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

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.

References in corpus (8)

• The density-matrix renormalization group in the age of matrix product states
• Surface codes: Towards practical large-scale quantum computation
• Suppressing quantum errors by scaling a surface code logical qubit
• Random Quantum Circuits
• Efficient Algorithms for Maximum Likelihood Decoding in the Surface Code
• Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes
• Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation
• Finite-rate sparse quantum codes aplenty

Cited by in corpus (5)

• Tensor networks for quantum computing
• Tensor Network Decoding Beyond 2D
• Constant Overhead Entanglement Distillation via Scrambling
• Decoding general error correcting codes and the role of complementarity
• Approximate Quantum Error Correction with 1D Log-Depth Circuits