Classical and Quantum Computing of Shear Viscosity for $2+1D$ SU(2) Gauge Theory

We perform a nonperturbative calculation of the shear viscosity for $(2+1)$-dimensional SU(2) gauge theory by using the lattice Hamiltonian formulation. The retarded Green's function of the stress-energy tensor is calculated from real time evolution via exact diagonalization of the lattice Hamiltonian with a local Hilbert space truncation, and the shear viscosity is obtained via the Kubo formula. When taking the continuum limit, we account for the renormalization group flow of the coupling but no additional operator renormalization. We find the ratio of the shear viscosity and the entropy density $\frac{\eta}{s}$ is consistent with a well-known holographic result $\frac{1}{4\pi}$ at several temperatures on a $4\times4$ honeycomb lattice with the local electric representation truncated at $j_{\rm max}=\frac{1}{2}$. We also find the ratio of the spectral function and frequency $\frac{\rho^{xy}(\omega)}{\omega}$ exhibits a peak structure when the frequency is small. Both the exact diagonalization method and simple matrix product state classical simulation method beyond $j_{\rm max}=\frac{1}{2}$ on bigger lattices require exponentially growing resources. So we develop a quantum computing method to calculate the retarded Green's function and analyze various systematics of the calculation including $j_{\rm max}$ truncation and finite size effects, Trotter errors and the thermal state preparation efficiency. Our thermal state preparation method still requires resources that grow exponentially with the lattice size, but with a very small prefactor at high temperature. We test our quantum circuit on both the Quantinuum emulator and the IBM simulator for a small lattice and obtain results consistent with the classical computing ones.