Primitive Quantum Gates for an $SU(3)$ Discrete Subgroup: $\Sigma(72\times3)$

Abstract

We construct a primitive gate set for the digital quantum simulation of a discrete subgroup of $SU(3)$: the 216-element $\Sigma(72\times3)$. The necessary primitives are the inversion gate, the group multiplication gate, the trace gate, and the group Fourier transform, for which we provide qubit decompositions. The resulting fault-tolerant T gate costs for a fiducial calculation of shear viscosity would require about $10^{12}$ T gates which compares favorably to other modern estimates.