Primitive Quantum Gates for an SU(3) Discrete Subgroup: $\Sigma(36\times3)$

TL;DR

This paper introduces the first primitive quantum gate set for simulating a nonabelian subgroup of SU(3), enabling more complex quantum simulations of group structures.

Contribution

It constructs the primitive gates for the 108-element  group, including implementations for both qubit and qutrit-based architectures, and develops a specialized compiler.

Findings

First nonabelian SU(3) subgroup constructed for quantum simulation

Primitive gates successfully implemented for different quantum architectures

A specialized compiler for unitary decomposition was developed

Abstract

We construct the primitive gate set for the digital quantum simulation of the 108-element $\Sigma(36\times3)$ group. This is the first time a nonabelian crystal-like subgroup of $SU(3)$ has been constructed for quantum simulation. The gauge link registers and necessary primitives -- the inversion gate, the group multiplication gate, the trace gate, and the $\Sigma(36\times3)$ Fourier transform -- are presented for both an eight-qubit encoding and a heterogeneous three-qutrit plus two-qubit register. For the latter, a specialized compiler was developed for decomposing arbitrary unitaries onto this architecture.