The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases

TL;DR

This paper introduces the XP stabiliser formalism, extending the Pauli stabiliser framework with fractional rotations, enabling representation of more complex states and codespaces, and explores their properties and classical simulability.

Contribution

It generalizes the stabiliser formalism to include fractional phases, broadening the scope of stabiliser states and codes, and provides algorithms for code analysis and simulation.

Findings

XP stabiliser states are equivalent to weighted hypergraph states

Algorithms for determining codespaces and logical operators are developed

Analysis of classical simulability of measurements on XP codes

Abstract

We propose an extension to the Pauli stabiliser formalism that includes fractional $2\pi/N$ rotations around the $Z$ axis for some integer $N$. The resulting generalised stabiliser formalism - denoted the XP stabiliser formalism - allows for a wider range of states and codespaces to be represented. We describe the states which arise in the formalism, and demonstrate an equivalence between XP stabiliser states and 'weighted hypergraph states' - a generalisation of both hypergraph and weighted graph states. Given an arbitrary set of XP operators, we present algorithms for determining the codespace and logical operators for an XP code. Finally, we consider whether measurements of XP operators on XP codes can be classically simulated.