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 eta/s is consistent with a well-known holographic result 1/(4 pi) at several temperatures on a  4×4 hexagonal lattice with the local electric representation truncated at j_max=1/2. We also find the ratio of the spectral function and frequency rho^(xy)(omega)/omega exhibits a peak structure when the frequency is small. Both exact diagonalization method and simple matrix product state classical simulation method beyond j_max=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_max truncation and finite size effects and Trotter errors. 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.