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Quantum cellular automata are a coarse homology theory

Matthias Ludewig

Abstract

We show that quantum cellular automata naturally form the degree-zero part of a coarse homology theory. The recent result of Ji and Yang that the space of QCA forms an Omega-spectrum in the sense of algebraic topology is a direct consequence of the formal properties of coarse homology theories.

Quantum cellular automata are a coarse homology theory

Abstract

We show that quantum cellular automata naturally form the degree-zero part of a coarse homology theory. The recent result of Ji and Yang that the space of QCA forms an Omega-spectrum in the sense of algebraic topology is a direct consequence of the formal properties of coarse homology theories.
Paper Structure (13 sections, 22 theorems, 108 equations)

This paper contains 13 sections, 22 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Nets and QCA on coarse spaces
  4. Preliminaries on coarse spaces
  5. Nets on coarse spaces
  6. Local nets and coarse maps
  7. Tensor factors and homomorphisms
  8. Quantum cellular automata and quantum circuits
  9. A coarse homology theory
  10. QCA and $K$-theory
  11. The Mayer-Vietoris axiom
  12. Definition of the coarse homology theory
  13. QCA and Azumaya nets

Key Result

Theorem 1.1

There is a coarse homology theory $Q$ such that the degree zero homotopy group yields quantum cellular automata for spaces of bounded geometry.

Theorems & Definitions (79)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: bornological coarse spaces BunkeEngel2020
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7: coarse maps
  • Definition 2.8: closeness
  • Example 2.9
  • ...and 69 more