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International Symposium on Information Theory and its Applications

Performance of Nonbinary Cubic Codes

Arun John Moncy, Pradeep Kiran Sarvepalli

Published Oct 1, 2018• 2 citations
Publisher

Abstract

Cubic codes were proposed by Haah as candidates for self-error correction in three dimensions (3D). While these codes are not self-correcting, Bravyi and Haah showed that they are partially self-correcting. In this paper we are interested in generalizations of the cubic code to prime alphabet. Kim initiated the study of such codes over prime alphabet. Haah also proposed a framework based on modules that enables the study of cubic codes over higher alphabet. However, there are many open questions remaining, especially those pertaining to the performance of nonbinary cubic codes. Building on Bravyi and Haah's decoder, we study the performance of nonbinary cubic codes over the quantum erasure and depolarizing channels. This is the first such study of nonbinary cubic codes.

DOI
10.23919/isita.2018.8664389

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Cubic codes were proposed by Haah as candidates for self-error correction in three dimensions (3D). While these codes are not self-correcting, Bravyi and Haah showed that they are partially self-correcting. In this paper we are interested in generalizations of the cubic code to prime alphabet. Kim initiated the study of such codes over prime alphabet. Haah also proposed a framework based on modules that enables the study of cubic codes over higher alphabet. However, there are many open questions remaining, especially those pertaining to the performance of nonbinary cubic codes. Building on Bravyi and Haah's decoder, we study the performance of nonbinary cubic codes over the quantum erasure and depolarizing channels. This is the first such study of nonbinary cubic codes.
A big open question in the quantum information theory concerns the feasibility of a self-correcting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction, if the memory is in contact with a cold enough thermal bath. Here we report analytic and numerical evidence for self-correcting behavior in the quantum spin lattice model known as the 3D cubic code. We prove that its memory time is at least L(cβ), where L is the lattice size, β is the inverse temperature of the bath, and c>0 is a constant coefficient. However, this bound applies only if the lattice size L does not exceed a critical value which grows exponentially with β. In that sense, the model can be called a partially self-correcting memory. We also report a Monte Carlo simulation indicating that our analytic bounds on the memory time are tight up to constant coefficients. To model the readout step we introduce a new decoding algorithm, which can be implemented efficiently for any topological stabilizer code. A longer version of this work can be found in Bravyi and Haah, arXiv:1112.3252.
Haah's 3D cubic code was introduced as a candidate for a self-correcting quantum memory in three dimensions, using the absence of string logical operators to obstruct thermal errors. It has been a long-standing open question whether it is a self-correcting quantum memory. Even though the memory time was explored numerically, no analytical upper bound was known. Here, we settle this question rigorously and show that the cubic code thermalizes rapidly at every positive temperature. Our proof relies on the framework of [Commun. Math. Phys. 407, 195 (2026)] to turn a decay-of-correlations condition into a modified logarithmic Sobolev inequality for the Davies generator. To the best of our knowledge, this is the first analytic proof of rapid mixing for all positive temperatures for any fracton code.
The codeword-stabilized (CWS) quantum code formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes [see A. Cross et al., e-print arXiv:0708.1021], but only for binary states. Here we generalize the CWS framework to the nonbinary case (of both prime and nonprime dimensions) and map the search for nonbinary quantum codes to a corresponding search problem for classical nonbinary codes with specific error patterns. We show that while the additivity properties of nonbinary CWS codes are similar to the binary case, the structural properties of the nonbinary codes differ substantially from the binary case, even for prime dimensions. In particular, we identify specific structure patterns of stabilizer groups, based on which efficient constructions might be possible for codes that encode more dimensions than any stabilizer codes of the same length and distance; similar methods cannot be applied in the binary case. Understanding of these structural properties can help prune the search space and facilitate the identification of good nonbinary CWS codes.
In the late 1950's and early 1960's, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for correcting random errors with algebraic decoding, such as Bose-Chaudhuri-Hocqenghem (BCH) and Reed-Solomon (RS) codes. Recently it has been shown that finite fields can also be used successfully to construct binary quasi-cyclic (QC)-LDPC codes that perform very well not only over the AWGN channel but also over the binary erasure channel with iterative decoding, besides being efficiently encodable. This paper is concerned with constructions of nonbinary QC-LDPC codes based on finite fields
Three code constructions generating new families of good nonbinary quantum codes are presented in this paper. The first two ones are derived from Hermitian self-orthogonal non-narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes. The third one is derived from qqq-ary ($q\ensuremath{\ne}2$ is a prime power) Steane's enlargement of Calderbank-Shor-Steane codes applied to Euclidean self-orthogonal non-narrow-sense BCH codes. The quantum nonbinary BCH codes presented here have parameters better than the ones available in the literature.