Genetic algorithm enhanced Solovay-Kitaev algorithm for quantum compiling of Fibonacci anyons

Quantum compiling, which aims to approximate target qubit gates by finding optimal sequences (braidwords) of basic braid operations, constitutes a fundamental challenge in quantum computing. We develop a genetic algorithm (GA)-enhanced Solovay-Kitaev algorithm (SKA) for approximating single-qubit gates using four elementary braiding matrices (EBMs) derived from Fibonacci anyons. The GA-enhanced SKA demonstrates robust performance, efficiently identifying optimal braidwords within exponentially large search spaces. Notably, the approximation precision achieved by our method surpasses that of Monte Carlo (MC)-enhanced SKA and becomes comparable to deep reinforcement learning (RL) approaches when braidword lengths exceed 25. Implementing 2- and 3-order approximations with the GA-enhanced SKA yields optimal braidword (initial braiding lengths l0=50 and 30 respectively) achieving gate distances of 5.9*10-7 - sufficient precision for most quantum computing applications. This work develops an optimized compilation framework for non-Abelian anyon gates, providing an essential methodology for enhancing future topological quantum computation architectures through gate optimization.