  Quantum computing is a promising way to systematically solve the longstanding
computational problem, the ground state of a many-body fermion system. Many
efforts have been made to realise certain forms of quantum advantage in this
problem, for instance, the development of variational quantum algorithms. A
recent work by Huggins et al. reports a novel candidate, i.e. a
quantum-classical hybrid Monte Carlo algorithm with a reduced bias in
comparison to its fully-classical counterpart. In this paper, we propose a
family of scalable quantum-assisted Monte Carlo algorithms where the quantum
computer is used at its minimal cost and still can reduce the bias. By
incorporating a Bayesian inference approach, we can achieve this
quantum-facilitated bias reduction with a much smaller quantum-computing cost
than taking empirical mean in amplitude estimation. Besides, we show that the
hybrid Monte Carlo framework is a general way to suppress errors in the ground
state obtained from classical algorithms. Our work provides a Monte Carlo
toolkit for achieving quantum-enhanced calculation of fermion systems on
near-term quantum devices.
