Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classification of locality preserving symmetries on spin chains

Alex Bols, Wojciech De Roeck, Michiel De Wilde, Bruno de O. Carvalho

Abstract

We consider the action of a finite group $G$ by locality preserving automorphisms (quantum cellular automata) on quantum spin chains. We refer to such group actions as ``symmetries''. The natural notion of equivalence for such symmetries is \emph{stable equivalence}, which allows for stacking with factorized group actions. Stacking also endows the set of equivalence classes with a group structure. We prove that the anomaly of such symmetries provides an isomorphism between the group of stable equivalence classes of symmetries with the cohomology group $H^3(G,U(1))$, consistent with previous conjectures. This amounts to a complete classification of locality preserving symmetries on spin chains. We further show that a locality preserving symmetry is stably equivalent to one that can be presented by finite depth quantum circuits with covariant gates if and only if the slant product of its anomaly is trivial in $H^2(G, U(1)[G])$.

Classification of locality preserving symmetries on spin chains

Abstract

We consider the action of a finite group by locality preserving automorphisms (quantum cellular automata) on quantum spin chains. We refer to such group actions as ``symmetries''. The natural notion of equivalence for such symmetries is \emph{stable equivalence}, which allows for stacking with factorized group actions. Stacking also endows the set of equivalence classes with a group structure. We prove that the anomaly of such symmetries provides an isomorphism between the group of stable equivalence classes of symmetries with the cohomology group , consistent with previous conjectures. This amounts to a complete classification of locality preserving symmetries on spin chains. We further show that a locality preserving symmetry is stably equivalent to one that can be presented by finite depth quantum circuits with covariant gates if and only if the slant product of its anomaly is trivial in .

Paper Structure

This paper contains 27 sections, 16 theorems, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setup and main result
  3. Spin chains, quantum cellular automata, and finite depth quantum circuits
  4. Locality preserving symmetries
  5. Equivalence and stable equivalence
  6. Main result
  7. The anomaly of a locality preserving symmetry
  8. Examples
  9. Right restrictions and covariance
  10. Right restrictions that are group homomorphisms
  11. Covariant right restrictions
  12. An invariant for symmetries that admit covariant right restrictions
  13. Covariant right restrictions that are group homomorphisms
  14. Proof of the main Theorem \ref{['thm:classification']}
  15. Injectivity of $\Omega$
  16. ...and 12 more sections

Key Result

Theorem 2.1

The monoid $(\mathop{\mathrm{\mathsf{Sym}}}\nolimits_G/\sim)$ is in fact a group. There is a map $\Omega : \mathop{\mathrm{\mathsf{Sym}}}\nolimits_G \rightarrow H^3(G, U(1))$ which assigns to each symmetry $\alpha$ a 3-cohomology class, which we will call its anomaly, and which lifts to an isomorphi Moreover, for each $[\omega] \in H^3(G, U(1))$ there exists a symmetry whose anomaly is $[\omega]$.

Figures (4)

  • Figure 1: The FDQC defining $\alpha^{(g)}$.
  • Figure 2: The FDQC (in red) defining $\alpha_I^{(g)}$ for $I = [a, b]$.
  • Figure 3: The symmetry $\alpha'$ can be seen as a conjugation by a FDQC $\prod\limits_{j\in \mathbb Z} U_j(g)$.
  • Figure 4: The unitaries $\tilde{U}_j(g)$ are defined as tensor products of $U_j(g)$ with projective representations $\overline U_j(g)$ defined on appropriate degrees of freedom $\mathcal{C}_j,\mathcal{C}_{j+1}$.

Theorems & Definitions (35)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • ...and 25 more