We study tensor network states defined on an underlying graph which is sparsely connected. Generic sparse graphs are expander graphs with a high probability, and one can represent volume law entangled states efficiently with only polynomial resources. We find that message-passing inference algorithms such as belief propagation can lead to efficient computation of local expectation values for a class of tensor network states defined on sparse graphs. As applications, we study local properties of square root states, graph states, and also employ this method to variationally prepare ground states of gapped Hamiltonians defined on generic graphs. Using the variational method we study the phase diagram of the transverse field quantum Ising model defined on sparse expander graphs.
We study tensor network states defined on an underlying graph which is sparsely connected. Generic sparse graphs are expander graphs with a high probability, and one can represent volume law entangled states efficiently with only polynomial resources. We find that message-passing inference algorithms such as belief propagation can lead to efficient computation of local expectation values for a class of tensor network states defined on sparse graphs. As applications, we study local properties of square root states, graph states, and also employ this method to variationally prepare ground states of gapped Hamiltonians defined on generic graphs. Using the variational method we study the phase diagram of the transverse field quantum Ising model defined on sparse expander graphs.
When studying interacting systems, computing their statistical properties is a fundamental problem in various fields such as physics, applied mathematics, and machine learning. However, this task can be quite challenging due to the exponential growth of the state space as the system size increases. Many standard methods have significant weaknesses. For instance, message-passing algorithms can be inaccurate and even fail to converge due to short loops, while tensor network methods can have exponential computational complexity in large graphs due to long loops. In this Letter, we propose a new method called "tensor network message passing." This approach allows us to compute local observables like marginal probabilities and correlations by combining the strengths of tensor networks in contracting small subgraphs with many short loops and the strengths of message-passing methods in globally sparse graphs, thus addressing the crucial weaknesses of both approaches. Our algorithm is exact for systems that are globally treelike and locally dense-connected when the dense local graphs have a limited tree width. We have conducted numerical experiments on synthetic and real-world graphs to compute magnetizations of Ising models and spin glasses, and have demonstrated the superiority of our approach over standard belief propagation and the recently proposed loopy message-passing algorithm. In addition, we discuss the potential applications of our method in inference problems in networks, combinatorial optimization problems, and decoding problems in quantum error correction.
The classical simulation of many-body quantum systems is an essential tool in understanding many fundamental aspects of condensed matter physics. But a major obstacle arises from the number of degrees of freedom involved in describing such systems, which is exponential in the system size. Recently, however, a class of numerical techniques based on structures called "tensor networks" has emerged, which allows many "typical" quantum states (such as the ground states of gapped, local Hamiltonians) to be represented much more efficiently. In this work we extend and apply these techniques to consider several central topics in quantum many-body physics. After reviewing the relevant background material from the field of tensor networks and tensor network states, we demonstrate a method for computing high order moments and cumulants of operators with respect to such states, including the so-called "Binder cumulant," a powerful tool for detecting phase transitions. Next, we employ tensor network algorithms to characterize the ground state phase diagram of a quantum spin model, including both symmetry-breaking phases and symmetry protected topological order, and find a signicant variety of phases and phase transitions. Finally, we consider the entanglement properties of quantum states exhibiting many-body localization, using a combination of exact diagonalization and tensor network techniques. | 231 pages
The study of quantum circuit simulation using classical computers is a key research topic that helps define the boundary of verifiable quantum advantage, solve quantum many-body problems, and inform development of quantum hardware and software. Tensor networks have become forefront mathematical tools for these tasks. Here we introduce a method to approximately simulate quantum circuits using sparsely-populated tensors. We describe a sparse tensor data structure that can represent quantum states with no underlying symmetry, and outline algorithms to efficiently contract and truncate these tensors. We show that the data structure and contraction algorithm are efficient, leading to expected runtime scalings versus qubit number and circuit depth. Our results motivate future research in optimization of sparse tensor networks for quantum simulation.
We introduce a change of perspective on tensor network states that is defined by the computational graph of the contraction of an amplitude. The resulting class of states, which we refer to as tensor network functions, inherit the conceptual advantages of tensor network states while removing computational restrictions arising from the need to converge approximate contractions. We use tensor network functions to compute strict variational estimates of the energy on loopy graphs, analyze their expressive power for ground states, show that we can capture aspects of volume law time evolution, and provide a mapping of general feed-forward neural nets onto efficient tensor network functions. Our work expands the realm of computable tensor networks to ones where accurate contraction methods are not available, and opens up new avenues to use tensor networks.
Tensor network states form a variational ansatz class widely used, both analytically and numerically, in the study of quantum many-body systems. It is known that if the underlying graph contains a cycle, e.g., as in projected entangled pair states, then the set of tensor network states of given bond dimension is not closed. Its closure is the tensor network variety. Recent work has shown that states on the boundary of this variety can yield more efficient representations for states of physical interest, but it remained unclear how to systematically find and optimize over such representations. We address this issue by defining an ansatz class of states that includes states at the boundary of the tensor network variety of given bond dimension. We show how to optimize over this class in order to find ground states of local Hamiltonians by only slightly modifying standard algorithms and code for tensor networks. We apply this method to different models and observe favorable energies and runtimes when compared with standard tensor network methods.
Belief propagation (BP) provides a scalable heuristic for contracting tensor networks on loopy graphs, but its success in quantum many-body settings has largely rested on empirical evidence. Developing upon a recently introduced cluster-expansion framework for tensor networks, we rigorously study the applicability of BP to many-body quantum systems. For a state represented as a PEPS satisfying a ``loop-decay" condition, we prove that BP supplemented by cluster corrections approximates local observables with exponentially small relative error, and we give explicit formulas expressing local expectation values as BP predictions dressed by connected clusters intersecting the observable region. This representation establishes a direct link between cluster corrections and physical correlation functions. As a result, we show that ``loop-decay" \emph{necessarily implies} exponential decay of connected correlations, yielding sharp, rigorous criteria for when BP can and cannot succeed, and ruling out its validity at critical points. Numerical simulations of the two- and three-dimensional transverse field Ising model at zero and finite temperature confirm our analytical predictions, demonstrating quantitative accuracy deep in gapped phases and systematic failure near criticality.