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

TL;DR

This paper proves that an exact internal $Z_2$ electromagnetic duality in the 2D $Z_2$ toric code cannot be Clifford, revealing a fundamental link between duality algebra and circuit complexity.

Contribution

It establishes a rigorous no-go theorem for Clifford realizations of exact internal electromagnetic duality in the toric code.

Findings

Clifford duality cannot realize exact internal $Z_2$ symmetry.

Exact internal $Z_2$ duality must be non-Clifford.

Clifford realizations imply a $Z_{2^m}$ algebra with $m extgreater 1$.

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.