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arXiv:1212.6703 · Physical Review A

Quantum Kronecker sum-product low-density parity-check codes with finite rate

Alexey A. Kovalev, Leonid P. Pryadko

Published Jul 11, 2013• 108 citations
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Abstract

We introduce an ansatz for quantum codes which gives the hypergraph-product (generalized toric) codes by Tillich and Z\'emor and generalized bicycle codes by MacKay et al. as limiting cases. The construction allows for both the lower and the upper bounds on the minimum distance; they scale as a square root of the block length. Many thus defined codes have a finite rate and limited-weight stabilizer generators, an analog of classical low-density parity-check (LDPC) codes. Compared to the hypergraph-product codes, hyperbicycle codes generally have a wider range of parameters; in particular, they can have a higher rate while preserving the estimated error threshold.

DOI
10.1103/physreva.88.012311

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We introduce an ansatz for quantum codes which gives the hypergraph-product (generalized toric) codes by Tillich and Z'emor and generalized bicycle codes by MacKay et al. as limiting cases. The construction allows for both the lower and the upper bounds on the minimum distance; they scale as a square root of the block length. Many thus defined codes have a finite rate and limited-weight stabilizer generators, an analog of classical low-density parity-check (LDPC) codes. Compared to the hypergraph-product codes, hyperbicycle codes generally have a wider range of parameters; in particular, they can have a higher rate while preserving the estimated error threshold.
We suggest several techniques to improve the toric codes and the finite-rate generalized toric codes (quantum hypergraph-product codes) recently introduced by Tillich and Zémor. For the usual toric codes, we introduce the rotated lattices specified by two integer-valued periodicity vectors. These codes include the checkerboard codes, and the family of minimal single-qubit-encoding toric codes with block length n = t2+ (t+1)2and distance d = 2t + 1, t = 1, 2, ... We also suggest several related algebraic constructions which increase the rate of the existing hypergraph-product codes by up to four times.
We study the fault tolerance of quantum low-density parity check (LDPC) codes, such as generalized toric codes with a finite rate suggested by Tillich and Z\'emor [in ISIT 2009: IEEE International Symposium on Information Theory (IEEE, New York, 2009)]. We show that any family of quantum LDPC codes where each syndrome measurement involves a limited number of qubits and each qubit is involved in a limited number of measurements (as well as any similarly limited family of classical LDPC codes), in which distance scales as a positive power $\ensuremath{\alpha}$ of the number of physical qubits ($\ensuremath{\alpha}<1$ for ``bad'' codes), has a finite error probability threshold. We conclude that for sufficiently large quantum computers, quantum LDPC codes can offer an advantage over the toric codes.
We construct quantum LDPC (qLDPC) and high-rate CSS codes from our recently introduced classical fair-density parity-check (FDPC) codes. To this end, we introduce a structured sparsification of FDPC parity-check matrices, which reduces their check weights while preserving the underlying combinatorial structure and distance guarantees. Combined with the hypergraph-product construction, this yields finite-length qLDPC codes with analytically controlled blocklength, dimension, certified distance, and stabilizer weight. For quantum blocklengths N<105N<10^5N<105, the constructions introduced here span guaranteed rates from approximately 0.35%0.35\%0.35% to 25.8%25.8\%25.8% and certified quantum distances from 121212 to 696969, with stabilizer weights between 888 and 161616. In the large-blocklength regime, allowing the FDPC order and sparsified check weight to scale moderately with blocklength yields a family of high-rate CSS codes, which we refer to as quantum FDPC (qFDPC) codes, with rate RQR_QRQ​ and minimum distance DDD satisfying RQ=1−O(1log⁡log⁡N),  D=Ω(N1/4), R_Q = 1-O\left(\frac{1}{\log\log N}\right), \ \ D=Ω(N^{1/4}), RQ​=1−O(loglogN1​),  D=Ω(N1/4), and stabilizer weight O(log⁡Nlog⁡log⁡N)O(\log N\log\log N)O(logNloglogN). Finally, the analytically available FDPC weight distribution provides explicit information about the logical operators of the resulting hypergraph-product codes. Over the quantum erasure channel, this structure yields rigorous first-order maximum likelihood (ML) expressions and a higher-order weight-distribution approximation to the ML logical block error probability. This enables estimation of finite-length operating points and the onset of the error-floor regime. To the best of our knowledge, beyond surface-code-type constructions, this is the first finite-rate qLDPC framework to provide both finite-length certified minimum-distance information and an analytical characterization of low-weight logical multiplicities.
Hypergraph product codes introduced by Tillich and Zémor are a class of quantum LDPC codes with constant rate and distance scaling with the square-root of the block size. Quantum expander codes, a subclass of these codes, can be decoded using the linear time small-set-flip algorithm of Leverrier, Tillich and Zémor. In this paper, we numerically estimate the performance for the hypergraph product codes under independent bit and phase flip noise. We focus on two families of hypergraph product codes. The first family has rate 1/61∼1.6%1/61 \sim 1.6\%1/61∼1.6%, has qubits of weight 101010 or 121212 and stabilizers of weight 111111. We report a threshold near 4.6%4.6\%4.6% for the small-set-flip decoder. We also show that for similar rate, the performance of the hypergraph product is better than the performance of the toric code as soon as we deal with more than 500500500 logical qubits and that for 360036003600 logical qubits, the logical error rate for the hypergraph product code is several orders of magnitude smaller. The second family has rate 0.20.20.2, qubits of weight 101010 and 202020 and stabilizers of weight 151515. We report a threshold near 2%2\%2% for the small-set-flip decoder.
In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes LkL_kLk​ and Lk+L_k^{+}Lk+​, defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters [[6k2, k2, d]][[6k^2,\, k^2,\, d]][[6k2,k2,d]]. The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies d=3d=3d=3 for k=3k=3k=3 and d=4d=4d=4 for k≥4k\ge4k≥4, establishing a new framework for structured quantum LDPC code design and optimization.