Verifiable Solutions to the Schrodinger Equation with Variational Quantum Imaginary-Time Evolution

Variational optimization of parameterized quantum states is a major use of current and future quantum technologies. While recent advancements has resulted in more robust algorithms, several challenges persist in the development of successful variational quantum eigensolvers (VQEs). In particular, parameterized quantum circuits can become stuck in local minima and fail to achieve the ground state, due to barren plateaus in the cost-function landscape. Here we propose a variance-based VQE, which results in verifiable solutions to the Schrodinger equation by direct estimation of the wavefunction variance. We connect previously reported variance-based techniques to the imaginary-time evolution (ITE) formalism. Finally, we show that with little alteration the present algorithm can be used to achieve excited eigenstates. While the measurement cost for the algorithm is a quadratic increase over energy-minimization VQEs, we argue that the verifiability and the simple extension to excited states makes variance-based VQEs a promising technique for finding eigenstates of structured Hamiltonians.