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

## TL;DR

This paper introduces a tensor network-based algorithm leveraging the structure of the influence functional and process tensor to efficiently simulate non-Markovian quantum dynamics, enabling accurate computation of phonon emission spectra in strongly coupled systems.

## Contribution

It establishes a novel connection between influence functionals and process tensors, leading to an efficient tensor network algorithm for non-Markovian quantum process simulation.

## Key findings

- Achieved orders-of-magnitude speedup in simulations.
- Accurately computed phonon emission spectra in the spin-boson model.
- Demonstrated significant deviations from memoryless assumptions.

## Abstract

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.