# Polynomial complexity of open quantum system problems

## TL;DR

This paper proves that the complexity of modeling open quantum systems grows at most polynomially with time and bath size, enabling efficient algorithms for simulating their dynamics using tensor networks.

## Contribution

It establishes a polynomial complexity theorem for open quantum systems and introduces tensor network algorithms with linear bond dimension growth.

## Key findings

- Number of equations grows linearly with time
- Tensor network bond dimension increases linearly with time
- Algorithms successfully applied to spin-boson and Gaudin models

## Abstract

Open quantum systems (OQS's) are ubiquitous in non-equilibrium quantum dynamics and in quantum science and technology. Solving the dynamics of an OQS in a quantum many-body bath has been considered a computationally hard problem because of the dimensionality curse. Here, considering that full knowledge of the bath dynamics is unnecessary for describing the reduced dynamics of an OQS, we prove a polynomial complexity theorem, that is, the number of independent equations required to fully describe the dynamics of an OQS increases at most linearly with the evolution time and polynomially with the bath size. Therefore, efficient computational algorithms exist for solving the dynamics of a small-sized OQS (such as a qubit or an atom). We further prove that, when the dynamics of an OQS and the bath is represented by a tensor network, a tensor contraction procedure can be specified such that the bond dimension (i.e., the range of tensor indices contracted in each step) increases only linearly (rather than exponentially) with the evolution time, providing explicitly efficient algorithms for a wide range of OQS's. We demonstrate the theorems and the tensor-network algorithm by solving two widely encountered OQS problems, namely, a spin in a Gaussian bath (the spin-boson model) and a central spin coupled to many environmental spins (the Gaudin model). This work provides approaches to understanding dynamics of OQS's, learning the environments via quantum sensors, and optimizing quantum information processing in noisy environments.