A note on optimality of quantum circuits over metaplectic basis

Metaplectic quantum basis is a universal multi-qutrit quantum basis, formed by the ternary Clifford group and the axial reflection gate R = |0ih0| + |1ih1| − |2ih2|. 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 twolevel Householder reflection to precision ε by a metaplectic circuit of R-count at most C log3 (1/ε) + O(log log 1/ε) 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 numbertheoretical 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.