Variational quantum eigensolvers by variance minimization

The original variational quantum eigensolver(VQE)typically minimizes energy with hybrid quantum-classical optimization that aims to find the ground state.Here,we propose a VQE based on minimizing energy variance and call it the variance-VQE,which treats the ground state and excited states on the same footing,since an arbitrary eigenstate for a Hamiltonian should have zero energy variance.We demonstrate the properties of the variance-VQE for solving a set of excited states in quantum chemistry problems.Remarkably,we show that optimization of a combination of energy and variance may be more efficient to find low-energy excited states than those of minimizing energy or variance alone.We further reveal that the optimization can be boosted with stochastic gradient descent by Hamiltonian sampling,which uses only a few terms of the Hamiltonian and thus significantly reduces the quantum resource for evaluating variance and its gradients.