Perturbation theory, irrep truncations, and state preparation methods for quantum simulations of SU(3) lattice gauge theory

TL;DR

The paper addresses efficient quantum-state preparation for $SU(3)$ lattice gauge theories by combining a refined reduced electric basis with a locally bounded site-singlet truncation ($B$) and symmetry-aware Clebsch–Gordan precomputation. Guided by strong-coupling perturbation theory, it develops VQE-based and adiabatic-state-preparation strategies, including hybrid approaches that switch between variational and adiabatic regimes to balance fidelity and circuit depth. Through classical simulations on small lattices, it benchmarks ground-state energies and fidelities across EM/EMEM and multi-Givens ansätze, demonstrating that PT-informed circuits can achieve percent-level fidelity near $g\sim1$ with shallow depths, and that hybrids substantially reduce resource costs. The work also releases exttt{ymcirc} and exttt{pyclebsch} to enable scalable circuit construction and CGC computation, enabling broader adoption and future extensions to larger lattices and more general truncations.

Abstract

We study methods for efficient preparation of approximate ground states of $SU(3)$ lattice gauge theory on quantum hardware. Working in a variant of the electric basis, we introduce a refinement of the irrep truncation based on the energy density of site singlets, which provides a finer gradation of simulation complexity. Using strong-coupling perturbation theory as a guide, we develop simple ansatz circuits for ground state preparation and test them via classical simulation on small lattices, including the $2\times 2$ plaquette lattice in $d=2$ and the cube in $d=3$. We contrast state fidelities and resource requirements of variational methods against adiabatic state preparation and introduce a method that hybridizes the two approaches. Finally, we report on the public release of \texttt{ymcirc} -- a package of tools for building $SU(3)$ circuits and processing measurements -- and \texttt{pyclebsch}, a package for efficiently computing $SU(N)$ Clebsch-Gordan coefficients.

Figures (11)

• Figure 1: A comparison of $\Box$ matrix element counts in the $B$ truncation scheme versus the $T_r$ truncation for: $d=3/2$ (Top), $d=2$ (Middle), $d=3$ (Bottom). Points marking a specific $T_r$ indicate that $B$ has unlocked all $r$-index tensor irreps. Intervals where $B$ counts are greater than $T_r$ counts are those where $B$ has unlocked some but not all $r+1$-index tensor irreps.
• Figure 2: The reduction in $\Box$ matrix element counts using symmetrized CGCs ("Sym") versus unsymmetrized CGCs ("Unsym"). Each curve shows the ratio $\frac{\text{Sym}}{\text{Unsym}}$ for different site-singlet energy cutoffs ($B$) in $d$ dimensions. Sym counts are never higher than Unsym counts, suggesting that symmetrized CGCs are more efficient for both matrix element calculations and quantum simulations.
• Figure 3: Comparison of strongly-coupled perturbation theory with exact diagonalization results. Note that as $B$ increases, the number of matrix elements for $\square$ and the number of $SU(3)$ irreps included increases as seen in Fig. \ref{['fig:unsym_counts']} and in Table \ref{['tab:3_2d_B_singlets']}. For example, in $d=3/2$, the 8 appears at $B=5.67$, the 6 appears at $B=6$ and the 15 appears at $B=9.67$.
• Figure 4: Energy contours for the EM ansatz states on the $2\times 2$ plaquette lattice with periodic boundary conditions and $B=4.0$ energy truncation. The black crosses denote the prediction of strong-coupling PT, while the colored dots denote the local minima. Solid and dashed contours denote 6% and 10% deviations from the minima, respectively.
• Figure 5: (a) Comparison of ground state energies obtained by VQE using the EM and EMEM ansätze to exact diagonalization for different values of energy truncation $B$. The solid lines depict the exact diagonalization energies, the triangles depict the energies obtained using the EM ansatz, and the stars depict the energies obtained using the EMEM ansatz. (b) Relative error in energies obtained by VQE with respect to exact diagonalization. The markers for EM and EMEM are the same as (a).