Classically optimized variational quantum eigensolver with applications to topological phases

Abstract

The variational quantum eigensolver (VQE) is regarded as a promising candidate of hybrid quantum-classical algorithms for near-term quantum computers. Meanwhile, VQE is confronted with a challenge that statistical error associated with measurement as well as systematic error could significantly hamper the optimization. To circumvent this issue, we propose the classically optimized VQE (CO-VQE), where the whole process of optimization is efficiently conducted on a classical computer. The efficacy of the method is guaranteed by the observation that quantum circuits with up to logarithmic depth are classically tractable via simulations of local subsystems with up to quasipolynomial cost (polynomial for constant depth). In CO-VQE, we only use quantum computers to measure nonlocal quantities after the parameters are optimized. As a proof of concept, we present numerical experiments on quantum spin models with topological phases. After the optimization, we identify the topological phases by nonlocal order parameters as well as unsupervised machine learning on inner products between quantum states. The proposed method maximizes the advantage of using quantum computers while avoiding strenuous optimization on noisy quantum devices. In addition, our paper indicates that clustering technique combined with the fidelity measured on quantum computers could be useful for phase classification in condensed-matter physics.