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arXiv:0812.2674 · Physical Review A

Degenerate quantum codes and the quantum Hamming bound

Pradeep Kiran Sarvepalli, Andreas Klappenecker

Published Mar 19, 2010• 22 citations

Abstract

The parameters of a nondegenerate quantum code must obey the Hamming bound. An important open problem in quantum coding theory is whether the parameters of a degenerate quantum code can violate this bound for nondegenerate quantum codes. In this article we show that Calderbank-Shor-Steane (CSS) codes, over a prime power alphabet $q\ensuremath{\geqslant}5$, cannot beat the quantum Hamming bound. We prove a quantum version of the Griesmer bound for the CSS codes, which allows us to strengthen the Rains' bound that an $[[n,k,d]{]}_{2}$ code cannot correct more than $\ensuremath{\lfloor}(n+1)/6\ensuremath{\rfloor}$ errors to $\ensuremath{\lfloor}(n\ensuremath{-}k+1)/6\ensuremath{\rfloor}$. Additionally, we also show that any $[[n,k,d]{]}_{q}$ quantum code with $k+d\ensuremath{\leqslant}(1\ensuremath{-}2{\mathit{eq}}^{\ensuremath{-}2})n$ cannot beat the quantum Hamming bound.

DOI: 10.1103/physreva.81.032318

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The parameters of a nondegenerate quantum code must obey the Hamming bound. An important open problem in quantum coding theory is whether the parameters of a degenerate quantum code can violate this bound for nondegenerate quantum codes. In this article we show that Calderbank-Shor-Steane (CSS) codes, over a prime power alphabet

q\ensuremath{\geqslant}5

, cannot beat the quantum Hamming bound. We prove a quantum version of the Griesmer bound for the CSS codes, which allows us to strengthen the Rains' bound that an

[[n,k,d]{]}_{2}

code cannot correct more than

\ensuremath{\lfloor}(n+1)/6\ensuremath{\rfloor}

errors to

\ensuremath{\lfloor}(n\ensuremath{-}k+1)/6\ensuremath{\rfloor}

. Additionally, we also show that any

[[n,k,d]{]}_{q}

quantum code with

k+d\ensuremath{\leqslant}(1\ensuremath{-}2{\mathit{eq}}^{\ensuremath{-}2})n

cannot beat the quantum Hamming bound.

No

((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))

code with

d< 127

can 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

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.

It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.