Theory Seminar: Taming the many-body Hilbert Space: Tensor Networks and Belief Propagation

Siddhant Midha
Princeton University
Quantum Info & Computing Seminar, at the Seminar Room (Online)
Wednesday, July 29, 14:30 CET

Many-body quantum systems derive their richness from an exponentially large Hilbert space, but this very structure makes them notoriously difficult to study. Tensor networks (TN) provide a language for reasoning about entanglement and correlations in many-body quantum systems, as well as state-of-the-art numerical tools for their study. Tensor network contraction on arbitrary graphs is thus a fundamental computational challenge with applications ranging from quantum simulation to quantum error correction. Belief propagation (BP), a technique originally developed in statistical inference, has recently been shown to be a candidate for TN contraction. Here, we develop a rigorous theoretical framework for BP in tensor networks, leveraging insights from statistical mechanics to devise a cluster expansion that systematically improves the BP approximation. We prove that the cluster expansion converges exponentially fast if an object called the “loop tensor” decays sufficiently fast with the loop size, giving a rigorous error bound on BP. We then provide efficient algorithms for computing local expectation values on tensors network states. Our work opens the door to a systematic theory of BP for tensor networks, bridging the gap between widely used numerical practice and provable algorithmic performance.