Dynamics of disordered quantum systems with two- and three-dimensional tensor networks (2503.05693v3)

Abstract: Quantum spin glasses form a good testbed for studying the performance of various quantum annealing and optimization algorithms. In this work we show how two- and three-dimensional tensor networks can accurately and efficiently simulate the quantum annealing dynamics of Ising spin glasses on a range of lattices. Such dynamics were recently simulated using D-Wave's Advantage$2$ system [A. D. King et al, Science, 10.1126/science.ado6285 (2025)] and, following extensive comparisons to existing numerical methods, claimed to be beyond the reach of classical computation. Here we show that by evolving lattice-specific tensor networks with simple belief propagation to keep up with the entanglement generated during the time evolution and then extracting expectation values with more sophisticated variants of belief propagation, state-of-the-art accuracies can be reached with modest computational resources. We exploit the scalability of our simulations and simulate a system of over $300$ qubits, allowing us to verify the universal physics present and extract a value for the associated Kibble-Zurek exponent which agrees with recent values obtained in literature. Our results demonstrate that tensor networks are a viable approach for simulating large scale quantum dynamics in two and three dimensions on classical computers, and algorithmic advancements are expected to expand their applicability going forward.

Overview of Dynamics of Disordered Quantum Systems with Two- and Three-Dimensional Tensor Networks

The paper under discussion addresses the challenges of simulating the quantum dynamics of disordered systems, specifically focusing on quantum annealing processes in Ising spin glasses across various lattice geometries. This work leverages two- and three-dimensional tensor networks, incorporating methods such as belief propagation (BP) and loop corrections, to achieve efficient and accurate simulations. These tensor networks capitalize on predefined lattice structures to model the complex behavior inherent in these quantum systems, providing a scalable method for classical computation of quantum dynamics without requiring prohibitive computational resources.

The authors begin by setting the stage for their research—highlighting that simulating the dynamics of many-body quantum systems in higher dimensions is a crucial challenge in physics. The paper contrasts the efficiency of classical computational methods with the scalability issues they typically encounter, particularly when addressing entanglement growth during simulations of non-equilibrium quantum systems. This work proposes an alternative solution utilizing tensor networks, which can mitigate some of these computational difficulties.

Methodology and Technical Contributions

The central methodological contribution involves utilizing tensor networks whose structure aligns with the geometry of underlying lattices—essentially encoding the wavefunction into a format amenable to simulation. The authors adopt a belief propagation-based approach to simulate these systems, evolving tensor networks with BP during time evolution and applying advanced BP variants like loop corrections to measure observables post-evolution.

The simulation strategy hinges on leveraging the flexibility and efficiency of BP algorithms to cope with and maintain fidelity despite entanglement growth, thereby efficiently managing the computational resources. This approach stands in contrast to other classical simulation techniques and recent quantum annealing results using D-Wave's system, ostensibly allowing simulations at scales previously deemed intractable for classical approaches.

Numerical Results and Implications

The authors present a compelling series of numerical results underscoring the efficacy of their approach. Key findings demonstrate that with appropriate computational resources, the errors in simulated correlators are significantly lower than those achieved by the D-Wave quantum annealer, particularly for the cylindrical and diamond cubic lattices. While the error for dimerized cubic lattices is comparable, the simulation still proves viable at larger scales.

Importantly, the paper leverages its scalable simulation capabilities to observe a collapse in the correlation function when crossing a dynamical phase transition, facilitating the extraction of the Kibble-Zurek exponent. This result aligns well with contemporary literature and underscores the potential of tensor network methods to yield insights into critical phenomena in disordered quantum systems.

Practical and Theoretical Implications

The theoretical implications of this research extend to validating tensor networks as a powerful tool for probing quantum dynamics in complex systems. Practically, the methodology can align with applications related to solving optimization problems through simulated quantum annealing, simulating local spin Hamiltonian dynamics, and exploring quantum circuit dynamics in two-dimensional superconducting processors.

Further, the approach suggests that the classical simulation landscape can continue to evolve through leveraging scalable, message-passing based tensor networks. The potential for future developments includes enhancements via optimized belief propagation algorithms, leveraging increased computational power, and utilizing GPUs or other accelerative computing technologies.

Conclusion

In summary, the paper provides a robust framework demonstrating that tensor networks, informed by effective belief propagation and loop correction techniques, are well-suited for simulating dynamics in higher-dimensional disordered quantum systems. This work not only advances the understanding of quantum simulation capabilities but also opens doors for exploring computationally efficient simulations of other complex quantum systems, potentially extending beyond the methodologies currently ascribed to quantum annealing benchmarks.