# Gaussian Quantum Trajectories for the Variational Simulation of Open Quantum-Optical Systems
## 6 Conclusions and Outlook  
We have shown how Gaussian variational quantum trajectory methods can provide a computationally-efficient description of open quantum systems. We have explicitly derived the dynamics for both \(XP\)- and \(N\Theta\)-Gaussian states and applied them as an example to a driven-dissipative cavity. \(XP\)-Gaussian states are always well-defined, though they may be too rigid to describe states with much phase-diffusion. \(N\Theta\)-states, on the other hand, exist only by approximation, but often provide a very good description for the true state as long as the density is sufficiently high. Computationally, the cost of a Gaussian trajectory scales similarly to TWA, but typically, less samples are needed. \(XP\)-Gaussian methods, like TWA, are limited to the semiclassical regime where the Wigner function is always positive. \(N\Theta\)-Gaussian methods on the other hand can describe the interference patterns similar to Fock-states (\(XP\)-Gaussian states are the only pure states with an entirely positive Wigner  
Figure 5: Same process as Figure 4 (corresponding legend) simulated with ten samples of exact/variational trajectories, as well as TWA. It is clear that trajectory methods are more tolerant for low statistics than TWA. Corresponding to the criterion (29), we see that photon-counting is still more tolerant than heterodyne detection, corresponding to the fact that \(\mathrm{Var}_{1}(X)\gg\mathrm{Var}_{2}(X)\) for this photon-counting unraveling, as opposed to heterodyne. Note that the initial state already has a deviation from the true value for TWA, but not for the trajectory methods: the criterion (29) for the needed ratio of samples diverges there.  
function [60]). As these variational methods, unlike TWA, ensure that the state remains physically well defined, we expect them to keep predicting accurate results to a broader class of problems such as coupled cavities. As we have shown, accuracy can be strongly improved by the proper choice of the ansatz. Furthermore, similar ansatzes, which can for example be Gaussian in other variables [73], may be suitable dependent on the problem. As for exact trajectories, the choice of unraveling greatly determines the amount of samples needed for a proper description. In addition, we have seen that this choice of unraveling can also influence the accuracy of the variational method. Besides the extension to larger systems, the application to systems with multiple Markovian jump processes is straightforward. Extensions towards more general non-Markovian noise are less clear-cut, but may be feasible through, e.g., a doubled Hilbert space framework [8].  
The following is available online at [[http://www.mdpi.com/2076-3417/8/9/1427/s1](http://www.mdpi.com/2076-3417/8/9/1427/s1)]([http://www.mdpi.com/2076-3417/8/9/1427/s1](http://www.mdpi.com/2076-3417/8/9/1427/s1)), Code S1: used code for time-evolution of exact trajectories, Gaussian trajectories and TWA.  
Conceptualization, M.W. and W.V. Funding acquisition, M.W.; Investigation, W.V. and M.W. Methodology, M.W. and W.V. Software, W.V. Supervision, M.W. Visualization, W.V. Writing, original draft, W.V. Writing, review and editing, M.W.  
This research was funded by the FWO Odysseus program.  
We thank [PERSON] for his important initial contributions to this project; and acknowledge discussion with [PERSON] and [PERSON]. Part of the computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation-Flanders (FWO) and the Flemish Government department EWI.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\(Xp\)-Gaussian state Gaussian in the quadrature variables  
\(N\Theta\)-Gaussian state Gaussian in density and phase  
TWA Truncated Wigner Approximation  
PC Photon-Counting  
Hom. (X) Homodyne detection of \(X\)-quadrature  
Het. Heterodyne detection  
'sym  
symmetrically ordered