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 oneway 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.

Experimental Realization of Multipartite Entanglement of 60 Modes of a Quantum Optical Frequency Comb

“…The vertical arrows mark the half-frequencies of the pumps; the curved arrows denote the zzz (bottom) and yyy (top) EPR pairs. (b) Quantum graph states [30]: The initial EPR pairs from the OPO (top) turn, after a single beam splitter (grey ellipses), into a dual-rail CV cluster state (bottom), whose ±1/2-weight edges are color-coded (contrary to the qubit case, weighted cluster CV states are still stabilizer states [30,31]). …”

“…1(b), bottom [30,31]. The measurement of these nullifiers requires homodyne detection at 3 different optical frequencies.…”

Hybrid quantum information processing

“…This is big advantage of CV QIP. There are some proposals and results to create large scale CV entanglement [4,6,5]. Since entanglement is the heart of QIP, CV QIP has a big advantage over qubit QIP.…”

It’s a Good Time for Time-Bin Qubits

“…Additionally, time-frequency encodings benefit from a relative insensitivity to inhomogeneities in transmission mediums [12,14]. These advantages have been recognized in works exploring the preparation of time-frequency entangled states [15][16][17][18], including their use in the violation of Bell inequalities [19,20], quantum key distribution [21], teleportation [22], and continuousvariable cluster states [23].Quantum computing based on time-frequency encoding has received comparatively little attention, but has become increasingly feasible with the advent of fast switchable integrated phase controllers [24,25]. This was highlighted by a recent classical simulation of a quantum random walk…”