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arXiv:1407.7909

Using arbitrary parity-check matrices for quantum error correction assisted by less noisy qubits

Yuichiro Fujiwara

Published Jul 30, 2014• 0 citations
arXivHTMLPDF

Abstract

Recently a framework for assisted quantum error correction was proposed in which a specific type of error is allowed to occur on auxiliary qubits, which is in contrast to standard entanglement assistance that requires noiseless auxiliary qubits. However, while the framework maintains the ability to import any binary or quaternary linear code without sacrificing active error correction power, it requires the code designer to turn a parity-check matrix of the underlying classical code into an equivalent one in standard form. This means that classical coding theoretic techniques that require parity-check matrices to be in specific form may not fully be exploitable. Another issue of the recently proposed scheme is that the error correction capabilities for bit errors and phase errors are generally equal, which is not ideal for asymmetric error models. This paper addresses these two problems. We generalize the framework in such a way that any parity-check matrix of any binary or quaternary linear code can be exploited. Our generalization also allows for importing a pair of distinct linear codes so that error correction capabilities become suitably asymmetric.

DOI
10.48550/arxiv.1407.7909

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Recently a framework for assisted quantum error correction was proposed in which a specific type of error is allowed to occur on auxiliary qubits, which is in contrast to standard entanglement assistance that requires noiseless auxiliary qubits. However, while the framework maintains the ability to import any binary or quaternary linear code without sacrificing active error correction power, it requires the code designer to turn a parity-check matrix of the underlying classical code into an equivalent one in standard form. This means that classical coding theoretic techniques that require parity-check matrices to be in specific form may not fully be exploitable. Another issue of the recently proposed scheme is that the error correction capabilities for bit errors and phase errors are generally equal, which is not ideal for asymmetric error models. This paper addresses these two problems. We generalize the framework in such a way that any parity-check matrix of any binary or quaternary linear code can be exploited. Our generalization also allows for importing a pair of distinct linear codes so that error correction capabilities become suitably asymmetric.
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The classical linear codes that construct quantum error correction codes need to meet certain specific conditions. The proposal of entanglement-assisted quantum error correcting codes eliminates the restriction of these conditions, so all classical linear codes can be constructed into entanglement-assisted quantum error correcting codes. The Hadamard matrix is a special type of orthogonal matrix. Hadamard codes, constructed based on the standard form of Hadamard matrices, are a type of linear complementary dual (LCD) codes. By extending the generator and parity-check matrices of Hadamard codes, two types of classical codes whose parameters are [(l+1)(n−1), n−1] and [3n−3, n−1] maintain the LCD property. Utilizing the relationship between their LCD property and the dimensions of the Euclidean hull, this paper constructs two types of EAQECCs based on Hadamard matrices. These EAQECCs can achieve maximal entanglement, resulting in higher code rates and net rates compared to other EAQECCs. Their parameters are [[3n−3, 2n−2, d≥1; n−1]] and [[2n−2, n−1, d≥1; n−1]], respectively.
Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum information science, where in principle any binary linear code can be turned into a quantum error-correcting code by assuming a small number of reliable quantum bits. This paper studies how best to take advantage of these latest ideas to construct desirable quantum error-correcting codes of very high information rate. Our methods exploit structured high-rate low-density parity-check codes available in the classical domain and provide quantum analogues that inherit their characteristic low decoding complexity and high error correction performance even at moderate code lengths. Our approach to designing high-rate quantum error-correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation where promising quantum analogues of low-density parity-check codes are difficult to find.
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