Process tensor networks for non-Markovian open quantum systems

Abstract

The advance of quantum technology relies heavily on an accurate understanding of the unavoidable interactions between quantum systems and their environment. While it is often adequate to account for the environment using approximate time-local (i.e. Markovian) equations of motion, in many scenarios such a description fails, and a more general non-Markovian theory becomes necessary. The failure of Markovian descriptions concerns not only quantitative aspects of the reduced dynamics of a quantum system, but also qualitative and conceptual aspects, such as the failure of the quantum regression formula relating the system's dynamics to its multi-time correlations. Despite considerable progress in recent years, the description and simulation of non-Markovian open quantum systems remains a conceptual and computational challenge. In this thesis we develop a versatile set of numerical methods for non-Markovian open quantum systems by combining the so-called process tensor framework with the numerical power of tensor network methods. The recently introduced process tensor is an alternative approach to open quantum systems and is - unlike the canonical approach based on dynamical maps - well suited for a rigorous characterisation of non-Markovian open quantum systems. We construct and apply process tensors in a matrix product operator form (PT-MPO) to yield a numerically exact, yet efficient representation of non-Markovian open quantum systems, which allows for a variety of practical applications. Building on the PT-MPO we introduce general methods to (1) efficiently find optimal control procedures for non-Markovian open quantum systems, (2) compute the dynamics and multi-time correlations of chains of non-Markovian open quantum systems, and (3) construct a time-translational invariant PT-MPO, which allows efficient computation of steady states even in non-equilibrium non-Markovian scenarios.