A Non-Commuting Stabilizer Formalism

TL;DR

This paper introduces a non-commutative extension of the stabilizer formalism to describe richer quantum states, including non-Abelian anyonic models, with efficient methods for analyzing their properties.

Contribution

It develops a novel non-commuting stabilizer framework that extends the standard Pauli stabilizer formalism, enabling the description of more complex quantum states.

Findings

Includes techniques for computing entanglement and expectation values

Provides methods for state preparation and Hamiltonian construction

Demonstrates states with non-Abelian anyons not possible in standard formalism

Abstract

We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group $\langle \alpha I, X,S\rangle$, where $\alpha=e^{i\pi/4}$ and $S=\operatorname{diag}(1,i)$. We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons.