Threshold theorem in isolated quantum dynamics with local stochastic control errors

Abstract

We investigate the effect of local stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schr\"odinger equation. For any local stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements required and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the required number of measurements increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing, adiabatic quantum computation, the quantum approximate optimization algorithm, and the quantum circuit model.Comment: 4 pages, 0 figur