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

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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.

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.

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Cited by

[1] Nithin Raveendran, Narayanan Rengaswamy, Asit Kumar Pradhan, and Bane Vasic, 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 275 (2022) ISBN:978-1-6654-9113-6.

[2] Yijia Xu, Yixu Wang, En-Jui Kuo, and Victor V. Albert, "Qubit-oscillator concatenated codes: decoding formalism & code comparison", arXiv:2209.04573.

[3] Nithin Raveendran, Narayanan Rengaswamy, Asit Kumar Pradhan, and Bane Vasić, "Soft Syndrome Decoding of Quantum LDPC Codes for Joint Correction of Data and Syndrome Errors", arXiv:2205.02341.

[4] Narayanan Rengaswamy, Nithin Raveendran, Ankur Raina, and Bane Vasić, "Entanglement Purification with Quantum LDPC Codes and Iterative Decoding", arXiv:2210.14143.

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