Interacting many-electron problems pose some of the greatest computational challenges in science, with essential applications across many fields. The solutions to these problems will offer accurate predictions of chemical reactivity and kinetics, and other properties of quantum systems. Fermionic quantum Monte Carlo (QMC) methods which use a statistical sampling of the ground state, are among the most powerful approaches to these problems. Controlling the fermionic sign problem with constraints ensures the efficiency of QMC at the expense of potentially significant biases owing to the limited flexibility of classical computation. Here we propose an approach that combines constrained QMC with quantum computation to reduce such biases. We implement our scheme experimentally using up to 16 qubits to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals. These experiments represent the largest chemistry simulations performed with the help of quantum computers, while achieving accuracy that is competitive with state-of-the-art classical methods without burdensome error mitigation. Compared with the popular variational quantum eigensolver our hybrid quantum-classical computational model offers an alternative path towards achieving a practical quantum advantage for the electronic structure problem without demanding exceedingly accurate preparation and measurement of the ground-state wavefunction.