Zhiyang He, Quynh T. Nguyen, Christopher A. Pattison (Aug 12 2025).

Abstract: Proving threshold theorems for fault-tolerant quantum computation is a burdensome endeavor with many moving parts that come together in relatively formulaic but lengthy ways. It is difficult and rare to combine elements from multiple papers into a single formal threshold proof, due to the use of different measures of fault-tolerance. In this work, we introduce composable fault-tolerance, a framework that decouples the probabilistic analysis of the noise distribution from the combinatorial analysis of circuit correctness, and enables threshold proofs to compose independently analyzed gadgets easily and rigorously. Within this framework, we provide a library of standard and commonly used gadgets such as memory and logic implemented by constant-depth circuits for quantum low-density parity check codes and distillation. As sample applications, we explicitly write down a threshold proof for computation with surface code and re-derive the constant space-overhead fault-tolerant scheme of Gottesman using gadgets from this library. We expect that future fault-tolerance proofs may focus on the analysis of novel techniques while leaving the standard components to the composable fault-tolerance framework, with the formal proof following the intuitive ``napkin math'' exactly.

Arxiv: https://arxiv.org/abs/2508.08246