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IEEE International Symposium on Information Theory (ISIT)

QBEL: Quantum Burst Error Locating Codes

Roni Con, Ryan Gabrys, Eitan Yaakobi

Published Jun 22, 2025• 2 citations

Abstract

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 of$2 b$qubits containing a burst of quantum errors of length at most$b$, 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 length$b$that do not end with a$Y$Pauli operator. We prove that the redundancy of this code is optimal with respect to this error set.

DOI: 10.1109/isit63088.2025.11195453

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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 of

2 b

qubits containing a burst of quantum errors of length at most

b

, 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 length

b

that do not end with a

Y

Pauli operator. We prove that the redundancy of this code is optimal with respect to this error set.

Quantum burst error correction codes (QBECCs) are of great importance to deal with the memory effect in quantum channels. As the most important family of QBECCs, quantum cyclic codes (QCCs) play a vital role in the correction of burst errors. In this work, we characterize the burst error correction ability of QCCs constructed from the Calderbank-Shor-Steane (CSS) and the Hermitian constructions. We determine the burst error correction limit of QCCs and quantum Reed-Solomon codes with algorithms in polynomial-time complexities. As a result, lots of QBECCs saturating the quantum Reiger bound are obtained. We show that quantum Reed-Solomon codes have better burst error correction abilities than the previous results. At last, we give the quantum error-trapping decoder (QETD) of QCCs for decoding burst errors. The decoder runs in linear time and can decode both degenerate and nondegenerate burst errors. What’s more, the numerical results show that QETD can decode much more degenerate burst errors than the nondegenerate ones.

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).

Quantum synchronizable codes (QSCs) are designed to correct both Pauli errors and synchronization misalignment. In this letter, we introduce a new framework for constructing QSCs based on classical coset codes. This construction is versatile, yielding codes for both random and, notably, burst errors, thereby extending the scope of existing QSC designs. The resulting codes exhibit two notable properties: they fully preserve the phase-error-correction capability of their parent Calderbank-Shor-Steane (CSS) codes, and in the absence of misalignment, they revert to the full error-correcting power of the parent codes.

Abstract We present the construction of quantum error-locating (QEL) codes based on classical error-locating (EL) codes. Similar to classical EL codes, QEL codes lie midway between quantum error-correcting codes and quantum error-detecting codes. Then QEL codes can locate qubit errors within one sub-block of the received qubit symbols but do not need to determine the exact locations of the erroneous qubits. We show that, an e-error-locating code derived from an arbitrary binary cyclic code with generator polynomial g ( x ) , can lead to a QEL code with e error-locating abilities, only if g ( x ) does not contain the ( 1 + x ) -factor .