A Note on Optimality of Quantum Circuits over Metaplectic Basis

TL;DR

This paper improves the constant factor in a quantum circuit approximation algorithm over the metaplectic basis from 8 to 5, under certain conjectures, and discusses efficient gate approximations and integer geometry methods for circuit synthesis.

Contribution

It reduces the approximation constant in metaplectic quantum circuit synthesis from 8 to 5 for certain reflections, enhancing potential optimality of quantum circuits.

Findings

Improved approximation constant from 8 to 5 for non-exceptional reflections.

Discussed efficient approximation methods for ternary quantum gates.

Demonstrated use of Lenstra's integer geometry algorithm for circuit synthesis.

Abstract

Metaplectic quantum basis is a universal multi-qutrit quantum basis, formed by the ternary Clifford group and the axial reflection gate $R=|0\rangle \langle 0| + |1\rangle \langle 1| - |2\rangle \langle 2|$. It is arguably, a ternary basis with the simplest geometry. Recently Cui, Kliuchnikov, Wang and the Author have proposed a compilation algorithm to approximate any two-level Householder reflection to precision $\varepsilon$ by a metaplectic circuit of $R$-count at most $C \, \log_3(1/\varepsilon) + O(\log \log 1/\varepsilon)$ with $C=8$. A new result in this note takes the constant down to $C=5$ for non-exceptional target reflections under a certain credible number-theoretical conjecture. The new method increases the chances of obtaining a truly optimal circuit but may not guarantee the true optimality. Efficient approximations of an important ternary quantum gate proposed by Howard, Campbell and others is also discussed. Apart from this, the note is mostly didactical: we demonstrate how to leverage Lenstra's integer geometry algorithm from 1983 for circuit synthesis.