 2004• 110 citations
Publisher

Abstract

A finite Gilbert-Varshamov (GV) bound for pure stabilizer (binary and nonbinary) quantum error correcting codes is presented in analogy to the GV bound for classical codes by using several enumerative results in finite unitary geometry. From this quantum GV bound we obtain several new binary quantum codes in a nonconstructive way having better parameters than the known codes.

DOI
10.1109/tit.2004.838088

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A finite Gilbert-Varshamov (GV) bound for pure stabilizer (binary and nonbinary) quantum error correcting codes is presented in analogy to the GV bound for classical codes by using several enumerative results in finite unitary geometry. From this quantum GV bound we obtain several new binary quantum codes in a nonconstructive way having better parameters than the known codes.
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.
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.
We generalize the random coding argument of stabilizer codes and derive a lower bound on the quantum capacity of an arbitrary discrete memoryless quantum channel. For the depolarizing channel, our lower bound coincides with that obtained by Bennett et al. We also slightly improve the quantum Gilbert–Varshamov bound for general stabilizer codes, and establish an analog of the quantum Gilbert–Varshamov bound for linear stabilizer codes. Our proof is restricted to the binary quantum channels, but its extension of to l-adic channels is straight forward.
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.