The resource theory of tensor networks

Abstract

Tensor networks provide succinct representations of quantum many-body states and are an important computational tool for strongly correlated quantum systems. Their expressive and computational power is characterized by an underlying entanglement structure, on a lattice or more generally a (hyper)graph, with virtual entangled pairs or multipartite entangled states associated to (hyper)edges. Changing this underlying entanglement structure into another can lead to both theoretical and computational benefits. We study a natural resource theory which generalizes the notion of bond dimension to entanglement structures using multipartite entanglement. It is a direct extension of resource theories of tensors studied in the context of multipartite entanglement and algebraic complexity theory, allowing for the application of the sophisticated methods developed in these fields to tensor networks. The resource theory of tensor networks concerns both the local entanglement structure of a quantum many-body state and the (algebraic) complexity of tensor network contractions using this entanglement structure. We show that there are transformations between entanglement structures which go beyond edge-by-edge conversions, highlighting efficiency gains of our resource theory that mirror those obtained in the search for better matrix multiplication algorithms. We also provide obstructions to the existence of such transformations by extending a variety of methods originally developed in algebraic complexity theory for obtaining complexity lower bounds. The resource theory of tensor networks allows to compare different entanglement structures and should lead to more efficient tensor network representations and contraction algorithms.

66 pages

References in corpus (77)

• Three qubits can be entangled in two inequivalent ways
• The density-matrix renormalization group
• Area laws for the entanglement entropy - a review
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Symmetry protected topological orders and the group cohomology of their symmetry group
• An Area Law for One Dimensional Quantum Systems
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
• Classifying quantum phases using Matrix Product States and PEPS
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Four qubits can be entangled in nine different ways
• Stripe order in the underdoped region of the two-dimensional Hubbard model
• Tensor networks for complex quantum systems
• Holographic duality from random tensor networks
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• Simulating quantum computation by contracting tensor networks
• Entropy scaling and simulability by Matrix Product States
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Gapless spin-liquid ground state in the $S=1/2$ kagome antiferromagnet
• The computational complexity of PEPS
• Towards Quantum Machine Learning with Tensor Networks
• PEPS as ground states: degeneracy and topology
• Simulating Large Quantum Circuits on a Small Quantum Computer
• Equivalence of restricted Boltzmann machines and tensor network states
• Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
• Algorithms for finite Projected Entangled Pair States
• Hyper-optimized tensor network contraction
• Tensor networks for Lattice Gauge Theories and Atomic Quantum Simulation
• Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
• Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm
• Efficient classical simulation of random shallow 2D quantum circuits
• Resonating valence bond states in the PEPS formalism
• Efficient tensor network simulation of IBM's Eagle kicked Ising experiment
• Tensor renormalization of quantum many-body systems using projected entangled simplex states
• Tensor Network Algorithms: a Route Map
• Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure
• Learning Relevant Features of Data with Multi-scale Tensor Networks
• Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States
• Supervised Learning with Quantum-Inspired Tensor Networks
• Tensor Networks and Quantum Error Correction
• A canonical form for Projected Entangled Pair States and applications
• Phase diagram of the Shastry-Sutherland Compound SrCu2(BO3)2 under extreme combined conditions of field and pressure
• Contracting Arbitrary Tensor Networks: General Approximate Algorithm and Applications in Graphical Models and Quantum Circuit Simulations
• Tensor rank is not multiplicative under the tensor product
• Supervised Learning with Projected Entangled Pair States
• Contracting projected entangled pair states is average-case hard
• Necessary and sufficient conditions for local manipulation of multipartite pure quantum states
• An area law for 2D frustration-free spin systems
• Fast counting with tensor networks
• A generalization of the injectivity condition for Projected Entangled Pair States
• Continuous Tensor Network States for Quantum Fields
• The Tensor Rank of the Tripartite State ${|W⟩}^{\otimes n}$}
• Tensor-network codes
• Hyper-optimized approximate contraction of tensor networks with arbitrary geometry
• Random tensor networks with nontrivial links
• General tensor network decoding of 2D Pauli codes
• Projected Entangled Pair States: Fundamental analytical and numerical limitations
• Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning
• Hypergraph min-cuts from quantum entropies
• Border rank is not multiplicative under the tensor product
• Entanglement distillation from Greenberger-Horne-Zeilinger shares
• Optimization at the boundary of the tensor network variety
• Barriers for Rank Methods in Arithmetic Complexity
• Tensor network representations from the geometry of entangled states
• The minimal canonical form of a tensor network
• Asymptotic tensor rank of graph tensors: beyond matrix multiplication
• Dimension of Tensor Network varieties
• From probabilistic graphical models to generalized tensor networks for supervised learning
• 2D Multipartite Valence Bond States in Quantum Antiferromagnets
• Tensor surgery and tensor rank
• Generative modeling with projected entangled-pair states
• A full dichotomy for Holant${}^c$, inspired by quantum computation
• Privacy-preserving machine learning with tensor networks
• The Linear Span of Uniform Matrix Product States
• Geometric rank of tensors and subrank of matrix multiplication

Cited by in corpus (2)

• The minimal canonical form of a tensor network
• The product structure of MPS-under-permutations