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An index for quantum cellular automata on fusion spin chains

Corey Jones, Junhwi Lim

Abstract

Interpreting the GNVW index for 1D quantum cellular automata (QCA) in terms of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, and as the boundary operators of 2D topological codes. We show that for the fusion spin chains built from the fusion category $\mathbf{Fib}$, the index is a complete invariant for the group of QCA modulo finite depth circuits.

An index for quantum cellular automata on fusion spin chains

Abstract

Interpreting the GNVW index for 1D quantum cellular automata (QCA) in terms of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, and as the boundary operators of 2D topological codes. We show that for the fusion spin chains built from the fusion category , the index is a complete invariant for the group of QCA modulo finite depth circuits.
Paper Structure (9 sections, 18 theorems, 56 equations)

This paper contains 9 sections, 18 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. QCA on abstract spin chains
  3. Physical motivation
  4. Index for QCA on abstract spin chains
  5. Generalized translations on fusion spin chains
  6. Concrete spin chains
  7. Generalized Kramers-Wannier translations from $G$-graded extensions
  8. Index as an isomorphism for the $\textbf{Fib}$ chain
  9. Toward general topological invariants of QCA

Key Result

Theorem A

Let $A$ be an abstract spin chain that satisfies the finite index property (Definition FiniteIndexProperty). Then from Definition indef is a homomorphism containing $\mathop{\mathrm{\mathbf{FDQC}}}\nolimits(A)$ in its kernel.

Theorems & Definitions (47)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • ...and 37 more