  Quantum computing and quantum Monte Carlo (QMC) are respectively the
state-of-the-art quantum and classical computing methods for understanding
many-body quantum systems. Here, we propose a hybrid quantum-classical
algorithm that integrates these two methods, inheriting their distinct features
in efficient representation and manipulation of quantum states and overcoming
their limitations. We first introduce non-stoquasticity indicators (NSIs) and
their upper bounds, which measure the sign problem, the most notable limitation
of QMC. We show that our algorithm could greatly mitigate the sign problem,
which decreases NSIs with the assistance of quantum computing. Meanwhile, the
use of quantum Monte Carlo also increases the expressivity of shallow quantum
circuits, allowing more accurate computation that is conventionally achievable
only with much deeper circuits. We numerically test and verify the method for
the N$_2$ molecule (12 qubits) and the Hubbard model (16 qubits). Our work
paves the way to solving practical problems with intermediate-scale and
early-fault tolerant quantum computers, with potential applications in
chemistry, condensed matter physics, materials, high energy physics, etc.
