Primitive Quantum Gates for an $SU(3)$ Discrete Subgroup: $Σ(216\times3)$

Abstract

We construct a primitive gate set for the digital quantum simulation of the $648$-element discrete subgroup of $SU(3)$, $\dsb$. The five algorithmic primitives are the inversion, group multiplication, trace, group Fourier transform, and kinetic-phase gates, for which we provide qubit and qutrit--qubit decompositions. For a fiducial shear-viscosity calculation on a $10^3$ spatial lattice with $50$ time steps and a total synthesis-error budget of $10^{-8}$, the estimated fault-tolerant costs are $2.3$--$6.8\times10^{10}$ $T$ gates in the qubit architecture and $1.3$--$5.9\times10^{12}$ $R$ gates in the qutrit--qubit architecture.