Analytical insights into the spin–boson model and optimal control via the redfield master equation

Abstract: In this work, we are interested in the dynamics of the driven spin–boson model. First, we derive a master equation describing the interaction of a two-level system (TLS) with an Ohmic heat bath. The system is subject to the action of two control fields, physically associated with the time-dependent bias energy and time-dependent tunneling. In the limit where tunneling is equal to zero, we derive an analytic solution of the master equation in the undriven case. Then, we apply quantum optimal control theory to the resolvent to compute the time-dependent bias energy and time-dependent tunneling which allows for creating a single-qubit quantum gate for the fully driven spin–boson model. The resolvent is the generalization of the evolution quantum propagator in dissipative dynamics. In order to solve the quantum optimal control problem, we use the technique of automatic differentiation. We are able to implement the X-gate and Z-gate with good fidelity. The advantage of applying optimal control theory to the resolvent is that the logic gate does not depend on the initial state of the qubit. Thus, the optimal solution is unique regardless of the initial state of the qubit.