Fault-tolerant noise guessing decoding of quantum random codes

TL;DR

This work extends fault-tolerant decoding for quantum random linear codes (QRLCs) by introducing a degeneracy-aware, maximum-likelihood decoder built on QGRAND principles. It accounts for preparation, measurement, and two-qubit gate errors during syndrome extraction and leverages reduced row echelon forms to map syndromes to deterministic coset leaders, with a parallelizable FE/FL framework. The analysis reveals a finite threshold $p_{ m threshold} \approx 2\times 10^{-5}$ in the asymptotic regime, arising from the high degeneracy of error patterns, and discusses the practical limitations of scalability under high-entropy noise. While not scalable to large codes or high-entropy regimes, the approach shows that QRLCs can be fault-tolerant under realistic noise when the entropy is low, opening avenues for versatile, high-rate QRLC implementations with appropriate decoding relaxations or heuristics.

Abstract

This work addresses the open question of implementing fault-tolerant QRLCs with feasible computational overhead. We present a new decoder for quantum random linear codes (QRLCs) capable of dealing with imperfect decoding operations. A first approach, introduced by Cruz et al., only considered channel errors, and perfect gates at the decoder. Here, we analyze the fault-tolerant characteristics of QRLCs with a new noise-guessing decoding technique, when considering preparation, measurement, and gate errors in the syndrome extraction procedure, while also accounting for error degeneracy. Our findings indicate a threshold error rate ($\pth$) of approximately $\pnum$ in the asymptotic limit, while considering realistic noise levels in the mentioned physical procedures.

Figures (12)

• Figure 1: Noise model considered.
• Figure 2: Simple example, with two syndrome extractions (SE). The error is not detected by the first syndrome extraction process, but it is detected by the second extraction.
• Figure 3: Relations between the different quantities of interest. For simplicity, every error represented is assumed to be a base error.
• Figure 4: Model considered to compute the asymptotic regime, where $q\rightarrow \infty$.
• Figure 5: Experimental $P_{\rm total}$, and resulting fit by \ref{['eq:fit']}. We observe the experimental data matching the qualitative description of the theoretical analysis, with $R^2>0.999$ for all cases.