Graphical calculus for Gaussian pure states

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.

(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)

References in corpus (21)

• Multi-party entanglement in graph states
• Universal Quantum Computation with Continuous-Variable Cluster States
• Resource-efficient linear optical quantum computation
• One-Way Quantum Computing in the Optical Frequency Comb
• Experimental generation of four-mode continuous-variable cluster states
• Building Gaussian Cluster States by Linear Optics
• Identifying phases of quantum many-body systems that are universal for quantum computation
• Ultracompact Generation of Continuous-Variable Cluster States
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• Entanglement in the stabilizer formalism
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• The Optical Frequency Comb as a One-Way Quantum Computer
• Gapped Two-Body Hamiltonian for continuous-variable quantum computation
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• Demonstration of Cluster State Shaping and Quantum Erasure for Continuous Variables
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• Weighing matrices and optical quantum computing
• Graph states and carrier-envelope phase squeezing

Cited by in corpus (2)

• Deterministic generation of Gaussian pure state in quasi-local dissipative system
• Multipartite Entanglement and Hypergraph states of three qubits