Neural Quantum States for Constrained Optimization

Solving constrained optimization problems using Variational Quantum Algorithms (VQAs) is a promising application of near-term quantum hardware. However, in order to achieve precise estimates of complex quantum observables, a large number of measurements is required. We propose a method that integrates neural network estimators with VQAs applied to the Binary Knapsack Problem. Trained on Pauli- Z measurements on a qubit and photon-number measurements on qumodes, the neural network estimator aims to provide precise estimates for a qubit-qumode Hamiltonian using a significantly smaller number of measurements than traditional post-processing techniques. The Binary Knapsack Problem is reformulated into a Quadratic Unconstrained Binary Optimization problem which is solved variationally by a hybrid qubit-qumode circuit that is enhanced by a neural network estimator. The result is an estimator that achieves precise estimates of complex quantum observables utilised in VQAs. This work explores the utility of integrating classical methods with quantum simulation algorithms and has relevance not just in solving constrained optimization problems, but also in enhancing Variational Monte Carlo simulations for quantum chemistry calculations and many-body problems among others.