Quantum Cellular Automata, Tensor Networks, and Area Laws

Abstract

Quantum Cellular Automata are unitary maps that preserve locality and respect causality. We identify them, in any dimension, with simple tensor networks (PEPU) whose bond dimension does not grow with the system size. As a result, they satisfy an area law for the entanglement entropy they can create. We define other classes of non-unitary maps, the so-called quantum channels, that either respect causality or preserve locality. We show that, whereas the latter obey an area law for the amount of quantum correlations they can create, as measured by the quantum mutual information, the former may violate it. We also show that neither of them can be expressed as tensor networks with a bond dimension that is independent of the system size.

arXiv comment5+2 pages, 2 figures; v3: minor revision

References in corpus (23)

• The density-matrix renormalization group in the age of matrix product states
• Area laws for the entanglement entropy - a review
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Area laws in quantum systems: mutual information and correlations
• Criticality, the area law, and the computational power of PEPS
• Complete classification of 1D gapped quantum phases in interacting spin systems
• Entanglement spectrum and boundary theories with projected entangled-pair states
• Topology and Broken Symmetry in Floquet Systems
• Lecture Notes of Tensor Network Contractions
• Index theory of one dimensional quantum walks and cellular automata
• Chiral Floquet Phases of Many-body Localized Bosons
• Matrix Product Unitaries: Structure, Symmetries, and Topological Invariants
• Floquet topological phases with symmetry in all dimensions
• Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories
• Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
• Interacting invariants for Floquet phases of fermions in two dimensions
• Dynamically enriched topological orders in driven two-dimensional systems
• Entanglement Dynamics in 1D Quantum Cellular Automata
• Classification of Quantum Cellular Automata
• Matrix Product Representation of Locality Preserving Unitaries
• Classification of Matrix-Product Unitaries with Symmetries
• Ergodicity of quantum cellular automata

Cited by in corpus (40)

• A review of Quantum Cellular Automata
• Exact thermalization dynamics in the "Rule 54" Quantum Cellular Automaton
• Quantum Circuits assisted by LOCC: Transformations and Phases of Matter
• Symmetry Protected Topological Order in Open Quantum Systems
• Preparation of matrix product states with log-depth quantum circuits
• A Noisy Approach to Intrinsically Mixed-State Topological Order
• Bounds in Nonequilibrium Quantum Dynamics
• Sequential Quantum Circuits as Maps between Gapped Phases
• Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond
• Topological Phases with Average Symmetries: the Decohered, the Disordered, and the Intrinsic
• Many Body Quantum Chaos and Dual Unitarity Round-a-Face
• Topological lower bound on quantum chaos by entanglement growth
• Crystalline Quantum Circuits
• Classification of interacting Floquet phases with $U(1)$ symmetry in two dimensions
• Fermionic quantum cellular automata and generalized matrix product unitaries
• Tensor networks for interpretable and efficient quantum-inspired machine learning
• A converse to Lieb-Robinson bounds in one dimension using index theory
• Entanglement in the Quantum Game of Life
• Engineering Yang-Lee anyons via Majorana bound states
• Integrable nonunitary quantum circuits
• Many-body entropies and entanglement from polynomially-many local measurements
• Numerical Simulation of Critical Quantum Dynamics without Finite Size Effects
• Coarse-grained Entanglement and Operator Growth in Anomalous Dynamics
• Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors
• Integrable spin chains and the Clifford group
• Gauging anomalous unitary operators
• Dissipative quantum many-body dynamics in (1+1)D quantum cellular automata and quantum neural networks
• Towards topological fixed-point models beyond gappable boundaries
• Matrix-product unitaries: Beyond quantum cellular automata
• Information Flow in Non-Unitary Quantum Cellular Automata
• Quantum cellular automata and categorical dualities of spin chains
• Topological Phases of Many-Body Localized Systems: Beyond Eigenstate Order
• Breaking global symmetries with locality-preserving operations
• A massless interacting Fermionic Cellular Automaton exhibiting bound states
• Photonic cellular automaton simulation of relativistic quantum fields: observation of Zitterbewegung
• Renormalisation of Quantum Cellular Automata
• Continuous matrix product operators for quantum fields
• Exact large deviations and emergent long-range correlations in sequential quantum East circuits
• Anomalous transport in U(1)-symmetric quantum circuits
• Tri-unitary quantum circuits