# Constructing Quantum Spin Liquids Using Combinatorial Gauge Symmetry

## TL;DR

This paper introduces combinatorial gauge symmetry, a new local transformation involving spin rotations and permutations, enabling the construction of quantum spin liquids with accessible two-body interactions and potential applications in topological qubits.

## Contribution

It defines combinatorial gauge symmetry and links it to Hadamard matrices, providing a novel framework for designing quantum spin liquids with simple interactions.

## Key findings

- Identifies combinatorial gauge symmetry in Hamiltonians with Hadamard coupling matrices.
- Shows how this symmetry can be used to construct quantum spin liquids.
- Connects the symmetry to automorphisms of Hadamard matrices and topological qubit design.

## Abstract

We introduce the notion of combinatorial gauge symmetry -- a local transformation that includes single spin rotations plus permutations of spins (or swaps of their quantum states) -- that preserve the commutation and anti-commutation relations among the spins. We show that Hamiltonians with simple two-body interactions contain this symmetry if the coupling matrix is a Hadamard matrix, with the combinatorial gauge symmetry being associated to the automorphism of these matrices with respect to monomial transformations. Armed with this symmetry, we address the physical problem of how to build quantum spin liquids with physically accessible interactions. In addition to its intrinsic physical significance, the problem is also tied to that of how to build topological qubits.