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1 Quantum Error Correction beyond the Bounded Distance Decoding Limit

Kenta Kasai, Manabu Hagiwara, Hideki Imai, Life Fellow, Kohichi Sakaniwa, Senior Member

Published Aug 16, 2016• 38 citations
OpenAlex

Abstract

In this paper, we consider quantum error correction over depolarizing channels with non-binary low-density parity-check codes defined over Galois field of size $2^p$ . The proposed quantum error correcting codes are based on the binary quasi-cyclic CSS (Calderbank, Shor and Steane) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., $2^p$, the error floors are considerably improved.

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In this paper, we consider quantum error correction over depolarizing channels with non-binary low-density parity-check codes defined over Galois field of size 2p2^p2p . The proposed quantum error correcting codes are based on the binary quasi-cyclic CSS (Calderbank, Shor and Steane) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., 2p2^p2p, the error floors are considerably improved.
In this paper, we consider quantum error correction over depolarizing channels with nonbinary low-density parity-check codes defined over Galois field of size 2p. The proposed quantum error correcting codes are based on the binary quasi-cyclic Calderbank, Shor, and Steane (CSS) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., 2p, the error floors are considerably improved.
In this paper, we propose a construction method for two-level quantum error-correcting codes via non-binary LDPC codes over an extended field of order 2p, p an integer p >; 1. The proposed quantum error-correcting codes are based on binary quasi-cyclic LDPC codes which have almost achieved a “Bounded Distance Decoding (BDD)” limit but have not surpassed the limit yet. Quantum codes constructed from the proposed method surpass the BDD limit. Furthermore the codes outperform the efficiently-decodable state-of-the-art quantum codes.
Quantum error correction codes are needed to protect quantum systems from interference. An important family of quantum error correction codes are stabilizer codes. Many stabilizer codes in the literature belong to a subclass of stabilizer codes, referred to as CSS codes since they are based on the so-called CSS formalism. CSS codes have received significant attention because the dual-containing condition for the CSS formalism is relatively straightforward to satisfy and also because it is easy to produce CSS codes from classical error correction codes. However, some stabilizer codes in the literature show superior error correction performance than existing CSS codes, and are based on binary low-density parity-check (LDPC) codes. Since it has been shown that CSS codes based on nonbinary LDPC codes have good performance over quantum qubit channels, in this paper we investigate stabilizer codes based on nonbinary LDPC codes. In particular, we propose constructions of stabilizer codes based on nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes for qubit channels. The simulation results show that our QC-LDPC stabilizer codes decoded by a nonbinary sum-product algorithm have better performance than their binary counterparts.
The near-capacity performance of classical low-density parity check (LDPC) codes and their efficient iterative decoding makes quantum LDPC (QLPDC) codes a promising candidate for quantum error correction. In this paper, we present a comprehensive survey of QLDPC codes from the perspective of code design as well as in terms of their decoding algorithms. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank-Shor-Steane-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-four cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their word error rate performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic uniformly reweighted belief propagation for the sake of achieving an improved performance.
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define XXX and ZZZ-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with XXX-, YYY-, and ZZZ-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the YYY-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2^22 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.
Quantum error correction is essential for realizing scalable quantum computation. Among various approaches, low-density parity-check codes over higher-order Galois fields have shown promising performance due to their structured sparsity and compatibility with iterative decoding algorithms whose computational complexity scales linearly with the number of physical qubits. In this work, we demonstrate that explicitly exploiting the degeneracy of quantum errors can significantly enhance the decoding performance. Simulation results over the depolarizing channel indicate that the proposed method, at a coding rate of 1/3, achieves a frame error rate as low as 10−410^{-4}10−4 at a physical error rate of 9.45% for a code with 104,000 logical qubits and 312,000 physical qubits, approaching the quantum hashing bound. These findings highlight the critical role of degeneracy in closing the gap to the fundamental limits of quantum error correction.