Convergence and schedule optimization of multi-channel quantum Zeno dragging with application to k-SAT problems, part I

We analyze a novel framework (quantum Zeno dragging) for quantum algorithms and quantum control that is driven by quantum measurement alone. Quantum Zeno dragging works by measuring a set of observables while gradually changing the measurement basis. This can be used to solve Boolean Satisfiability problems (k-SAT), where each clause corresponds to a k-local observable. We study such measurement-driven quantum dynamics with generalized measurement and prove a theorem similar to adiabatic computing. Specifically, we show that the convergence towards the target state is guaranteed if i) the measurement basis is moving sufficiently slowly; and ii) the measurement channel is applied sufficiently many times compared to the spectral gap of some cost operator. We treat all scales between projective measurement and continuous-time weak measurement, and show how this choice impacts the prefactor without affecting the asympotic scaling of the convergence time.