Exploiting the causal tensor network structure of quantum processes to efficiently simulate non-Markovian path integrals

In the path integral formulation of the evolution of an open quantum system coupled to a Gaussian, non-interacting environment, the dynamical contribution of the latter is encoded in an object called the influence functional. Here, we relate the influence functional to the process tensor -- a more general representation of a quantum stochastic process -- describing the evolution. We then use this connection to motivate a tensor network algorithm for the simulation of multi-time correlations in open systems, building on recent work where the influence functional is represented in terms of time evolving matrix product operators. By exploiting the symmetries of the influence functional, we are able to use our algorithm to achieve orders-of-magnitude improvement in the efficiency of the resulting numerical simulation. Our improved algorithm is then applied to compute exact phonon emission spectra for the spin-boson model with strong coupling, demonstrating a significant divergence from spectra derived under commonly used assumptions of memorylessness.Comment: 6+5 pages, 4 figure

A discrete memory-kernel for multi-time correlations in non-Markovian quantum processes

Efficient simulations of the dynamics of open systems is of wide importance for quantum science and tech-nology. Here, we introduce a generalization of the transfer-tensor, or discrete-time memory kernel, formalism to multi-time measurement scenarios. The transfer-tensor method sets out to compute the state of an open few-body quantum system at long times, given that only short-time system trajectories are available. Here, we showthat the transfer-tensor method can be extended to processes which include multiple interrogations (e.g. measurements) of the open system dynamics as it evolves, allowing us to propagate high order short-time correlation functions to later times, without further recourse to the underlying system-environment evolution. Our approach exploits the process-tensor description of open quantum processes to represent and propagate the dynamics in terms of an object from which any multi-time correlation can be extracted. As an illustration of the utility of the method, we study the build-up of system-environment correlations in the paradigmatic spin-boson model, and compute steady-state emission spectra, taking fully into account system-environment correlations present in the steady state.Comment: 9 pages, 2 figure