Optimizing Variational Quantum Algorithms using Pontryagin's Minimum Principle

TL;DR

This paper applies Pontryagin's minimum principle to optimize variational quantum algorithms, revealing that bang-bang control protocols are optimal and outperform traditional quantum annealing under certain noise conditions.

Contribution

It demonstrates the optimality of bang-bang protocols for variational quantum algorithms and characterizes their time scales, informing more efficient hybrid quantum-classical optimization strategies.

Findings

Bang-bang protocols are optimal for fixed computation time.

Characteristic pulse durations are system-size independent.

Optimal protocols outperform quantum annealing with noise.

Abstract

We use Pontryagin's minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bang-bang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed Quantum Approximate Optimization Algorithm. Focusing on the Sherrington-Kirkpatrick spin-glass as an example, we find a system-size independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bang-bang protocols and the characteristic time scale of the pulses provide an efficient parameterization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. For the particular systems we study, we find numerically that the optimal nonadiabatic bang-bang protocols outperform conventional quantum annealing in the presence of weak white additive external noise and weak coupling to a thermal bath modeled with the Redfield master equation.