Shorter quantum circuits via single-qubit gate approximation

Vadym Kliuchnikov1,2, Kristin Lauter3, Romy Minko4,5, Adam Paetznick1, and Christophe Petit6,7

1Microsoft Quantum, Redmond, WA, US
2Microsoft Quantum, Toronto, ON, CA
3Facebook AI Research, Seattle, WA, US
4University of Oxford, Oxford, UK
5Heilbronn Institute for Mathematical Research, University of Bristol, Bristol, UK
6University of Birmingham, Birmingham, UK
7Université Libre de Bruxelles, Brussels, Belgium

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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+$\sqrt{\mathrm{T}}$ gate set 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 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+$\sqrt{\mathrm{T}}$). We also provide detailed numerical results for Clifford+T and Clifford+$\sqrt{\mathrm{T}}$ 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.

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