Training Variational Quantum Algorithms Is NP-Hard

Variational quantum algorithms are proposed to solve relevant computational problems on near term devices. Popular versions variational eigensolvers and ap- proximate optimization that ground state from chemistry binary problems, respectively. They based the idea of using a classical computer train parameterized circuit. We show corresponding NP-hard. Moreover, hardness is robust in sense that, for every polynomial time algorithm, there instances which relative error resulting problem can be arbitrarily large assuming P $\neq$ NP. Even classically tractable systems composed only logarithmically many qubits or free fermions, we This elucidates intrinsically hard does not merely inherit problem. Our analysis shows training landscape have far optimal persistent local minima. means gradient higher order descent will generally converge solutions.