Optimal Control for Quantum Optimization of Closed and Open Systems

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 $s(t)B+[1\ensuremath{-}s(t)]C$ of two noncommuting Hamiltonians $B$ and $C$, where $C$ encodes the solution to the optimization problem in its ground state, $B$ is a Hamiltonian whose ground state is easy to prepare, and $s(t)\ensuremath{\in}[0,1]$ is the bounded ``switching schedule'' or ``path,'' with $t\ensuremath{\in}[0,{t}_{f}]$. This approach encompasses two of the most widely studied quantum-optimization algorithms: quantum annealing [QA; continuous $s(t)$] and the quantum approximate optimization algorithm [QAOA; piecewise constant $s(t)$]. 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 $s(t)$. 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 $s(t)=0$ and $s(t)=1$ in finite subintervals of $[0,{t}_{f}]$, in particular, $s(0)=0$ and $s({t}_{f})=1$, in contrast to the standard prescription $s(0)=1$ and $s({t}_{f})=0$ of QA. As an example, we prove that for a single spin-$1/2$, the optimal solution in the closed-system setting is the bang-bang schedule, switching midway from $s\ensuremath{\equiv}0$ to $s\ensuremath{\equiv}1$. 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 $s(0)=0$ and $s({t}_{f})=1$. 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 $s({t}_{f})=1$, 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.