[Paper Review] General tensor network decoding of 2D Pauli codes

Christopher T. Chubb|arXiv (Cornell University)|Jan 11, 2021

Quantum Computing Algorithms and Architecture72 references26 citations

TL;DR

Introduces a general tensor network decoder for 2D Pauli codes that approximates maximum-likelihood decoding with runtime O(n log n + n χ^3) and demonstrates near-optimal thresholds across several code families under multiple Pauli noise models; provides SweepContractor.jl implementation.

ABSTRACT

In this work we develop a general tensor network decoder for 2D codes. Specifically, we propose a decoder that approximates maximally likelihood decoding for 2D stabiliser and subsystem codes subject to Pauli noise. For a code consisting of $n$ qubits our decoder has a runtime of $O(n\log n+nχ^3)$, where $χ$ is an approximation parameter. We numerically demonstrate the power of this decoder by studying four classes of codes under three noise models, namely regular surface codes, irregular surface codes, subsystem surface codes and colour codes, under bit-flip, phase-flip and depolarising noise. We show that the thresholds yielded by our decoder are state-of-the-art, and numerically consistent with optimal thresholds where available, suggesting that the tensor network decoder well approximates optimal decoding in all these cases. Novel to our decoder is an efficient and effective approximate contraction scheme for arbitrary 2D tensor networks, which may be of independent interest. We have also released an implementation of this algorithm as a stand-alone Julia package: SweepContractor.jl.

Motivation & Objective

• Motivate robust quantum error correction by mitigating noise in 2D quantum codes.
• Develop a general-purpose, approximately optimal decoder for 2D stabiliser and subsystem codes under Pauli noise.
• Achieve scalable decoding with tunable speed-accuracy via tensor network contraction and MPS compression.
• Demonstrate decoder performance on regular/irregular surface codes, subsystem surface codes, and colour codes under various noise models.

Proposed method

• Construct a general tensor network representation of maximum likelihood decoding for arbitrary 2D Pauli codes under Pauli noise.
• Introduce a 2D tensor network contraction algorithm based on a sweep-line approach that maintains an MPS approximation of the contracted network.
• Use a two-stage Contract and Compress procedure to perform approximate contractions with a controllable bond dimension χ and a cap χ′.
• Provide a precise complexity analysis: Time = O(n log n + n χ^3); Space = O(n + sqrt(n) χ^2).
• Apply the decoder to four code families (regular surface codes, irregular surface codes, subsystem surface codes, colour codes) under bit-flip, phase-flip, and depolarising noise; release a Julia package SweepContractor.jl for implementation.

Experimental results

Research questions

• RQ1 How close can a tensor network decoder get to maximum likelihood decoding for 2D Pauli codes under Pauli noise?
• RQ2 What are the runtime and memory costs of an approximate 2D TN contraction decoder on irregular and regular 2D codes?
• RQ3 Do TN-based decoders achieve thresholds comparable to or saturating known upper bounds for various 2D code families and noise models?
• RQ4 Can the contraction scheme be extended to arbitrary 2D tensor networks beyond regular lattices, and how does performance vary across code families?

Key findings

• The TN decoder attains thresholds that are state-of-the-art and consistent with optimal/upper-bound thresholds where available.
• For tested codes under bit-flip, phase-flip, and depolarising noise, the TN thresholds saturate or closely approach known hashing-bound upper limits.
• The swept-line contraction with MPS compression enables approximate contraction on arbitrary 2D TNs with time O(n log n + n χ^3) and space O(n + sqrt(n) χ^2).
• The decoder performs well across regular surface codes, irregular surface codes, subsystem surface codes, and colour codes, often matching or approaching optimal decoding.
• An open-source Julia package SweepContractor.jl is released to enable practical use of the algorithm.