Variational Quantum Eigensolver with Mutual Variance-Hamiltonian Optimization

The zero-energy variance principle can be exploited in variational quantum eigensolvers for solving general eigenstates but its capacity for obtaining a specified eigenstate, such as ground state, is limited as all eigenstates are of zero energy variance. We propose a variance-based variational quantum eigensolver for solving the ground state by searching in an enlarged space of wavefunction and Hamiltonian. With a mutual variance-Hamiltonian optimization procedure, the Hamiltonian is iteratively updated to guild the state towards to the ground state of the target Hamiltonian by minimizing the energy variance in each iteration. We demonstrate the performance and properties of the algorithm with numeral simulations. Our work suggests an avenue for utilizing guided Hamiltonian in hybrid quantum-classical algorithms.