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We give a novel procedure for approximating general single-qubit unitariesfrom a finite universal gate set by reducing the problem to a novel magnitudeapproximation problem, achieving an immediate improvement in sequence length bya factor of 7/9. Extending the works arXiv:1612.01011 and arXiv:1612.02689, weshow that taking probabilistic mixtures of channels to solve fallback(arXiv:1409.3552) and magnitude approximation problems saves factor of two inapproximation costs. In particular, over the Clifford+$\sqrt{\mathrm{T}}$ gateset we achieve an average non-Clifford gate count of$0.23\log_2(1/\varepsilon)+2.13$ and T-count $0.56\log_2(1/\varepsilon)+5.3$with mixed fallback approximations for diamond norm accuracy $\varepsilon$.  This paper provides a holistic overview of gate approximation, in addition tothese new insights. We give an end-to-end procedure for gate approximation forgeneral gate sets related to some quaternion algebras, providing pedagogicalexamples using common fault-tolerant gate sets (V, Clifford+T andClifford+$\sqrt{\mathrm{T}}$). We also provide detailed numerical results forClifford+T and Clifford+$\sqrt{\mathrm{T}}$ gate sets. In an effort to keep thepaper self-contained, we include an overview of the relevant algorithms forinteger point enumeration and relative norm equation solving. We provide anumber of further applications of the magnitude approximation problems, as wellas improved algorithms for exact synthesis, in the Appendices.

Shorter quantum circuits via single-qubit gate approximation