New Algorithms Unlock Faster Sampling of Complex Systems with Tensor Networks

Scientists are tackling the challenge of efficiently sampling probability distributions within complex systems, a crucial step for advancements in areas like Monte Carlo simulations and quantum advantage experiments. Alec Dektor, Eugene Dumitrescu, and Chao Yang, from Lawrence Berkeley and Oak Ridge National Laboratories, present novel algorithms designed to sample two-dimensional isometric tensor network states (isoTNS). Building upon existing methods for one-dimensional tensor networks, their work introduces both an independent sampling technique, providing single configurations with associated probabilities, and a greedy search strategy to identify multiple high-probability configurations.

This research is significant because it extends powerful tensor network methods to higher dimensions, potentially unlocking more accurate and efficient simulations of strongly correlated quantum systems. Researchers have now introduced two novel algorithms for sampling two-dimensional isometric tensor network states, or isoTNS, extending established techniques used for one-dimensional tensor networks. These algorithms address a critical need for efficiently modelling complex quantum systems with limited entanglement. The first algorithm independently samples configurations, yielding a single state alongside its associated probability. Simultaneously, a second algorithm employs a greedy search strategy to pinpoint K high-probability configurations and their corresponding probabilities. Numerical results confirm the effectiveness of both approaches across quantum states exhibiting varying degrees of entanglement and differing system sizes. This work builds upon the mature field of one-dimensional tensor networks, specifically the matrix product state formalism, and adapts its sampling algorithms for application to isoTNS. IsoTNS offer a promising structure for simulating quantum systems beyond one dimension, possessing favourable computational scaling compared to other methods like projected-entangled pair states. Despite advances in isoTNS algorithms for computing norms and local observables, efficient sampling methods have remained unexplored until now. The newly developed algorithms share desirable properties with their 1D counterparts, scaling polynomially with virtual bond dimensions, although they introduce a new source of potential truncation error.

This research quantifies this error through numerical experiments, validating the proposed algorithms and analysing their scaling behaviour. By enabling efficient sampling, this work opens avenues for applications in quantum advantage experiments, quantum Monte Carlo simulations, and the development of quantum digital twins. Furthermore, it provides a crucial subroutine for modelling finite-temperature quantum systems using the minimally entangled typical thermal states algorithm, pushing the boundaries of quantum simulation and computational efficiency. Independent and greedy sampling of two-dimensional isoTNS probability distributions Isometric tensor network states (isoTNS) form the basis of our new sampling algorithms for two-dimensional quantum systems. We developed two distinct approaches to efficiently sample probability distributions encoded within these isoTNS states, extending existing methods used for one-dimensional tensor networks. The first algorithm performs independent sampling, directly yielding a single configuration alongside its corresponding probability value. This method allows for straightforward evaluation of system properties from the generated samples. Alternatively, we implemented a greedy search strategy designed to identify K high-probability configurations and their associated probabilities. This approach prioritises configurations with the greatest likelihood, offering a focused exploration of the most probable system states. Numerical tests were conducted to validate the effectiveness of both algorithms across isoTNS states exhibiting varying degrees of entanglement and differing system sizes. These tests confirmed the algorithms’ ability to accurately represent the underlying probability distributions. The study leverages the computational advantages of isoTNS, specifically its efficient computation of norms, marginal distributions, and local observables once the orthogonality center is correctly positioned. While shifting the orthogonality center in 2D isoTNS is more complex than in 1D matrix product states, requiring multiple singular value decompositions, our algorithms minimise approximation errors inherent in these procedures. This work addresses a gap in the field, as efficient sampling algorithms for isoTNS have previously remained unexplored, despite advances in isoTNS simulations of dynamics, excited states, and thermal states. The algorithms presented here offer a valuable tool for applications ranging from quantum advantage experiments to quantum Monte Carlo simulations and the evaluation of quantum supremacy claims. Novel algorithms efficiently sample probability distributions from two-dimensional isometric tensor networks for machine learning applications Sampling from probability distributions encoded by quantum systems is a crucial computational task with both practical and theoretical implications. The research introduces two novel sampling algorithms for two-dimensional isometric tensor network states (isoTNS), extending established methods for one-dimensional tensor networks. The first algorithm independently samples to yield a single configuration alongside its associated probability. The second algorithm utilises a greedy search strategy to identify K high-probability configurations and their corresponding probabilities. These algorithms share properties with their 1D counterparts, exhibiting polynomial scaling in virtual bond-dimensions. However, the 2D isoTNS algorithms introduce an additional source of truncation error, which the study quantifies through numerical experiments. The work builds upon tensor network notation and algorithms used in the matrix product state (MPS) formalism for generating both independent and clusters of samples. Numerical results demonstrate the effectiveness of these algorithms across quantum states with varying entanglement and system size. The proposed methods enable efficient sampling of high-dimensional probability distributions represented by 2D isoTNS, a tensor network structure with favourable computational scaling compared to general projected-entangled pair states (PEPS), specifically χ7 versus χ10. These advancements facilitate simulations of dynamics, excited states, fermionic systems, thermal states, and string-net liquids. Despite the introduction of truncation error, the algorithms represent a significant step towards efficient sampling within the isoTNS framework. Efficient sampling of two-dimensional isometric tensor network states with novel algorithms is crucial for many-body physics simulations Researchers have developed new algorithms for sampling probability distributions represented by two-dimensional isometric tensor network states (isoTNS). These algorithms extend existing methods used for one-dimensional tensor networks to the more complex two-dimensional case, addressing a key challenge in efficiently simulating large quantum systems. The first algorithm independently samples configurations and calculates their probabilities, while the second employs a greedy search to identify the K most probable configurations alongside their probabilities. Numerical results demonstrate the effectiveness of both algorithms across states exhibiting varying levels of entanglement and different system sizes, including calculations performed on up to 256 qubits for GHZ- and W-states. The study acknowledges that truncation error, arising from limitations in the virtual bond dimension, can affect the accuracy of the estimated probabilities. The authors also note that while the algorithms scale polynomially with system size, virtual bond-dimensions, and K, further refinements to sampling and probability estimation are possible. Future research directions include extending the algorithms to three-dimensional states and exploring techniques like importance sampling, potentially using Metropolis algorithms to improve efficiency. The work also suggests potential applications in Monte Carlo sampling within tensor network quantum many-body methods, including both conventional and tensor-network-based Monte Carlo algorithms such as METTS. These advancements contribute to pushing the boundaries of classical sampling for distributions considered computationally challenging and may play a role in verifying and extending quantum supremacy experiments. 👉 More information 🗞 Sampling two-dimensional isometric tensor network states 🧠 ArXiv: https://arxiv.org/abs/2602.02245 Tags: