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

TL;DR

This work presents a polylog-time MWPM decoder that detects overflow in finite-bit representations by employing an algebraic framework over a truncated polynomial ring, and reduces the arithmetic bit length required to represent intermediate values in the determinant computation by more than $99.9\% while preserving its polylogarithmic runtime scaling.

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 promising 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 $99.9\%$, while preserving its polylogarithmic runtime scaling. These results open the possibility of a proof-of-principle demonstration of the polylog-time MPWM decoding in the early FTQC regime.

Figures (4)

• Figure 1: Conceptual comparison between Takada-Yamasaki takada2025doubly and the proposed bitwise algorithm.
• Figure 2: MWPM error rate as a function of the code distance for different discretization precisions of integerized edge weights. Each curve corresponds to a different number that represents a binary digit of the minimum discretized edge weight of a detector graph. The error rate indicates the fraction of trials where the MWPM solution obtained from the integer-weight graph differs from that obtained using the floating-point weights. Error bars indicate the standard error computed over the $10^7$ trials.
• Figure 3: Required $w_{\mathrm{th}}$ to obtain the correct MWPM for each number $\{2,4,\dots,28\}$ with a code distance of 5, and a high binary precision of 8 bits and a low binary precision of 4 bits. The three curves represent the lower bounds of $w_\mathrm{th}$ obtained from MWPM weights using the respective edge-weight scalings, where the plotted values correspond to the maximum $w_\mathrm{th}$ observed over $10^7$ trials
• Figure 4: Failure probability as a function of the threshold bit length $w_{\mathrm{th}}$ for a code distance of 5, a high binary precision of 8 bits, and a low binary precision of 4 bits. The failure probability represents the fraction of trials, among those with path graph size at most 28, in which the decoding process fails, or the decoded MWPM of the perturbed path graph does not coincide with any of the correct MWPMs of the original path graph. Error bars indicate the standard error computed over $10^6$ trials. (a) Isolation-based bit-reduction method using high-binary-precision MWPM candidate generation. (b) Extended method using low-binary-precision MWPM candidate generation with high-binary-precision verification. For each panel, the dashed horizontal line indicates the logical error rate. The blue curve denotes the baseline number of perturbation samples required by the conventional method, while orange, green, and red denote 2×, 4×, and 8× that number, respectively. Our method sufficiently suppresses the decoding failure probability with arithmetic bit lengths of $3\times 10^2$–$5\times 10^2$, as indicated by the green curve.

Theorems & Definitions (8)

• Theorem 1: Bit-Level Computability
• proof
• Theorem 2: Parallelizability
• proof
• Theorem 3: Correctness
• proof
• Theorem 4: Failure Detection
• proof