Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound
1Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA
2Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637, USA
3Department of Electrical Engineering and Computer Sciences, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India
Published: | 2022-07-20, volume 6, page 767 |
Eprint: | arXiv:2111.07029v2 |
Doi: | https://doi.org/10.22331/q-2022-07-20-767 |
Citation: | Quantum 6, 767 (2022). |
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Abstract
Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. In particular, several researchers have considered the individual code qubits being encoded with the continuous variable GottesmanKitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder – a hardware-friendly min-sum algorithm (MSA) – utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis.

Featured image: GKP code concatenated with LP-QLDPC codes obtained from lifting regular quasi-cyclic LDPC codes of increasing code length and minimum distance. The dashed curves correspond to the sequential MSA decoder without using the analog information (from the inner GKP error correction), whereas the solid curves make use of the analog information for decoding the outer code. The transition of the curves with increasing σ signifies an error threshold, which is improved when decoding utilizes the analog information. Furthermore, the improved threshold for the code family surpasses the CSS Hamming bound highlighted as gray vertical lines at $\sigma$ = 0.524 demonstrating the advantage of GKP-QLDPC concatenation scheme.
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