Classification of Qubit Cellular Automata on Hypercubic Lattices

Area of Science:

• Quantum Information Science
• Condensed Matter Physics
• Theoretical Computer Science

Background:

• Quantum cellular automata (QCA) are quantum dynamical systems on lattices.
• Understanding their structure and dynamics is crucial for quantum computing and simulation.
• Previous work has focused on specific QCA models, lacking a general classification.

Purpose of the Study:

• To classify quantum cellular automata (QCA) on hypercubic lattices (Z^s) with von Neumann neighborhoods.
• To determine the conditions under which these automata are realizable as finite-depth quantum circuits.
• To analyze the entanglement dynamics generated by these QCA.

Main Methods:

• Classification of QCA based on their structure and realizability as quantum circuits.
• Simulation of QCA evolution for a few time steps.
• Quantification of entanglement production between a central cell and its neighborhood.

Main Results:

• The most general structure of QCA realizable as finite-depth quantum circuits is characterized.
• Simulations reveal the dynamics of QCA evolution.
• The rate of entanglement production between a cell and its surroundings is evaluated.

Conclusions:

• The study provides a comprehensive classification of a significant class of quantum cellular automata.
• The findings offer insights into the construction and behavior of quantum dynamical systems.
• This work contributes to understanding entanglement generation in quantum many-body systems.