Simulations of Open Quantum Systems and Decoherence-Free Subspaces
M.W. Weerheim · Research Repository (Delft University of Technology) · 2020
Complex quantum systems, such as a quantum computer, will always be coupled in some way to the environment. This can cause what's called decoherence, a destructive process by which information is lost from the system into the environment. In this bachelor thesis paper, we discuss decoherence-free subspaces within networks of coupled quantum harmonic oscillators (or QHOs). We investigate where such noiseless subspaces (or NSs) occur most frequently in an ensemble of Erdos-Renyi networks, for which we do not yet consider the influence of the bath. We then proceed by adding the bath into the equation, using some of the theory of open quantum systems. Specifically, we derive the Lindblad master equation and show its form for the case of our networks. Consequently, we simulate the behavior of the moments of the position operators for the graphs with 3 nodes, both by means of the full Lindblad equation, and by first tracing out those moments to obtain their differential equations. We compare those two results to each other, and also look back to the situation before adding the bath to see if the NSs are still present. From the results of the simulations, we can conclude several things. Firstly, we see that for ensembles with probability of connection p very close to either 0 or 1, both to number of noiseless modes and the probability of finding at least one is largest. This is credited to their relatively high degrees of symmetry. Secondly, in the results of the density matrix and moment simulations, we see that, indeed, the noiseless modes are preserved when considering the influence of the bath. Furthermore, we can conclude that simulation of the density matrix for the case of coupled QHOs in a network is in many cases not stable; the cutoff at a finite level s needed to simulate an otherwise infinite- dimensional operator leads to non-positivity of the density matrix. Therefore, it is best to simulate the moments from their respective differential equations, as opposed to the full Lindblad master equation. Finally, the differential equations for the moments and their solutions show that there are indeed noiseless clusters for eigenmodes perpendicular to the center of mass, as predicted.