Abstract
Optimization is one of the key applications of quantum computing where a quantum speedup has been an eagerly anticipated outcome. A promising approach to optimization using quantum dynamics is to consider a linear combination of two noncommuting Hamiltonians and , where encodes the solution to the optimization problem in its ground state, is a Hamiltonian whose ground state is easy to prepare, and is the bounded “switching schedule” or “path,” with . This approach encompasses two of the most widely studied quantum-optimization algorithms: quantum annealing [QA; continuous ] and the quantum approximate optimization algorithm [QAOA; piecewise constant ]. While it is notoriously difficult to prove a quantum advantage for either algorithm, it is possible to compare and contrast them by finding the optimal . Here we provide a rigorous analysis of this quantum optimal control problem, entirely within the geometric framework of Pontryagin’s maximum principle of optimal control theory. We extend earlier results, derived in a purely closed-system setting, to open systems. This is the natural setting for experimental realizations of QA and QAOA. In the closed-system setting it was shown that the optimal solution is a “bang-anneal-bang” schedule, with the bangs characterized by and in finite subintervals of , in particular, and , in contrast to the standard prescription and of QA. As an example, we prove that for a single spin-, the optimal solution in the closed-system setting is the bang-bang schedule, switching midway from to . For finite-dimensional environments and without any approximations we identify sufficient conditions ensuring that either the bang-anneal, anneal-bang, or bang-anneal-bang schedules are optimal, and recover the optimality of and . However, for infinite-dimensional environments and a system described by an adiabatic Redfield master equation we do not recover the bang-type optimal solution. In fact we can only identify conditions under which , and even this result is not recovered in the fully Markovian limit, suggesting that the pure anneal-type schedule is optimal. Our open-system results have implications for the use of experimental quantum-information processors, which are by necessity noisy, and suggest that in this practical sense the optimal schedules for quantum optimization are likely to be continuous.
- Received 15 July 2021
- Revised 9 October 2021
- Accepted 15 October 2021
DOI:https://doi.org/10.1103/PhysRevApplied.16.054023
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