Partial Syndrome Measurement for Hypergraph Product Codes

Noah Berthusen and Daniel Gottesman

Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA

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Abstract

Hypergraph product codes are a promising avenue to achieving fault-tolerant quantum computation with constant overhead. When embedding these and other constant-rate qLDPC codes into 2D, a significant number of nonlocal connections are required, posing difficulties for some quantum computing architectures. In this work, we introduce a fault-tolerance scheme that aims to alleviate the effects of implementing this nonlocality by measuring generators acting on spatially distant qubits less frequently than those which do not. We investigate the performance of a simplified version of this scheme, where the measured generators are randomly selected. When applied to hypergraph product codes and a modified small-set-flip decoding algorithm, we prove that for a sufficiently high percentage of generators being measured, a threshold still exists. We also find numerical evidence that the logical error rate is exponentially suppressed even when a large constant fraction of generators are not measured.

The surface code, despite showing theoretical and experimental promise, is a poor choice for large-scale quantum computations due to its unfavorable asymptotic scaling. Quantum low-density parity-check (qLDPC) codes have been introduced as performant alternatives; however, they are more difficult to implement on hardware. In this paper, we propose a method that aims to lessen the difficulty of implementing these qLDPC codes on quantum architectures that do not have access to long-range gates. This is done by measuring the generators whose qubits are far apart less frequently than those whose qubits are close together. We provide analytical and numerical evidence that shows quantum expander codes still perform well under a related toy model.

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Cited by

[1] Yifan Hong, Matteo Marinelli, Adam M. Kaufman, and Andrew Lucas, "Long-range-enhanced surface codes", arXiv:2309.11719, (2023).

[2] Noah Berthusen, Dhruv Devulapalli, Eddie Schoute, Andrew M. Childs, Michael J. Gullans, Alexey V. Gorshkov, and Daniel Gottesman, "Toward a 2D Local Implementation of Quantum LDPC Codes", arXiv:2404.17676, (2024).

[3] Craig Gidney, Michael Newman, Peter Brooks, and Cody Jones, "Yoked surface codes", arXiv:2312.04522, (2023).

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