Training Variational Quantum Algorithms Is NP-Hard

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

Automated summary

Variational quantum algorithms are proposed to solve computational problems on quantum devices, with classical optimization problems shown to be NP-hard. The landscape of training can have many persistent local minima, leading gradient and higher order algorithms to converge to suboptimal solutions. Even classically tractable systems composed of logarithmically many qubits or free fermions exhibit NP-hard optimization.