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

TL;DR

This work develops a primitive gate set for the 216-element discrete subgroup $|\\Sigma(72\\times3)|$ of $SU(3)$ to enable digital quantum simulation of nonperturbative gauge dynamics on quantum hardware. It constructs qudit/qubit decompositions for a full gate palette consisting of inversion, multiplication, trace, and Fourier-transform gates, with a fast quantum FFT along the subgroup lattice guiding the basis transformation between group-element and irrep representations. The authors provide detailed circuit decompositions, analyze fault-tolerant resource costs, and show that, for a fixed lattice, the T-gate budget to simulate shear viscosity is on the order of $10^{12}$–$10^{13}$ gates when using FFT-based transforms, a substantial improvement over prior estimates. They also project costs for larger crystal-like subgroups, discuss comparisons with other discretization approaches, and outline future directions including fermion incorporation and broader subgroup exploration. Overall, the work demonstrates a scalable path to higher-accuracy quantum simulations of $SU(3)$ gauge theories using discrete subgroups and structured primitive gates.

Abstract

We construct a primitive gate set for the digital quantum simulation of a discrete subgroup of $SU(3)$: the 216-element $Σ(72\times3)$. 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.

Figures (17)

• Figure 1: Euclidean calculations of the expectation value of the plaquette $\langle E_0\rangle$ as a function of Wilson coupling $\beta$ on $8^d$ lattices for (top) $(2+1)$-dimensions (bottom) $(3+1)$-dimensions. The shaded region indicates the scaling regime $\beta\geq \beta_s$.
• Figure 2: Lattice of subgroups of $SU(3)$. Each path represents a possible decomposition of $\mathfrak{U}_{FFT}^{G}$. The double-line path indicates the one explored in this work.
• Figure 3: Schematic of the $\mathfrak{U}_{FFT}^{G}$ provided in Ref. Pueschel:1998zzo.
• Figure 4: Diagrammatic formulation of $\mathfrak{U}_{-1}^{\Sigma(72\times3)}$ with $\mathfrak{U}_{-1}^{\mathop{\mathrm{\Sigma(36\times3)}}\nolimits}$ from Gustafson:2024kym and the remaining gates defined in Fig. \ref{['fig:inversedecomp']} and \ref{['fig:qubitdsainverse']}.
• Figure 5: Decomposition for (top) $\mathfrak{U}_{-1,\times}(C^2V^2)$ and (bottom) $\mathfrak{U}_{-1,\times}(X)$ (right) into qubit-qutrit gates which appear in the circuit decomposition of Fig. \ref{['fig:diagramaticinverse']}.