Efficient Sampling Scheme with Trotter Gates for Evaluating Thermal Expectation Values on Quantum Computers*

Thermal expectation values of quantum many-body systems can be evaluated from expectation values obtained by sampled pure states. The efficiency of this sampling approach strongly depends on how one samples pure states. When a pure state is sampled Haar randomly, sample dependence decreases with the system size, and expectation values from a single sampled pure state represent thermal expectation values in sufficiently large systems. However, the preparation of Haar random states is difficult even on quantum computers because of exponentially many random numbers. In this study, we propose an alternative to a Haar random state: Pure states generated by applying Trotter gates to random product states. The number of applications is restricted to be proportional to the system size for the feasibility of the sampling on quantum computers. From the numerical investigation on sampling efficiency, we observe that the efficiency increases with system size like Haar random states when Trotter gates are made from a nonintegrable Hamiltonian. We also find some cases where the sampling efficiency of the proposed scheme is almost equal to that of the Haar random sampling.