Optimal Control for Quantum Optimization of Closed and Open Systems

We provide a rigorous analysis of the quantum optimal control problem in setting linear combination $s(t)B+(1-s(t))C$ two noncommuting Hamiltonians $B$ and $C$. This includes both annealing (QA) approximate optimization algorithm (QAOA). The target is to minimize energy final ``problem'' Hamiltonian $C$, for time-dependent bounded schedule $s(t)\in [0,1]$ $t\in \mc{I}:= [0,t_f]$. It was recently shown, purely closed system setting, that solution this ``bang-anneal-bang'' schedule, with bangs characterized by $s(t)= 0$ 1$ finite subintervals $\mc{I}$, particular $s(0)=0$ $s(t_f)=1$, contrast standard prescription $s(0)=1$ $s(t_f)=0$ annealing. Here we extend result open where described density matrix rather than pure state. natural experimental realizations QA QAOA. For finite-dimensional environments without any approximations identify sufficient conditions ensuring either bang-anneal, anneal-bang, or bang-anneal-bang schedules are optimal, recover optimality $s(t_f)=1$. However, infinite-dimensional an adiabatic Redfield master equation do not bang-type solution. In fact can only under which even recovered fully Markovian limit. analysis, carry out entirely within geometric framework Pontryagin Maximum Principle, simplifies using formulation compared state vector formulation.