Classical variational methods are used to simulate the dynamics of two-dimensional quantum models with continuous degrees of freedom using neural networks, pushing the limits of classical computation in simulating these systems.
Dynamics in correlated quantum matter is a hard problem, as its exact solution generally involves a computational effort that grows exponentially with the number of constituents. While remarkable progress has been witnessed in recent years for one-dimensional systems, much less has been achieved for interacting quantum models in higher dimensions, since they incorporate an additional layer of complexity. In this work, we employ a variational method that allows for an efficient and controlled computation of the dynamics of quantum many-body systems in one and higher dimensions. The approach presented here introduces a variational class of wave functions based on complex networks of classical spins akin to artificial neural networks, which can be constructed in a controlled fashion. We provide a detailed prescription for such constructions and illustrate their performance by studying quantum quenches in one- and two-dimensional models. In particular, we investigate the nonequilibrium dynamics of a genuinely interacting two-dimensional lattice gauge theory, the quantum link model, for which we have recently shown---employing the technique discussed thoroughly in this paper---that it features disorder-free localization dynamics [P. Karpov et al., Phys. Rev. Lett. 126, 130401 (2021)]. The present work not only supplies a framework to address purely theoretical questions but also could be used to provide a theoretical description of experiments in quantum simulators, which have recently seen an increased effort targeting two-dimensional geometries. Importantly, our method can be applied to any quantum many-body system with a well-defined classical limit.
We present a variational approach to quantum field theory based on wavefunctions parameterized by neural networks. While variational methods have a celebrated history across many fields, their application to quantum field theory has been limited by well-known challenges. We show that neural-network wavefunctions, combined with modern machine-learning techniques, enable competitive variational calculations in nontrivial field theories. As a demonstration, we reproduce the essential features of the two-dimensional nonlinear σ-model: asymptotic freedom, dynamical mass generation and the model's step-scaling function.
The possibility to simulate the properties of many-body open quantum systems with a large number of degrees of freedom (d.o.f.) is the premise to the solution of several outstanding problems in quantum science and quantum information. The challenge posed by this task lies in the complexity of the density matrix increasing exponentially with the system size. Here, we develop a variational method to efficiently simulate the nonequilibrium steady state of Markovian open quantum systems based on variational Monte Carlo methods and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the quantum master equation. We test the effectiveness of the method by modeling the two-dimensional dissipative XYZ spin model on a lattice.
The quantum dynamics of many-qubit systems is an outstanding problem that has recently driven significant advances in both numerical methods and programmable quantum processing units. In this work, we employ a comprehensive toolbox of state-of-the-art numerical approaches to classically simulate the dynamics of the two-dimensional transverse field Ising model. Our methods include three different tensor network techniques -- matrix product states, tree-tensor networks, and two-dimensional tensor-networks under the belief propagation approximation -- as well as time-dependent variational Monte Carlo with Neural Quantum States. We focus on two paradigmatic dynamical protocols: (i) quantum annealing through a critical point and (ii) post-quench dynamics. Our extensive results show the quantitative predictions of various state-of-the-art numerical methods providing a benchmark for future numerical investigations and experimental studies with the aim to push the limitations on classical and QPUs. In particular, our work connects classical simulability to different regimes associated with quantum dynamics in Rydberg arrays - namely, quasi-adiabatic dynamics, the Kibble-Zurek mechanism, and quantum quenches.
This thesis investigates quantum many-body systems in and out of equilibrium, with a dual focus on open-system theory and advanced computational methods. Taken together, the results articulate a unifying perspective: in both classical/analog simulations and digital algorithms for many-body quantum systems, the relevant physics often concentrates in small, dynamically selected pockets of the Liouville or Hilbert space. By exposing this structure - whether as noise-protected logical manifolds in bosonic modes, low-rank sectors of the density operator during open evolution, or expressive submanifolds learned by neural networks - and by designing algorithms that operate within these reduced subspaces, the thesis advances hardware-aware encoding strategies, scalable open-system simulation, and dependable neural-network-based variational Monte Carlo methods. The manuscript presents three original contributions: (i) a Liouville-space framework that leverages symmetries and spectral diagnostics to engineer dissipation as a resource for bosonic quantum information; (ii) a low-rank, time-dependent variational method for simulating Lindblad dynamics with adaptive error control; and (iii) methodological advances for Neural Quantum States that stabilize Monte-Carloâ driven optimization and enable reliable ground-state and dynamical calculations. These three directions structure the three main chapters of the thesis. Each chapter includes a focused, self-contained review of the necessary background, not as a textbook survey but as a coherent theoretical scaffold framing the proposed advances.