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.Comment: (v3) shortened title, very minor corrections (v2) minor corrections,
reference added, new figures for CZ gate and beamsplitter graph rules; (v1)
25 pages, 11 figures (made with TikZ
