# Simulation methods for open quantum many-body systems

It is very difficult to obtain exact solutions to systems involving interactions between more than two bodies, using either classical mechanics or quantum mechanics. To understand the physics of many-body systems, it is necessary to make use of approximation techniques or model systems that capture the essential physics of the problem. The complexity of the problem scales exponentially with the size of the system, requiring the use of sophisticated simulation methods to obtain useful results This is compounded by the fact that no system in nature is perfectly isolated. They are constantly interacting with their environment in the form of heat transfer, decoherence, etc. Actually, all real systems are open systems.

Open quantum many-body systems have witnessed a surge of interest in recent years, chiefly for two reasons. On the one hand, these systems offer the possibility of using controlled dissipation channels to engineer interesting quantum many-body states as the stationary state of their dynamics. On the other hand, open quantum many-body systems are attractive from a fundamental perspective, as their dynamics exhibits a wide range of features not found in equilibrium systems.

Open quantum many-body systems are even harder to simulate on classical computers than closed systems, while at the same time the stationary state of an open quantum system is much easier to experimentally prepare than the ground state of a closed system. These properties make open quantum systems one of the prime candidates to show a quantum advantage of quantum simulators over classical methods within noisy intermediate-scale quantum devices. However, this requires a thorough assessment of the capabilities of classical simulation methods, which a team of researchers now provide in a new review ^{1}.

The substantial effort to develop novel simulation methods to investigate open quantum many-body systems has produced a variety of numerical methods. Specifically, in this review methods for the Markovian quantum master equation (assuming a weak-coupling limit), including mean-field stochastic methods, tensor networks, variational methods, quantum Monte Carlo methods, a truncated Wigner approximation, BBGKY hierarchy equations, and linked-cluster expansions are considered.

While no method has yet emerged that is universally optimal for all cases, there have been several promising developments with different methods for different regimes. Even with the major technical advances recently achieved, there are still many open problems that are inaccessible with these state-of-the-art numerical techniques.

The authors come to very interesting conclusions. The first is that mean-field methods are considerably less reliable for open systems than their counterparts for closed systems, although the reason for this discrepancy is still an open question. Secondly, tensor network methods have demonstrated their ability to successfully tackle many hard problems surrounding open many-body systems and resolve long-standing open questions. A particularly interesting and promising case is that of open 2D systems, which are unexplored territory to a large extent. Finally, for the variational methods discussed in this review, there appears to be a trade-off between the formal suitability of the norm and its efficient computability.

Progress in recent years in simulating open quantum systems has brought the field to a level where one has a wide range of tools at hand to systematically make a comparison to experimental results, particularly in the context of quantum simulations. Combined with the experimental ease of preparing the steady state of an open quantum system, these are good reasons to believe that the study of strongly correlated open quantum many-body systems will become a research topic with an impact on other areas of science, such as material design and quantum computation.

*Author: César Tomé López is a science writer and the editor of Mapping Ignorance*

*Disclaimer: Parts of this article may be copied verbatim or almost verbatim from the referenced research paper.*

## References

- Hendrik Weimer, Augustine Kshetrimayum, and Román Orús (2021) Simulation methods for open quantum many-body systems
*Reviews of Modern Physics*doi: 10.1103/RevModPhys.93.015008 ↩