Liouville-Fock-state lattices: From environment-induced frustration to synthetic lattice dynamics

We introduce Liouville-Fock-state lattices (LFSLs) as a framework for visualizing open quantum systems through matrix representations of the Lindblad master equation. These generalize Hamiltonian Fock-state lattices, which have been considered for realizing synthetic dimensions in quantum simulators. By vectorizing the Lindblad master equation, the dynamics map to a doubled Hilbert space, naturally forming a synthetic lattice. Unlike the unitary evolution of pure states, LFSLs feature non-Hermitian dynamics with drifts, sources, and sinks—paralleling stochastic classical lattices. As such, we suggest that these systems can serve as potential quantum simulators of classical lattice models, with a computational advantage arising from large state spaces encoded in quantum models. For this purpose, we analyze both the Fock representation and alternative positive-semidefinite representations that yield real, non-negative populations, akin to classical probability distributions. Within this formalism, we also demonstrate a phenomenon, which we term environment-induced frustration, in which competing terms in the LFSL give rise to geometric frustration and a massively degenerate steady-state manifold.