Continuous-Space Quantum Simulation: A Discretization-Free Approach with Hybrid Quantum-Classical Ansatze

Most quantum many-body systems including those of electronic structure and materials are natively described in first quantization. However, simulating continuous-space systems on quantum computers is challenging as the systems have to be discretized and mapped onto qubits while respecting the underlying exchange statistics. The discretization usually amounts to either discretizing space itself or choosing a suitable finite basis set which introduces errors and scales poorly with the number of particles. In this talk, I propose an alternative approach of harnessing quantum resources within Variational Monte Carlo simulations for continuous space problems which does not require any form of discretization. To that end, the wavefunction ansatz is partly represented by a parameterized quantum circuit and optimized similarly to the Neural Quantum States framework. I apply our hybrid quantum-classical algorithm to the paradigmatic 1d quantum rotor model and show how the accuracy of the ground state energy can be controlled by the circuit depth. The results are compared to typical classical ansatze such as Jastrow wavefunctions and MPS calculations on a discretized approximation of the model. In terms of the latter, I demonstrate a reduction in the number of qubits when simulating the system using our continuous space formalism. Finally, I show how this framework can be applied to fermionic systems in first quantization.