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IEEE Transactions on Information Theory

On a Problem Concerning the Quantum Hamming Bound for Impure Quantum Codes

Zhuo Li, Lijuan Xing

Published Aug 19, 2010• 9 citations
Publisher

Abstract

A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate that there exists a thresholdN(d,m) such that an arbitrary((n,K,d))mquantum code must obey the quantum Hamming bound whenevern≥N(d,m). We list some values ofN(d,m) for smalldand binary quantum codes.

DOI
10.1109/tit.2010.2054610

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A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate that there exists a thresholdN(d,m) such that an arbitrary((n,K,d))mquantum code must obey the quantum Hamming bound whenevern≥N(d,m). We list some values ofN(d,m) for smalldand binary quantum codes.
No ((n,K,d<127))((n,K,d< 127))((n,K,d<127)) Code Can Violate the Quantum Hamming Bound
It is well-known that non-degenerate quantum error correcting codes (QECCs) are constrained by a quantum version of the Hamming bound. Whether degenerate codes also obey such a bound, however, remains a long-standing question with practical implications for the efficacy of QECCs. We employ a combination of previously derived bounds on QECCs to demonstrate that a subset of all codes must obey the quantum Hamming bound. Specifically, we combine an analytical bound due to Rains with a numerical bound due to Li and Xing to show that no((n,K,d))((n,K,d))((n,K,d))code withd<127d< 127d<127can violate the quantum Hamming bound.
The quantum Hamming bound was originally put forward as an upper bound on the parameters of nondegenerate quantum codes, but over the past few decades much work has been done to show that many degenerate quantum codes must also obey this bound. In this paper, we show that there is a Hamming-like bound stricter than the quantum Hamming bound that applies to degenerate ttt-error-correcting stabilizer codes of length greater than some positive integer N(t)N(t)N(t). We show that this bound holds for all single-error-correcting degenerate stabilizer codes, forcing all but a handful of optimal distance-3 stabilizer codes to be nondegenerate.
The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension q≥3q\ge3q≥3. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For q≥4q\ge4q≥4, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.
We prove a conjecture of Yu, Lai, and Oh: their strengthened quantum Hamming bounds, a family that interpolates between Hamming-type bounds and the Singleton bound, hold for all quantum codes of dimension K>1K>1K>1, including impure and nonadditive codes. The result covers every integer local dimension p≥2p\ge2p≥2, not necessarily a prime power, every integer t≥1t\ge1t≥1, both distances 2t+12t+12t+1 and 2t+22t+22t+2, and every interpolation parameter e∈{0,…,t}e\in\{0,\ldots,t\}e∈{0,…,t}. The case e=0e=0e=0 implies the ordinary quantum Hamming bound for general codes, recently proved by Zhang and Chen. Yu, Lai, and Oh proved their bounds for pure codes and for general codes of distance three or four. More generally, for every integer p≥2p\ge2p≥2 and every quantum code of length nnn, dimension K>1K>1K>1 and distance d≥3d\ge3d≥3, we prove Kf(0)≤pnEf(X)Kf(0)\le p^n\mathbb{E} f(X)Kf(0)≤pnEf(X), where X∼Bin⁡(n,1−p−2)X\sim\operatorname{Bin}(n,1-p^{-2})X∼Bin(n,1−p−2), for every real polynomial fff of degree less than ddd that is nonnegative on {0,…,n}\{0,\ldots,n\}{0,…,n} and positive at zero. For pure codes this inequality is implicit in the argument of Yu, Lai, and Oh. The proof uses the linear-programming method with quantum weight enumerators.