Efficient Time Evolution of 2D Open-Quantum Lattice Models with Long-Range Interactions using Tensor Networks

TL;DR

This work presents a construction of the time-evolution operator, as a projected entangled pair operator (denoted tePEPO), that can be used to evolve a tensor network ansatz through time, demonstrating the applicability of tensor networks to two-dimensional systems widely studied in experiments, but previously inaccessible to non-semi-classical methods.

Abstract

Simulating many-body open quantum systems is an extremely challenging problem, with methods often restricted to either models with nearest-neighbor interactions or semi-classical approximations. In particular, modeling two-dimensional systems with realistic long-range interactions, in addition to dissipation, is of vital importance to the development of modern quantum computing and simulation platforms. In this paper, we present a construction of the time-evolution operator, as a projected entangled pair operator (denoted tePEPO), that can be used to evolve a tensor network ansatz through time. Interactions beyond nearest-neighbor, including interactions between sites not collinear in the lattice, can be represented efficiently as a tePEPO. Furthermore, we obtain approximations to realistic radial long-range interactions decaying with a power-law, that give accurate results with small tePEPO bond dimension. Finally, we consider a physical example of a Rydberg atom Hamiltonian with long-range dipolar interactions, and show evidence of a dipole-dipole blockading effect in presence of dissipation. This work demonstrates the applicability of tensor networks to two-dimensional systems widely studied in experiments, but previously inaccessible to non-semi-classical methods.

Figures (4)

• Figure 1: Example of a many-body open quantum system. Atoms in excited Rydberg states $\ket{r}$ with strong, long-range repulsive interactions decaying as a power law $r^{-\alpha}$, typically of van der Waals $\alpha = 6$ or dipolar $\alpha = 3$ form. Atoms are promoted from the ground state $\ket{g}$ to a highly-excited Rydberg state $\ket{r}$ via a laser drive with frequency $\Omega$ and detuning $\Delta$. Atoms can then decay $\ket{r}\to \ket{g}$ incoherently at a rate $\gamma$.
• Figure 2: Examples of two terms (a) and (b) in the Hamiltonian \ref{['eq:ham-j1j2']} that are accepted by the fsa and another term (c) not in \ref{['eq:ham-j1j2']} that is correctly rejected due to the additional (blue) level. This term would incorrectly be included in the sum had the blue level been replaced by the red level.
• Figure 3: (a) and (b) example of products of fsa rules that do not require additional levels, and can therefore be included at no cost to the operator bond dimension. In (c), the product includes overlapping non-trivial levels so we require an additional level to transmit both the blue and red signals simultaneously. Including such product rules allows the fsa to also generate all the second order terms that intersect only once.
• Figure 4: