Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0907.3802

Progress on Problem about Quantum Hamming Bound for Impure Quantum Codes

Zhuo Li, Lijuan Xing

Published Jul 22, 2009• 0 citations
arXivHTMLPDF

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 threshold such that an arbitrary quantum code must obey the quantum Hamming bound whenever . We list some values of for small d and binary quantum codes.

DOI
10.48550/arxiv.0907.3802

Related papers

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

Zhuo Li, Lijuan Xing

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.

IEEE Transactions on Information Theory•Aug 19, 2010•9 citations

Asymptotics of the quantum Hamming bound for subsystem codes

Andreas Klappenecker, Pradeep Kiran Sarvepalli

Ashikhmin and Litsyn showed that all binary stabilizer codes - pure or impure - of sufficiently large length obey the quantum Hamming bound, ruling out the possibility that impure codes of large length can outperform pure codes with respect to sphere packing. In contrast we show that impure subsystem codes do not obey the quantum Hamming bound for pure subsystem codes, not even asymptotically. We show that there exist arbitrarily long Bacon-Shor codes that violate the quantum Hamming bound.

arXiv:0710.4271•Oct 23, 2007•1 citation

A Hamming-Like Bound for Degenerate Stabilizer Codes

Andrew Nemec, Theerapat Tansuwannont

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 $t$-error-correcting stabilizer codes of length greater than some positive integer $N(t)$. 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.

arXiv:2306.00048•May 31, 2023•0 citations

A Note on Quantum Hamming Bound

Salah A. Aly

Proving the quantum Hamming bound for degenerate nonbinary stabilizer codes has been an open problem for a decade. In this note, I prove this bound for double error-correcting degenerate stabilizer codes. Also, I compute the maximum length of single and double error-correcting MDS stabilizer codes over finite fields.

arXiv:0711.4603•Nov 29, 2007•2 citations

The strengthened quantum Hamming bound of Yu, Lai, and Oh

Valery, Grishin, Aryeh Lev Zabokritskiy

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 $K>1$, including impure and nonadditive codes. The result covers every integer local dimension $p\ge2$, not necessarily a prime power, every integer $t\ge1$, both distances $2t+1$ and $2t+2$, and every interpolation parameter $e\in\{0,\ldots,t\}$. The case $e=0$ 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 $p\ge2$ and every quantum code of length $n$, dimension $K>1$ and distance $d\ge3$, we prove $Kf(0)\le p^n\mathbb{E} f(X)$, where $X\sim\operatorname{Bin}(n,1-p^{-2})$, for every real polynomial $f$ of degree less than $d$ that is nonnegative on $\{0,\ldots,n\}$ 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.

quant-pharXiv:2610.30088•Oct 22, 2026•0 citations

The Quantum Hamming Bound in Arbitrary Local Dimension

Yu-Xuan Zhang, Jing-Ling Chen

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 $q\ge3$. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For $q\ge4$, 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.

quant-pharXiv:2606.22538•Jun 23, 2026•0 citations
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 threshold such that an arbitrary quantum code must obey the quantum Hamming bound whenever . We list some values of for small d and binary quantum codes.
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.
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.
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.