2010/07/31 by Nicolas C. Menicucci, Steven T. Flammia, Peter van Loock · 4 citations
Computer Science · Mathematics · Physics and Astronomy · #Bipartite graph #Cluster state #Computer science #Discrete mathematics #Formalism (music) #Gaussian #Graph #Graphical model #Mathematics #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum entanglement #Quantum mechanics #Statistical physics #Unitary state #quant-ph
paper · pdf · doi:10.1103/physreva.83.042335
published as Phys. Rev. A 83, 042335 (2011) · (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)
arxiv created 2011/02/04 · openalex publication_date 2011/04/28 · arxiv updated 2011/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We provide a unified graphical calculus for all Gaussian pure states, including graph transformation rules for all local and semilocal 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 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.