Graph states in phase space

TL;DR

This paper explores how graph states of n qubits can be represented as non-singular curves in a specially structured phase space, offering algebraic tools for quantum operations.

Contribution

It introduces a novel geometric and algebraic framework for representing and manipulating graph states in phase space.

Findings

Graph states correspond to non-singular curves in phase space.

Algebraic representation of quantum operations on these states.

Advantages include simplified analysis and potential for new quantum protocols.

Abstract

The phase space for a system of $n$ qubits is a discrete grid of $2^{n} \times 2^{n}$ points, whose axes are labeled in terms of the elements of the finite field $\Gal{2^n}$ to endow it with proper geometrical properties. We analyze the representation of graph states in that phase space, showing that these states can be identified with a class of non-singular curves. We provide an algebraic representation of the most relevant quantum operations acting on these states and discuss the advantages of this approach.