Spin Hamiltonian for Quantum Simulation of Minimally Truncated $3+1$D SU(2) Lattice Gauge Theory and Testing Eigenstate Thermalization

We construct the spatial-gauge-singlet Hamiltonian of pure SU(2) lattice gauge theory in $3+1$ dimensions ($3+1$D) on qubits under the minimal truncation $j_{\rm max}=1/2$, building on previous work on the hyperhoneycomb lattice, and test the eigenstate thermalization hypothesis (ETH). By enforcing Gauss's law and exploiting the redundancy among gauge-invariant loop operators, we find the minimal set of independent loop variables that can be bijectively mapped onto qubits, and write the Hamiltonian as a sum of Pauli strings. An $n\times n\times n$ lattice with fixed boundary conditions requires $2(n-1)^3+3(n-1)^2$ qubits. We exactly diagonalize the Hamiltonian on a $2\times 3\times 3$ lattice and identify the coupling range of quantum ergodicity from the level statistics. At the ergodic coupling $g=0.3$, we calculate the matrix elements of local energy density operators and the entanglement entropies of all eigenstates and find them consistent with the ETH. These results show that the $3+1$D truncated SU(2) lattice gauge theory thermalizes in accordance with the ETH and provide a benchmark for future real-time quantum simulations of non-equilibrium dynamics.