Quantum operations for Kramers-Wannier duality

TL;DR

This paper develops a quantum operation framework for the Kramers-Wannier duality in the transverse-field Ising model, enabling explicit duality implementation on quantum computers with a detailed operator construction.

Contribution

It introduces a superoperator formalism for the duality, providing an explicit operator-sum representation and connecting duality with quantum operations.

Findings

Constructed a superoperator mapping Ising to dual-Ising operators

Reproduced known fusion rules within the quantum operation framework

Provided a protocol for implementing duality on quantum computers

Abstract

We study the Kramers-Wannier duality for the transverse-field Ising lattice on a ring. A careful consideration of the ring boundary conditions shows that the duality has to be implemented with a proper treatment of different charge sectors of both the twisted and untwisted Ising and the dual-Ising Hilbert spaces. We construct a superoperator that explicitly maps the Ising operators to the dual-Ising operators. The superoperator naturally acts on the tensor product of the Ising and the dual-Ising Hilbert space. We then show that the relation between our superoperator and the Kramers-Wannier duality operator that maps the Ising Hilbert space to the dual-Ising Hilbert space is naturally provided by quantum operations and the duality can be understood as a quantum operation that we construct. We provide the operator-sum representation for the Kramers-Wannier quantum operations and reproduce the well-known fusion rules. In addition to providing the quantum information perspective on the Kramers-Wannier duality, our explicit protocol will also be useful in implementing the Kramers-Wannier duality on a quantum computer.