Cryptology ePrint Archive: Report 2022/372

Shorter quantum circuits

Vadym Kliuchnikov and Kristin Lauter and Romy Minko and Christophe Petit and Adam Paetznick

Abstract: 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 arXiv:1612.01011 and arXiv:1612.02689, we show that taking probabilistic mixtures of channels to solve fallback (arXiv:1409.3552) and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+$\sqrt{T}$ gate set we achieve an average non-Clifford gate count of 0.23log2(1/$\varepsilon$)+2.13 and T-count 0.56log2(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+$\sqrt{T}$). We also provide detailed numerical results for Clifford+T and Clifford+$\sqrt{T}$ 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.

Category / Keywords: applications / quantum information, applications, path finding

Date: received 21 Mar 2022, last revised 24 Mar 2022

Contact author: romy minko at bristol ac uk

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Version: 20220324:074220 (All versions of this report)

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