Hierarchical quantum decoders

TL;DR

This paper introduces a hierarchical quantum decoding method using the Lasserre SOS hierarchy, balancing speed and accuracy, and demonstrating near-optimal performance on quantum error correction codes.

Contribution

It proposes a novel hierarchical decoding framework based on SOS relaxations, providing tunable trade-offs and improved performance over standard heuristics.

Findings

Low levels of the hierarchy outperform LP relaxations.

Levels 2 and 3 nearly match exact decoding performance.

The method offers a flexible balance between speed and accuracy.

Abstract

Decoders are a critical component of fault-tolerant quantum computing. They must identify errors based on syndrome measurements to correct quantum states. While finding the optimal correction is NP-hard and thus extremely difficult, approximate decoders with faster runtime often rely on uncontrolled heuristics. In this work, we propose a family of hierarchical quantum decoders with a tunable trade-off between speed and accuracy while retaining guarantees of optimality. We use the Lasserre Sum-of-Squares (SOS) hierarchy from optimization theory to relax the decoding problem. This approach creates a sequence of Semidefinite Programs (SDPs). Lower levels of the hierarchy are faster but approximate, while higher levels are slower but more accurate. We demonstrate that even low levels of this hierarchy significantly outperform standard Linear Programming relaxations. Our results on rotated surface codes and honeycomb color codes show that the SOS decoder approaches the performance of exact decoding. We find that Levels 2 and 3 of our hierarchy perform nearly as well as the exact solver. We analyze the convergence using rank-loop criteria and compare the method against other relaxation schemes. This work bridges the gap between fast heuristics and rigorous optimal decoding.