Optimality of the Howard-Vala T-gate in stabilizer quantum computation

Abstract

In a remarkable work [Phys. Rev. A 86, 022316 (2012)], Howard and Vala introduced a qudit version of the qubit T -gate (i.e., π/8-gate) for any prime dimensional system. This non-Clifford gate is a key ingredient of the paradigm “Clifford+T ”, which are widely employed in the stabilizer formalism of universal and fault-tolerant quantum computation. Considering the applications and significance of the T -gate, it is desirable to characterize it from various angles. Here we prove that in any prime dimensional system, the Howard-Vala T -gate is optimal, among all diagonal gates, for generating magic resources from stabilizer states when the magic is quantified via the L1-norm of characteristic functions (Fourier transforms) of quantum states. The quadratic Gaussian sum in number theory plays a key role in establishing this optimality. This highlights an extreme feature of the Howard-Vala T -gate. We further reveal an intrinsic relation between the Howard-Vala T -gate and the Watson-Campbell-Anwar-Browne T -gate [Phys. Rev. A 92, 022312 (2015)] for any prime dimensional system.dimensional system.