Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact $\mathbb{Z}_2$ electromagnetic duality of $\mathbb{Z}_2$ toric code is non-Clifford

Ryohei Kobayashi

Abstract

The 2D $\mathbb{Z}_2$ toric code admits a global symmetry exchanging electric and magnetic quasiparticles, known as electromagnetic duality. Known realizations include lattice translation symmetry, an exact $\mathbb{Z}_4$ symmetry generated by a Clifford circuit, and an exact $\mathbb{Z}_2$ symmetry generated by a non-Clifford circuit. We show that a Clifford electromagnetic duality cannot realize an exact internal $\mathbb{Z}_2$ symmetry. This is proved rigorously for symmetries with coarse translation invariance by $l$ lattice units for generic odd $l$. Therefore an exact internal $\mathbb{Z}_2$ electromagnetic duality must be non-Clifford, whereas generic internal Clifford realization necessarily has $\mathbb{Z}_{2^m}$ algebra with $m\ge 2$. Our result suggests an unexpected connection between the algebra of exact electromagnetic duality and Clifford hierarchy of circuits.

Exact $\mathbb{Z}_2$ electromagnetic duality of $\mathbb{Z}_2$ toric code is non-Clifford

Abstract

The 2D toric code admits a global symmetry exchanging electric and magnetic quasiparticles, known as electromagnetic duality. Known realizations include lattice translation symmetry, an exact symmetry generated by a Clifford circuit, and an exact symmetry generated by a non-Clifford circuit. We show that a Clifford electromagnetic duality cannot realize an exact internal symmetry. This is proved rigorously for symmetries with coarse translation invariance by lattice units for generic odd . Therefore an exact internal electromagnetic duality must be non-Clifford, whereas generic internal Clifford realization necessarily has algebra with . Our result suggests an unexpected connection between the algebra of exact electromagnetic duality and Clifford hierarchy of circuits.

Paper Structure

This paper contains 1 section, 1 theorem, 75 equations, 2 figures.

Table of Contents

  1. Condensation defect of ${\mathbb Z}_2$ gauge theory and electromagnetic duality

Key Result

Theorem 1

Let $l$ be odd. There exists no Clifford $\mathbb{Z}_2$ symmetry of the square-lattice toric code Hamiltonian that Equivalently, for odd $l$ there is no matrix such that where and $U\in \mathrm{Mat}_{l^2}(R_l)$ is the blocked representation of multiplication by a monomial $u=x^a y^b$.

Figures (2)

  • Figure 1: A finite-depth circuit $U=U_B U_G U_R$ of the $\mathbb{Z}_2$ toric code generates an exact $\mathbb{Z}_4$ Clifford symmetry exchanging $e$ and $m$Barkeshli:2022wuz. On the checkerboard lattice, each plaquette supports a toric code stabilizer, as shown in the lower-right. Plaquettes of one color support $X$ stabilizers, while plaquettes of the other color support $Z$ stabilizers. Under conjugation by $U$, the $X$ and $Z$ stabilizers are exchanged. The $\mathbb{Z}_4$ algebra of this operator is discussed explicitly in Ref. shirley2025QCA.
  • Figure 2: A square lattice with a single qubit on each edge.

Theorems & Definitions (2)

  • Theorem 1
  • proof