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 works [28] and [15], we show that taking probabilistic mixtures of channels to solve fallback [13] and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+√T gate set we achieve an average non-Clifford gate count of 0.23log2(1/ε)+2.13 and T-count 0.56log2(1/ε)+5.3 with mixed fallback approximations for diamond norm accuracy ε.
This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+√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 include 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.