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arXiv:quant-ph/9704019 · IEEE Transactions on Information Theory

Spatially correlated qubit errors and burst-correcting quantum codes

Farrokh Vatan, Vwani Roychowdhury, M. P. Anantram

Published Jul 1, 1999• 28 citations
arXivHTMLPDF

Abstract

We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of burst errors in classical coding theory. We present several different efficient schemes for constructing families of such codes. For example, one can find one-dimensional quantum codes of length n=13 and 15 that correct burst errors of width b<3; as a comparison, a random-error correcting quantum code that corrects t=3 errors must have length n/spl ges/17. In general, we show that it is possible to build quantum burst-correcting codes that have near optimal dimension. For example, we show that for any constant b, there exist b-burst-correcting quantum codes with length n, and dimension k=n-log n-O(1); as a comparison, the Hamming bound for the case with t (constant) random errors yields k/spl les/n-tlog n+O(1).

DOI
10.1109/18.771250

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We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of burst errors in classical coding theory. We present several different efficient schemes for constructing families of such codes. For example, one can find one-dimensional quantum codes of length n=13 and 15 that correct burst errors of width b<3; as a comparison, a random-error correcting quantum code that corrects t=3 errors must have length n/spl ges/17. In general, we show that it is possible to build quantum burst-correcting codes that have near optimal dimension. For example, we show that for any constant b, there exist b-burst-correcting quantum codes with length n, and dimension k=n-log n-O(1); as a comparison, the Hamming bound for the case with t (constant) random errors yields k/spl les/n-tlog n+O(1).
In practical communication and computation systems, errors occur predominantly in adjacent positions rather than in a random manner. In this paper, we develop a stabilizer formalism for quantum burst error correction codes (QBECC) to combat such error patterns in the quantum regime. Our contributions are as follows. Firstly, we derive an upper bound for the correctable burst errors of QBECCs, the quantum Reiger bound (QRB). Secondly, we propose two constructions of QBECCs: one by heuristic computer search and the other by concatenating two quantum tensor product codes (QTPCs). We obtain several new QBECCs with better parameters than existing codes with the same coding length. Moreover, some of the constructed codes can saturate the quantum Reiger bounds. Finally, we perform numerical experiments for our constructed codes over Markovian correlated depolarizing quantum memory channels, and show that QBECCs indeed outperform standard QECCs in this scenario.
Burst errors, which involve the corruption of consecutive symbols, are a more realistic and common type of error in many communication and storage systems. While classical codes for burst correction are well-established, quantum errorcorrecting codes that handle burst errors and error localization have been less explored. In this paper, we present constructions of quantum codes designed to locate and correct burst errors. We introduce a nearly optimal quantum error-locating burst code, which can identify an interval of2b2 b2bqubits containing a burst of quantum errors of length at mostbbb, improving upon previous constructions in terms of redundancy. This code leverages a stabilizer framework that is not based on the CSS (Calderbank-Shor-Steane) construction, offering a more efficient error-locating capability. Additionally, our construction can correct a large structured set of burst errors, specifically those of lengthbbbthat do not end with aYYYPauli operator. We prove that the redundancy of this code is optimal with respect to this error set.
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