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

• URL: http://arxiv.org/abs/2509.05295v2
• Date: Fri, 07 Nov 2025 16:24:01 GMT
• Title: On the convergence of the variational quantum eigensolver and quantum optimal control
• Authors: Marco Wiedmann, Daniel Burgarth, Gunther Dirr, Thomas Schulte-Herbrüggen, Emanuel Malvetti, Christian Arenz,
• Abstract summary: We develop a convergence theory for the variational quantum eigensolver (VQE)<n>We prove a sufficient criterion that characterizes when convergence to a ground state of a Hamiltonian can be guaranteed.<n>We analyze two commonly employed families of quantum circuit ans"atze.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: When does a variational quantum algorithm converge to a globally optimal solution? Despite the large literature around variational approaches to quantum computing, the answer is largely unknown. We address this open question by developing a convergence theory for the variational quantum eigensolver (VQE). By leveraging the terminology of quantum control landscapes, we prove a sufficient criterion that characterizes when convergence to a ground state of a Hamiltonian can be guaranteed for almost all initial parameter settings. More specifically, we show that if (i) a parameterized unitary transformation allows for moving in all tangent-space directions (local surjectivity) in a bounded manner and (ii) the gradient descent used for the parameter update terminates, then the VQE converges to a ground state almost surely. We develop constructions that satisfy both aspects of condition (i) and analyze two commonly employed families of quantum circuit ans\"atze. Finally, we discuss regularization techniques for guaranteeing gradient descent to terminate, as for condition (ii), and draw connections to the halting problem.

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