Unifying methods for optimal control in non-Markovian quantum systems via process tensors

The large dimensionality of environments is the limiting factor in applying optimal control to open quantum systems beyond the Markovian approximation. Various methods exist to simulate non-Markovian systems, which effectively reduce the environment to a number of active degrees of freedom. Here, we show that several of these methods can be expressed in terms of a process tensor in the form of a matrix-product-operator, which serves as a unifying framework to show how they can be used in optimal control and to compare their performance. The matrix-product-operator form provides a general scheme for computing gradients using back propagation and allows the efficiency of the different methods to be compared via the bond dimensions of their respective process tensors.