Fig. 6: Comparison of accuracy of the approximation vs. the full Gibbs state.
Results of measuring the imaginary part of the correlation function \({C}_{2,1}^{zz}(t)\) against Jt, where t is time, for the ρ4 Gibbs state are presented. The Hamiltonian employed is the XY model at the inverse temperature β = 0.8. Results were obtained using ibm_algiers, with the qubits' initial layout of [2,1,4,7] for the direct measurements, along with setting t = 5, the number of time steps = 51, dt ≈ 0.1, and the number of shots = 4000. In panel (a), the analytical result of each order of the approximation as illustrated in Eq. (18) is presented, as well as the analytical result of simulating the full Gibbs state (gray). In panel (b), the corresponding quantum hardware results are presented. The implementation of the full Gibbs state simulation on hardware is shown in orange. In both panels, the first order is in blue, the second order is in green, and the third order is in pink. We observe that all orders of the approximation we have developed with our sampling algorithm perform significantly better than simulating the full Gibbs state on hardware. It should be noted that the analytical result of the second order of the approximation in (a) demonstrates an overshoot, especially compared to the third order. We can see there is a general amplitude damping caused by the noisy hardware with the hardware results in (b). Hence, it would seem that the second-order result on hardware is more accurate than the third, but it is just a coincidence for this correlation function. In general, the third-order results on hardware and analytically are more accurate.
