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

TL;DR

This paper demonstrates that low-depth (logarithmic in size) random Clifford circuits can effectively correct Pauli noise in quantum error correction, using a tensor-network decoder, achieving near-optimal rates previously thought possible only with deeper circuits.

Contribution

The authors show that a tensor-network maximum-likelihood decoder enables effective quantum error correction with logarithmic-depth circuits under Pauli noise, extending prior results from erasure to more realistic noise models.

Findings

Logarithmic-depth circuits achieve the hashing bound under Pauli noise.

Tensor-network decoder efficiently handles log-depth encoding circuits.

Error correction performance approaches that of deeper circuits.

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.