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We use Pontryagin's minimum principle to optimize variational quantumalgorithms. We show that for a fixed computation time, the optimal evolutionhas a bang-bang (square pulse) form, both for closed and open quantum systemswith Markovian decoherence. Our findings support the choice of evolution ansatzin the recently proposed Quantum Approximate Optimization Algorithm. Focusingon the Sherrington-Kirkpatrick spin-glass as an example, we find a system-sizeindependent distribution of the duration of pulses, with characteristic timescale set by the inverse of the coupling constants in the Hamiltonian. Theoptimality of the bang-bang protocols and the characteristic time scale of thepulses provide an efficient parameterization of the protocol and inform thesearch for effective hybrid (classical and quantum) schemes for tacklingcombinatorial optimization problems. For the particular systems we study, wefind numerically that the optimal nonadiabatic bang-bang protocols outperformconventional quantum annealing in the presence of weak white additive externalnoise and weak coupling to a thermal bath modeled with the Redfield masterequation.

Optimizing Variational Quantum Algorithms using Pontryagin's Minimum Principle