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arXiv:2510.01543 · Communications Physics

Variational approach to open quantum systems with long-range competing interactions

Dawid A. Hryniuk, Marzena Hanna Szymańska

Published Jan 10, 2026• 1 citation
arXivHTMLPDF

Abstract

Competition between short- and long-range interactions underpins many emergent phenomena in nature. Despite rapid progress in their experimental control, computational methods capable of accurately simulating open quantum many-body systems with complex long-ranged interactions at scale remain scarce. Here, we address this limitation by introducing an efficient and scalable approach to dissipative quantum lattices in one and two dimensions, combining matrix product operators and time-dependent variational Monte Carlo. We showcase the versatility, effectiveness, and unique methodological advantages of our algorithm by simulating the non-equilibrium dynamics and steady states of spin-$$\frac{1}{2}$$ lattices with competing algebraically-decaying interactions for as many as N = 200 sites, revealing the emergence of spatially-modulated magnetic order far from equilibrium. This approach offers promising prospects for advancing our understanding of the complex non-equilibrium properties of a diverse variety of experimentally-realizable quantum systems with long-ranged interactions, including Rydberg atoms, ultracold dipolar molecules, and trapped ions. Far-from-equilibrium many-body quantum systems with complex non-local interactions are of significant fundamental and practical importance. Here the authors demonstrate how variational Monte Carlo in combination with compact tensor network parameterization of the density matrix enables accurate and scalable simulation of such open quantum systems.

DOI
10.1038/s42005-025-02479-2

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We introduce a novel method of efficiently simulating the non-equilibrium steady state of large many-body open quantum systems with highly non-local interactions, based on a variational Monte Carlo optimization of a matrix product operator ansatz. Our approach outperforms and offers several advantages over comparable algorithms, such as an improved scaling of the computational cost with respect to the bond dimension for periodic systems. We showcase the versatility of our approach by studying the phase diagrams and correlation functions of the dissipative quantum Ising model with collective dephasing and long-ranged power law interactions for spin chains of up to N=100 spins.
In variational Monte Carlo (VMC) calculations of NNN-site quantum systems with arbitrary all-to-all two-body interactions, evaluating the local energy generally costs O(N3)O(N^3)O(N3). We introduce a new framework that reduces this cost to O(N)O(N)O(N) for tensor network states, capable of scalable and accurate computation of real-time dynamics and ground states. As a result, we obtain accurate simulations of the adiabatic real-time protocol of a 10×1010\times1010×10 dipolar XY model realized in a Rydberg simulator [C. Chen et al., Nature 616, 691 (2023)], which was previously beyond the reach of classical simulation. Going beyond quantum experiments, we also directly perform ground state VMC to compare with the adiabatic state preparation. Our work demonstrates tensor network VMC as a powerful classical simulator for long-range quantum platforms such as Rydberg and ion-trap simulators, which are currently in urgent need of scalable classical benchmarking tools. As a separate technical contribution, we resolve the pathology of evolving from product states within of tensor network VMC.
We introduce a method to simulate open quantum many-body dynamics by combining time-dependent variational Monte Carlo (tVMC) with quantum trajectory techniques. Our approach unravels the Lindblad master equation into an ensemble of stochastic Schrödinger equations for a variational ansatz, avoiding the exponential cost of density matrix evolution. The method is compatible with generic ansatzë, including expressive neural-network wavefunctions. We derive the nonlinear stochastic equations of motion for the variational parameters and employ suitable Stratonovich numerical solvers. To validate our approach, we simulate quenches in the locally dissipative long-range Ising model in a transverse field, accurately capturing non-equilibrium magnetization and spin squeezing dynamics relevant to trapped-ion and Rydberg atom experiments. The framework is computationally efficient, scalable on high-performance computing platforms, and can be readily integrated into existing tVMC implementations. This work enables large-scale simulations of complex, dissipative quantum systems in higher dimensions, with broad implications for quantum technology and fundamental science.
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.
Many-body open quantum systems are exposed to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. As the exact treatment of such models is generally unfeasible, it is favourable to formulate an approximate description by means of the dynamics of the reduced density operator of the system. When studying models with a weak coupling to a memoryless environment, the existence of such solution is granted and the reduced dynamics is governed by the Lindblad quantum master equation. In recent years, open quantum systems have evolved into a major field of studies. Focus of these studies are the characterization of emergent phenomena and dissipative phase transitions, as well as the ongoing debate about whether quantum computing schemes are still hard to simulate classically --- and thus achieve quantum supremacy --- when in presence of some degree of noise-induced decoherence. However, the challenge posed by this task lies in the complexity of the density matrix that increases exponentially with the system size, and the quest for efficient numerical methods is a research field that is still in its infancy. Here, we first develop a real-time full configuration interaction quantum Monte Carlo technique that stems from a class of methods generally known as projector Monte Carlo. The approach enables the stochastic sampling of the Lindblad time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the non-equilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce statistical error. We demonstrate the efficiency of the approach by applying it to the dissipative two-dimensional XYZ spin-1/2 model on a lattice. As the importance sampling of projector approaches is often combined with variational results, we then introduce a novel method that is based on the 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 Lindblad quantum master equation. We test the effectiveness of the method by modeling the steady state of the dissipative two-dimensional XYZ spin-1/2 model through a dissipative phase transition, and also the real-time dynamics of the dissipative Ising model. In addition, we discuss the application of the developed methods and the open questions of the field.
The rapid development of quantum computers has enabled demonstrations of quantum advantages on various tasks. However, real quantum systems are always dissipative due to their inevitable interaction with the environment, and the resulting non-unitary dynamics make quantum simulation challenging with only unitary quantum gates. In this work, we present an innovative and scalable method to simulate open quantum systems using quantum computers. We define an adjoint density matrix as a counterpart of the true density matrix, which reduces to a mixed-unitary quantum channel and thus can be effectively sampled using quantum computers. This method has several benefits, including no need for auxiliary qubits and noteworthy scalability. Moreover, some long-time properties like steady states and the thermal equilibrium can also be investigated as the adjoint density matrix and the true dissipated one converge to the same state. Finally, we present deployments of this theory in the dissipative quantum XY model for the evolution of correlation and entropy with short-time dynamics and the disordered Heisenberg model for many-body localization with long-time dynamics. This work promotes the study of real-world many-body dynamics with quantum computers, highlighting the potential to demonstrate practical quantum advantages.