Sub-Cubic Quantum Gate Synthesis via Stochastic Commutator Decomposition

• URL: http://arxiv.org/abs/2605.02051v1
• Date: Sun, 03 May 2026 20:41:21 GMT
• Title: Sub-Cubic Quantum Gate Synthesis via Stochastic Commutator Decomposition
• Abstract summary: Commutator Synthesis is a quantum gate compilation framework.<n>It integrates Kuperberg's sub-cubic Solovay-Kitaev exponent c near 1.44042 with the error-tailoring machinery of randomized compilation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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