GNarsil: Splitting Stabilizers into Gauges (2404.18302v1)
Abstract: Quantum subsystem codes have been shown to improve error-correction performance, ease the implementation of logical operations on codes, and make stabilizer measurements easier by decomposing stabilizers into smaller-weight gauge operators. In this paper, we present two algorithms that produce new subsystem codes from a "seed" CSS code. They replace some stabilizers of a given CSS code with smaller-weight gauge operators that split the remaining stabilizers, while being compatible with the logical Pauli operators of the code. The algorithms recover the well-known Bacon-Shor code computationally as well as produce a new $\left[\left[ 9,1,2,2 \right]\right]$ rotated surface subsystem code with weight-$3$ gauges and weight-$4$ stabilizers. We illustrate using a $\left[\left[ 100,25,3 \right]\right]$ subsystem hypergraph product (SHP) code that the algorithms can produce more efficient gauge operators than the closed-form expressions of the SHP construction. However, we observe that the stabilizers of the lifted product quantum LDPC codes are more challenging to split into small-weight gauge operators. Hence, we introduce the subsystem lifted product (SLP) code construction and develop a new $\left[\left[ 775, 124, 20 \right]\right]$ code from Tanner's classical quasi-cyclic LDPC code. The code has high-weight stabilizers but all gauge operators that split stabilizers have weight $5$, except one. In contrast, the LP stabilizer code from Tanner's code has parameters $\left[\left[ 1054, 124, 20 \right]\right]$. This serves as a novel example of new subsystem codes that outperform stabilizer versions of them. Finally, based on our experiments, we share some general insights about non-locality's effects on the performance of splitting stabilizers into small-weight gauges.
- D. Kribs, R. Laflamme, and D. Poulin, “Unified and generalized approach to quantum error correction,” Physical review letters, vol. 94, no. 18, p. 180501, 2005. [Online]. Available: https://arxiv.org/abs/quant-ph/0412076
- D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories,” Physical Review A, vol. 73, no. 1, p. 012340, 2006. [Online]. Available: https://arxiv.org/abs/quant-ph/0506023
- N. P. Breuckmann, “Quantum subsystem codes: Their theory and use,” 2011.
- J. Huang, S. M. Li, L. Yeh, A. Kissinger, M. Mosca, and M. Vasmer, “Graphical CSS code transformation using ZX calculus,” arXiv preprint arXiv:2307.02437, 2023. [Online]. Available: https://arxiv.org/abs/2307.02437
- O. Higgott and N. P. Breuckmann, “Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead,” Physical Review X, vol. 11, no. 3, p. 031039, 2021. [Online]. Available: https://arxiv.org/abs/2010.09626
- M. Li and T. J. Yoder, “A numerical study of Bravyi-Bacon-Shor and subsystem hypergraph product codes,” in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE). IEEE, 2020, pp. 109–119. [Online]. Available: https://arxiv.org/abs/2002.06257
- M. M. Wilde, “Logical operators of quantum codes,” Physical Review A, vol. 79, no. 6, p. 062322, 2009. [Online]. Available: https://arxiv.org/abs/0903.5256
- G. M. Crosswhite and D. Bacon, “Automated searching for quantum subsystem codes,” Physical Review A, vol. 83, no. 2, p. 022307, 2011. [Online]. Available: https://arxiv.org/abs/1009.2203
- S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, “Subsystem surface codes with three-qubit check operators,” Quant. Inf. Comp., vol. 13, pp. 0963–0985, 2013. [Online]. Available: https://arxiv.org/abs/1207.1443
- N. Rengaswamy, R. Calderbank, S. Kadhe, and H. D. Pfister, “Logical Clifford synthesis for stabilizer codes,” IEEE Transactions on Quantum Engineering, vol. 1, pp. 1–17, 2020. [Online]. Available: https://arxiv.org/abs/1907.00310
- S. Aaronson and D. Gottesman, “Improved simulation of stabilizer circuits,” Physical Review A, vol. 70, no. 5, p. 052328, 2004. [Online]. Available: https://arxiv.org/abs/quant-ph/0406196
- J. Dehaene and B. De Moor, “Clifford group, stabilizer states, and linear and quadratic operations over GF(2),” Physical Review A, vol. 68, no. 4, p. 042318, 2003. [Online]. Available: https://arxiv.org/abs/quant-ph/0304125
- W. E. Ryan et al., “An introduction to LDPC codes,” CRC Handbook for Coding and Signal Processing for Recording Systems, vol. 5, no. 2, pp. 1–23, 2004.
- P. Panteleev and G. Kalachev, “Quantum LDPC codes with almost linear minimum distance,” IEEE Transactions on Information Theory, vol. 68, no. 1, pp. 213–229, 2021. [Online]. Available: https://arxiv.org/abs/2012.04068
- N. Raveendran, N. Rengaswamy, F. Rozpędek, A. Raina, L. Jiang, and B. Vasić, “Finite rate QLDPC-GKP coding scheme that surpasses the CSS hamming bound,” Quantum, vol. 6, p. 767, 2022. [Online]. Available: https://arxiv.org/abs/2111.07029
- R. Smarandache, A. Gómez-Fonseca, and D. G. Mitchell, “Using minors to construct generator matrices for quasi-cyclic ldpc codes,” in 2022 IEEE International Symposium on Information Theory (ISIT). IEEE, 2022, pp. 548–553.
- H. Chimal-Dzul, J. Lieb, and J. Rosenthal, “Generator matrices of quasi-cyclic codes over extension fields obtained from Gröbner basis,” IFAC-PapersOnLine, vol. 55, no. 30, pp. 61–66, 2022.
- M. B. Hastings and J. Haah, “Dynamically generated logical qubits,” Quantum, vol. 5, p. 564, 2021. [Online]. Available: https://arxiv.org/abs/2107.02194
- M. Davydova, N. Tantivasadakarn, and S. Balasubramanian, “Floquet codes without parent subsystem codes,” PRX Quantum, vol. 4, no. 2, p. 020341, 2023. [Online]. Available: https://arxiv.org/abs/2210.02468