Shorter quantum circuits via single-qubit gate approximation

We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set by reducing the problem to a novel magnitude approximation problem, achieving an immediate improvement in sequence length by a factor of 7/9. Extending the work of [Has17; Cam17], we show that taking probabilistic mixtures of channels to solve fallback [BRS15b] and magnitude approximation problems saves factor of two in approximation costs. In root particular, over the Clifford+ T gate set we achieve an average non-Clifford gate count of 0.23 log2(1/epsilon) + 2.13 and T-count 0.56 log2(1/epsilon) + 5.3 with mixed fallback approximations for diamond norm accuracy epsilon.This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approxi-mation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T root and Clifford+root T). We also provide detailed numerical results for Clifford+T and Clifford+ T gate sets. In an effort to keep the paper self-contained, we in-clude an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices.