Categorical Tensor Network States

Abstract

We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state S, we present a new and general method to factor S into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states.

39 pages, 31 figures, published version

References in corpus (24)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• A class of quantum many-body states that can be efficiently simulated
• Renormalization and tensor product states in spin chains and lattices
• Simulation of strongly correlated fermions in two spatial dimensions with fermionic Projected Entangled-Pair States
• Fermionic Projected Entangled Pair States
• Measurement-based quantum computation beyond the one-way model
• Fermionic multi-scale entanglement renormalization ansatz
• Contraction of fermionic operator circuits and the simulation of strongly correlated fermions
• Simulation of interacting fermions with entanglement renormalization
• Sequential Generation of Matrix-Product States in Cavity QED
• Strings, Projected Entangled Pair States, and variational Monte Carlo methods
• The maximally entangled symmetric state in terms of the geometric measure
• Approximating strongly correlated spin and fermion wavefunctions with correlator product states
• Non-perturbative k-body to two-body commuting conversion Hamiltonians and embedding problem instances into Ising spins
• Ground-State Properties of Quantum Many-Body Systems: Entangled-Plaquette States and Variational Monte Carlo
• Density Matrix Renormalization Group in the Heisenberg Picture
• Simulation of anyons with tensor network algorithms
• Exact matrix product solutions in the Heisenberg picture of an open quantum spin chain
• Dynamical simulations of classical stochastic systems using matrix product states
• Anyonic entanglement renormalization
• Sequential Implementation of Global Quantum Operations
• Categorical Quantum Circuits
• Phonon resonances in atomic currents through Bose-Fermi mixtures in optical lattices
• Time-dependent currents of 1D bosons in an optical lattice

Cited by in corpus (22)

• Quantum Entanglement in Deep Learning Architectures
• Tensor Networks for Lattice Gauge Theories with continuous groups
• Tensor network states and algorithms in the presence of a global SU(2) symmetry
• Beyond mean-field bistability in driven-dissipative lattices: bunching-antibunching transition and quantum simulation
• Unifying Neural-network Quantum States and Correlator Product States via Tensor Networks
• Encoding Hypergraphs into Quantum States
• Tensor Network Contractions for #SAT
• Solving search problems by strongly simulating quantum circuits
• Quantum Annealing Algorithms for Boolean Tensor Networks
• Undecidability in Tensor Network States
• Holographic Software for Quantum Networks
• Tensor Network Methods for Invariant Theory
• Computing the Tutte Polynomial of Lattice Path Matroids Using Determinantal Circuits
• Compact Neural-network Quantum State representations of Jastrow and Stabilizer states
• Charged String Tensor Networks
• Neural-network Quantum States for Spin-1 systems: spin-basis and parameterization effects on compactness of representations
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Bell non-locality using tensor networks and sparse recovery
• Seeding neural network quantum states with tensor network states
• Representation Theorem for Matrix Product States
• Tensor networks for quantum causal histories
• Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model