On the convergence of the variational quantum eigensolver and quantum optimal control

Abstract:When do variational quantum algorithms converge to a globally optimal solution? Despite extensive work, this question remains largely open. We develop a convergence theory for the variational quantum eigensolver (VQE), using quantum control landscape terminology to prove a sufficient criterion guaranteeing convergence to a Hamiltonian's ground state for almost all initial parameters. Specifically, we show that if (i) the parameterized unitary transformation allows movement in all tangent-space directions in a bounded manner (local surjectivity), and (ii) the gradient descent terminates, then VQE converges almost surely to a ground state. Under similar assumptions, we also guarantee almost-sure convergence to a global optimum on a smaller, closed unitary Lie subgroup, with representation theory providing a criterion for when this optimum is a ground state. We construct examples satisfying condition (i) and analyze two common circuit ansatz families, then discuss regularization techniques that ensure gradient descent terminates and connect to the halting problem.

Quantum Physics (quant-ph); Optimization and Control (math.OC)

arXiv:2509.05295 [quant-ph]

(or arXiv:2509.05295v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2509.05295 Focus to learn more arXiv-issued DOI via DataCite

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