Quantum many-body simulation of finite-temperature systems with sampling a series expansion of a quantum imaginary-time evolution

Simulating thermal-equilibrium properties at finite temperature is crucial for studying quantum many-body systems. Quantum computers are expected to enable us to simulate large systems at finite temperatures, overcoming challenges faced by classical computers, like the sign problem of the quantum Monte-Carlo technique. Conventional methods suitable for fault-tolerant quantum computing (FTQC) devices are designed for studying large-scale quantum many-body systems but require a large number of ancilla qubits and a deep quantum circuit with many basic gates, making them unsuitable for the early stage of the FTQC era, at which the availability of qubits and quantum gates is limited. In this paper, we propose a method suitable for quantum devices in this early stage to calculate the thermal-equilibrium expectation value of an observable at finite temperatures. Our proposal, named the Markov-chain Monte Carlo with sampled pairs of unitaries (MCMC-SPU) algorithm, involves sampling simple quantum circuits and generating the corresponding statistical ensembles. This approach addresses the issues of resource demand and the decay in probability associated with postselection of measurement outcomes on ancilla qubits. We validate our proposal with numerical simulation on the one-dimensional transverse-field Ising model as an illustrative example.