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arXiv:1009.3539

Is A Quantum Stabilizer Code Degenerate or Nondegenerate for Pauli Channel?

Fangying Xiao, Hanwu Chen

Published Sep 18, 2010• 0 citations
arXivHTMLPDF

Abstract

Mapping an error syndrome to the error operator is the core of quantum decoding network and is also the key step of recovery. The definitions of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix were presented, and then the error syndromes of quantum errors were expressed in terms of the columns of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix. It also showed that the error syndrome matrices of a stabilizer code are determined by its check matrix, which is similar to the classical case. So, the error-detection and recovery techniques of classical linear codes can be applied to quantum stabilizer codes after some modifications. Some necessary and/or sufficient conditions for the stabilizer code over GF(2) is degenerate or nondegenerate for Pauli channel based on the relationship between the error syndrome matrices and the check matrix was presented. A new way to find the minimum distance of the quantum stabilizer codes based on their check matrices was presented, and followed from which we proved that the performance of degenerate quantum code outperform (at least have the same performance) nondegenerate quantum code for Pauli channel.

DOI
10.48550/arxiv.1009.3539

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Mapping an error syndrome to the error operator is the core of quantum decoding network and is also the key step of recovery. The definitions of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix were presented, and then the error syndromes of quantum errors were expressed in terms of the columns of the bit-flip error syndrome matrix and the phase-flip error syndrome matrix. It also showed that the error syndrome matrices of a stabilizer code are determined by its check matrix, which is similar to the classical case. So, the error-detection and recovery techniques of classical linear codes can be applied to quantum stabilizer codes after some modifications. Some necessary and/or sufficient conditions for the stabilizer code over GF(2) is degenerate or nondegenerate for Pauli channel based on the relationship between the error syndrome matrices and the check matrix was presented. A new way to find the minimum distance of the quantum stabilizer codes based on their check matrices was presented, and followed from which we proved that the performance of degenerate quantum code outperform (at least have the same performance) nondegenerate quantum code for Pauli channel.
Mapping the error syndromes to error operators is the core of quantum decoding network and the key step to realize quantum error correction. The definitions of the bit flip error syndrome matrix and the phase flip error syndrome matrix are presented, and then the error syndromes of Pauli errors are expressed in terms of the columns of the bit flip error syndrome matrix and the phase flip error syndrome matrix. It is also shown that the error syndrome matrix of a stabilizer code is determined by its check matrix, which is similar to the relationship between the classical error and the parity check matrix of classical codes. So, the techniques of error detection and error correction for classical linear codes can be applied to quantum stabilizer codes after some modifications. The error correction circuits are constructed based on the relationship between the error operator and error syndrom. The decoding circuit is constructed by reversing the encoding circuit because the encoding operators are unitary.
In this article we address the computational hardness of optimally decoding a quantum stabilizer code. Much like classical linear codes, errors are detected by measuring certain check operators which yield an error syndrome, and the decoding problem consists of determining the most likely recovery given the syndrome. The corresponding classical problem is known to be NP-complete, and a similar decoding problem for quantum codes is also known to be NP-complete. However, this decoding strategy is not optimal in the quantum setting as it does not take into account error degeneracy, which causes distinct errors to have the same effect on the code. Here, we show that optimal decoding of stabilizer codes is computationally much harder than optimal decoding of classical linear codes, it is #P.
In this paper, we address the computational hardness of optimally decoding a quantum stabilizer code. Much like classical linear codes, errors are detected by measuring certain check operators which yield an error syndrome, and the decoding problem consists of determining the most likely recovery given the syndrome. The corresponding classical problem is known to be NP-complete, and are appropriate a similar decoding problem for quantum codes is also known to be NP-complete. However, this decoding strategy is not optimal in the quantum setting as it does not consider error degeneracy, which causes distinct errors to have the same effect on the code. Here, we show that optimal decoding of stabilizer codes (previously known to be NP-hard) is in fact computationally much harder than optimal decoding of classical linear codes, it is #P-complete.
In this paper, a new but simple construction of stabilizer codes and related entanglement-assisted quantum error-correcting codes is proposed based on syndrome assignment by classical parity-check matrices. This method turns the construction of quantum stabilizer codes to the construction of classical parity-check matrices satisfying a specific commutative condition. The designed minimum distance 2t*+1 of the constructed quantum stabilizer codes can be achieved by a commutative classical parity-check matrix with classical minimum distance 4t*-m, where the parameterm, 0 ≤m≤ 2t*, depends on a property of the parity-check matrix. Asmdecreases, there is an increasing set of additional correctable error operators beyond the designed error correcting capability t*. The (asymptotic) coding efficiency is at least comparable to that of CSS codes. A class of quantum Reed-Muller codes is constructed and codes in this class have a larger set of correctable error operators than that of the quantum Reed-Muller codes previously developed in the literature. Quantum circulant codes are also constructed and many of them are optimal in terms of their coding parameters.
Stabilizer codes are a fundamental class of error-correcting codes for quantum information that allow for syndrome decoding in the quantum domain. One of the substantial challenges in quantum error correction is that the quantum gates that perform error correction themselves are faulty in practice, which makes fault-tolerant implementation a vital component. Recently, a coding theoretic technique is proposed that makes it possible for syndrome decoding to help correct imperfectly extracted syndromes instead of fully relying on an external fault-tolerant mechanism. In particular, it was proved that any single-error-correcting stabilizer code can be made robust against single errors on either a data qubit or a syndrome bit by using at most one more stabilizer operator than necessary for standard syndrome decoding, while analogous overhead for making double-error-correcting stabilizer codes robust against double errors involving data qubits and/or syndrome bits was shown to be at most logarithmic. We generalize this result to t-error-correcting stabilizer codes and show that the overhead for achieving analogously defined global t-error correction for data qubits and/or syndrome bits is also at most logarithmic. The proof exploits combinatorial arrays that may be seen as parity-check matrices that detect errors of even weight but may overlook errors of odd weight.
A striking feature of quantum error correcting codes is that they can sometimes be used to correct more errors than they can uniquely identify. Such degenerate codes have long been known, but have remained poorly understood. We provide a heuristic for designing degenerate quantum codes for high noise rates, which is applied to generate codes that can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code. The gap between nondegenerate and degenerate code performance is quite large, in contrast to the tiny magnitude of the only previous demonstration of this effect. We also identify a channel for which none of our codes outperform the best nondegenerate code and show that it is nevertheless quite unlike any channel for which nondegenerate codes are known to be optimal.