Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound

Nithin Raveendran; Narayanan Rengaswamy; Filip Rozpędek; Ankur Raina; Liang Jiang; Bane Vasić

doi:10.22331/q-2022-07-20-767

Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound

Nithin Raveendran¹, Narayanan Rengaswamy¹, Filip Rozpędek², Ankur Raina³, Liang Jiang², and Bane Vasić¹

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.

Popular summary

We consider quantum error correction in a concatenated setting where the inner code is the continuous-variable square lattice Gottesman-Kitaev-Preskill (GKP) code and the outer code is chosen from a discrete-variable finite rate quantum low-density parity-check (QLDPC) code family. In this first work considering generic QLDPC codes as outer codes, we consider a simple noise model where the only source of noise is the Gaussian random displacement channel on each GKP data qubit, and all GKP ancillas used for syndrome measurements are infinitely squeezed (noiseless). The decoder for the outer code is the hardware-friendly min-sum algorithm (MSA), which is an approximation of the belief-propagation algorithm and is widely deployed in classical error correction applications. We show that the noise threshold for two lifted product QLDPC code families increase significantly when the MSA decoder for the outer QLDPC code appropriately leverages the GKP analog information from the inner GKP error correction. When the MSA decoder also uses a sequential node update schedule in addition to using the GKP analog information, our concatenated scheme surpasses the CSS Hamming bound (of C(p) = 1-2h(p), where p is related to the variance of the Gaussian random displacement) for these QLDPC code families. Usually BP based decoders for classical LDPC and QLDPC codes exhibit an error floor phenomenon where the logical error rate saturates at low channel error rates. But surprisingly, our simulations indicate that the GKP analog information seems to significantly lower the error floor or potentially eliminate it completely. These observations demonstrate the benefits of the finite rate QLDPC-GKP coding scheme. We also discuss several interesting questions that this work highlights, including the Hamming bound for degenerate codes, and the contribution of degeneracy of our considered QLDPC codes to surpassing the CSS Hamming bound.

► BibTeX data

@article{Raveendran2022finiterateqldpcgkp,
  doi = {10.22331/q-2022-07-20-767},
  url = {https://doi.org/10.22331/q-2022-07-20-767},
  title = {Finite {R}ate {QLDPC}-{GKP} {C}oding {S}cheme that {S}urpasses the {CSS} {H}amming {B}ound},
  author = {Raveendran, Nithin and Rengaswamy, Narayanan and Rozp{\k{e}}dek, Filip and Raina, Ankur and Jiang, Liang and Vasi{\'{c}}, Bane},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {767},
  month = jul,
  year = {2022}
}

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