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arXiv:2503.22047 · Physical Review Research

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

Caio B. Naves, Jonas Larson

Published Dec 1, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/5xq7-l8j8

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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.
In this licenciate thesis, based on two articles, we discuss the properties of open quantum systems and investigate how their dynamics can be used to design a synthetic lattice framework and how their spectrabehave based on two. The first article presents an extension of the Fock state lattice (FSL) framework to open quantum systems. Using the Liouville space representation, density matrices can be vectorized and the dynamics are generated by a linear operator, the Liouvillian, allowing for the introduction of Liouville-Fock state lattices (LFSLs). This framework provides an intuitive and visual understanding of the dynamics of open quantum systems and shows how environmental effects can give rise to phenomena such as directional transport, sources and sinks, typical phenomena occurring in classical processes. As an example, we show how frustration in LFSL can be induced by the environment. The second article in the work focuses on non-normality of Liouvillians and spectral diagnostics of chaos. Non-normal operators are characterized by the fact that their eigenvectors are not orthogonal, which leads to the spectrum exhibiting strong sensitivity to perturbations. We show that strong non-normality can give rise to random matrix-like spectral statistics without the system dynamics exhibiting chaotic signatures. Therefore, we conclude that this violates established assumptions and points to the need for alternative methods to characterize chaotic dynamics in open quantum systems.
In this work we introduce discrete-time quantum walks in state space, more precisely on Fock-state lattices. Fock-state lattices provide a natural and clean setting for implementing lattice models, particularly in quantum optical systems. Thus, contrary to the common setting where the walker resides in real space or phase space, here the walk takes place in a synthetic space. We present a general formalism based on Lie algebras and their properties. For each Lie algebra one can associate both a phase space and a Fock-state lattice, and by understanding how these spaces are related, together with the action of generalized displacement operators, we construct the discrete unitary operator that generates the walk. In this framework the displacement operators replace the usual nearest-neighbor shifts and lead to state-dependent tunneling on the lattice. By considering several examples we demonstrate ballistic spreading and other characteristic features of discrete-time quantum walks, such as coin-walker entanglement and symmetry-induced interference patterns. We also show that different algebraic structures can give rise to qualitatively different dynamics, including anomalous behavior such as super-ballistic spreading as well as localization effects.
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a two-fold way. The first part is devoted in establishing an exact correspondence between quantum evolution and classical Hamiltonian flow on a Kahler manifold. This correspondence enables a geometric quantization scheme that identifies a family of classical Hamiltonian systems admitting exponentially compressed quantum representations-appropriate for quantum simulation. In the second part we demonstrate that Liouville-integrable Hamiltonian dynamics induce finite-dimensional unitary evolution through action-angle variables and Koopman-von Neumann encoding. This allows efficient quantum representation and parallel evolution of large phase-space ensembles, where entangled encodings provide exponential compression in ensemble size and enable quantum speed-ups in observable estimation via amplitude estimation techniques. For non-integrable systems, Lie canonical perturbation theory is incorporated to construct near-symplectic transformations that map dynamics to approximately integrable forms, preserving unitary evolution up to a controlled error. We derive the resulting quantum computational complexity of the proposed quantum-symplectic scheme, revealing both an exponential compression in memory requirements and a potential polynomial speed-up with respect to the system size. Finally, the transport evolution equation governing the quantum phase-space observables is obtained.
Quantum simulation of dynamics in open quantum systems is crucial but poses a significant challenge due to the non-Hermitian nature leading to nonunitary evolution and the limited quantum resources on current quantum computers. Here we introduce a variational hybrid quantum-classical algorithm designed for simulating the time evolution governed by the Lindblad master equation. Our approach involves on a stochastic unveiling of the density matrix, transforming the Lindblad equation into a wave function-based quantum state diffusion (QSD) method with the aim of reducing qubit requirements. We then apply variational quantum simulation (VQS) to efficiently capture the nonunitary evolution in QSD. We demonstrate our QSD-VQS algorithm by investigating the quantum dynamics in a two-level system subjected to an amplitude damping channel and a four-level transverse field Ising model within a dissipative environment including time-independent and periodic Hamiltonian cases. The results reveal its promising utility with upcoming hardware in the near future.
We analyze a set of models frequently appearing in quantum optical settings by expressing their Hamiltonians in terms of Fock-state lattices. The few degrees-of-freedom of such models, together with the system symmetries, make the emerging Fock-state lattices rather simple such that they can be linked to known lattice models from the condensed matter community. This sheds new light on known quantum optical systems. While we provide a rather long list of models and their corresponding Fock-state lattices, we pick a few ones in order to demonstrate the strength of the method. The three-mode boson model, for example, is shown to display a fractal spectrum, and chiral evolution in the Fock-state lattice characterized by localized distributions traversing along symmetric trajectories. In a second example we consider the central spin model which generates a Fock-state lattice reminiscent of the SSH-model hosting topological edge states. We further demonstrate how the phenomena of flat bands in lattice models can manifest in related Fock-state lattices, which can be linked to so called dark states.
Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian CPTP evolution, many physical platforms display non-Markovian features such as algebraic relaxation and coherence backflow. Fractional calculus provides a natural way to model such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we develop a unified framework that embeds fractional master equations within the broader hierarchy of open-system formalisms. The fractional equation forms a structured subclass of memory-kernel models, reduces to the Lindblad form at unit order, and, through Bochner--Phillips subordination, admits a CPTP representation as an average over Lindblad semigroups. Its resolvent structure further connects fractional dynamics to established non-Markovian approaches, including Nakajima--Zwanzig kernels and hierarchical equations of motion, providing a compact surrogate for long-memory effects. This formulation positions fractional calculus as a rigorous and practical language for quantum dynamics with intrinsic memory, supporting both analytical insight and efficient quantum simulation.