Characterising semi-Clifford gates using algebraic sets

Motivated by their central role in fault-tolerant quantum computation, we study the sets of gates of the third-level of the Clifford hierarchy and their distinguished subsets of `nearly diagonal' semi-Clifford gates. The Clifford hierarchy gates can be implemented via gate teleportation given appropriate magic states. The vast quantity of these resource states required for achieving fault-tolerance is a significant bottleneck for the practical realisation of universal quantum computers. Semi-Clifford gates are important because they can be implemented with far more efficient use of these resource states. We prove that every third-level gate of up to two qudits is semi-Clifford. We thus generalise results of Zeng-Chen-Chuang (2008) in the qubit case and of the second author (2020) in the qutrit case to the case of qudits of arbitrary prime dimension $d$. Earlier results relied on exhaustive computations whereas our present work leverages tools of algebraic geometry. Specifically, we construct two schemes corresponding to the sets of third-level Clifford hierarchy gates and third-level semi-Clifford gates. We then show that the two algebraic sets resulting from reducing these schemes modulo $d$ share the same set of rational points.