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

TL;DR

This paper develops a set of quantum gates tailored for simulating a specific discrete subgroup of SU(3), enabling efficient quantum computations relevant to physical properties like shear viscosity.

Contribution

It introduces a primitive gate set for the $ ext{SU}(3)$ subgroup $ ext{Σ}(72×3)$ with explicit qubit decompositions, advancing quantum simulation capabilities.

Findings

Primitive gate set constructed for $ ext{Σ}(72×3)$

Fault-tolerant T gate cost estimated at about 10^12 gates

Efficient quantum simulation of physical properties like shear viscosity

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.