Overflow-Safe Polylog-Time Parallel Minimum-Weight Perfect Matching Decoder: Toward Experimental Demonstration

TL;DR

This paper introduces a new overflow-safe, polylogarithmic-time quantum error decoder based on a determinant approach, optimized for practical finite-bit machine implementation.

Contribution

It presents an overflow detection method using algebra over a truncated polynomial ring and reduces the required bit length for intermediate values by over 99.9%.

Findings

Achieves polylogarithmic parallel runtime for MWPM decoding.

Reduces intermediate value bit length by more than 99.9%.

Enables implementation on 512-bit machines for quantum error correction.

Abstract

Fault-tolerant quantum computation (FTQC) requires fast and accurate decoding of quantum errors, which is often formulated as a minimum-weight perfect matching (MWPM) problem. A determinant-based approach has been proposed as a novel method to surpass the conventional polynomial runtime of MWPM decoding via the blossom algorithm, asymptotically achieving polylogarithmic parallel runtime. However, the existing approach requires an impractically large bit length to represent intermediate values during the computation of the matrix determinant; moreover, when implemented on a finite-bit machine, the algorithm cannot detect overflow, and therefore, the mathematical correctness of such algorithms cannot be guaranteed. In this work, we address these issues by presenting a polylog-time MWPM decoder that detects overflow in finite-bit representations by employing an algebraic framework over a truncated polynomial ring. Within this framework, all arithmetic operations are implemented using bitwise XOR and shift operations, enabling efficient and hardware-friendly implementation. Furthermore, with algorithmic optimizations tailored to the structure of the determinant-based approach, we reduce the arithmetic bit length required to represent intermediate values in the determinant computation by more than $99.9\%$, making it possible to implement it on machines supporting $512$-bit computing while preserving the decoder's polylog runtime scaling. These results open the possibility of a proof-of-principle demonstration of the polylog-time MWPM decoding in the early FTQC regime.