Sub-Cubic Quantum Gate Synthesis via Stochastic Commutator Decomposition

TL;DR

This paper introduces Stochastic Commutator Synthesis, a quantum gate compilation method combining advanced decomposition techniques with stochastic choices to reduce gate counts and improve fidelity in quantum circuits.

Contribution

It integrates Kuperberg's sub-cubic decomposition with stochastic methods to enhance quantum gate synthesis, achieving significant T-count reductions and fidelity improvements.

Findings

Achieves 10-25% T-count reduction in quantum circuits.

Demonstrates up to 35% fidelity gains on trapped-ion hardware.

Provides a practical approach towards demonstrating complexity-theoretic separations.

Abstract

We present Stochastic Commutator Synthesis, a hybrid quantum gate compilation framework that integrates Kuperberg's sub-cubic Solovay-Kitaev exponent c near 1.44042 with the error-tailoring machinery of randomized compilation. Classical Solovay-Kitaev implementations produce known word lengths and accumulate coherent approximation errors that degrade fault-tolerant threshold estimates. Kuperberg's 2023-2025 result reduces this via doubly exponential convergence and higher-order commutator decompositions. SCS augments this geometric backbone with a Gibbs-sampled stochastic choice of commutator factors at each recursion level, converting coherent synthesis residuals into incoherent, Pauli-twirl-compatible noise -- a property exploited by RC. Combined with RL-guided pre-synthesis, SCS achieves consistent T-count reductions of 10-25 percent and demonstrates fidelity gains of up to 35 percent on multi-fold Forrelation circuits on trapped-ion hardware such as Sandia QSCOUT. We situate SCS within the complexity-theoretic landscape established by the Raz-Tal oracle separation, arguing that low-error, noise-robust compilation of Forrelation-type circuits constitutes a practical pathway toward demonstrating this separation on physical hardware.