Graphical calculus for Gaussian pure states

TL;DR

This paper introduces a comprehensive graphical calculus for Gaussian pure states, enabling unified analysis and transformations of CV cluster states, and addressing limitations of previous approaches that relied on infinite squeezing assumptions.

Contribution

The authors develop a universal graphical formalism for all Gaussian pure states, unifying various definitions of CV graph states and extending transformation rules beyond the infinite squeezing limit.

Findings

Unified graphical calculus for Gaussian pure states.

New definition of CV graph states encompassing existing variants.

Applications include error quantification, state generation analysis, and entanglement measurement.

Abstract

We provide a unified graphical calculus for all Gaussian pure states, including graph transformation rules for all local and semi-local Gaussian unitary operations, as well as local quadrature measurements. We then use this graphical calculus to analyze continuous-variable (CV) cluster states, the essential resource for one-way quantum computing with CV systems. Current graphical approaches to CV cluster states are only valid in the unphysical limit of infinite squeezing, and the associated graph transformation rules only apply when the initial and final states are of this form. Our formalism applies to all Gaussian pure states and subsumes these rules in a natural way. In addition, the term "CV graph state" currently has several inequivalent definitions in use. Using this formalism we provide a single unifying definition that encompasses all of them. We provide many examples of how the formalism may be used in the context of CV cluster states: defining the "closest" CV cluster state to a given Gaussian pure state and quantifying the error in the approximation due to finite squeezing; analyzing the optimality of certain methods of generating CV cluster states; drawing connections between this new graphical formalism and bosonic Hamiltonians with Gaussian ground states, including those useful for CV one-way quantum computing; and deriving a graphical measure of bipartite entanglement for certain classes of CV cluster states. We mention other possible applications of this formalism and conclude with a brief note on fault tolerance in CV one-way quantum computing.