Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder

• URL: http://arxiv.org/abs/2212.05071v1
• Date: Fri, 9 Dec 2022 19:00:00 GMT
• Title: Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder
• Authors: Andrew S. Darmawan, Yoshifumi Nakata, Shiro Tamiya, Hayata Yamasaki
• Abstract summary: We numerically demonstrate that the hashing bound, i.e., a rate known to be achieved with $d=mathcalO(n)$-depth random encoding circuits, can be attained even when the circuit depth is restricted to $d=mathcalO(log n)$ in 1D.
• Score: 2.867517731896504
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recent work [M. J. Gullans et al., Physical Review X, 11(3):031066 (2021)] has shown that quantum error correcting codes defined by random Clifford encoding circuits can achieve a non-zero encoding rate in correcting errors even if the random circuits on $n$ qubits, embedded in one spatial dimension (1D), have a logarithmic depth $d=\mathcal{O}(\log{n})$. However, this was demonstrated only for a simple erasure noise model. In this work, we discover that this desired property indeed holds for the conventional Pauli noise model. Specifically, we numerically demonstrate that the hashing bound, i.e., a rate known to be achieved with $d=\mathcal{O}(n)$-depth random encoding circuits, can be attained even when the circuit depth is restricted to $d=\mathcal{O}(\log n)$ in 1D for depolarizing noise of various strengths. This analysis is made possible with our development of a tensor-network maximum-likelihood decoding algorithm that works efficiently for $\log$-depth encoding circuits in 1D.

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