Quantum computation with continuous variables: requirement of nonlinearity for photon counting

Summary form only given. We show that universal continuous-variable (CV) quantum computation can be obtained using linear optics (phase-space displacements and squeezing), homodyne measurement with classical feed-forward, and a realization of the photon counting projective value measurements (PVM). We describe the PVM for current (ideal) photodetectors, and demonstrate that such detectors cannot be used to implement the photon counting PVM with linear optics alone. Specifically, proposals to implement photon counting using arrays of beamsplitters to distribute the photons across several modes is insufficient for universal CV quantum computation. The photon counting PVM carries with it an implicit nonlinearity, and we discuss how it can be implemented in a CV system using a Kerr interaction (or another nonlinear Hamiltonian) and homodyne measurement. The resource requirements of this measurement scheme compared with using linear optics and current photodetectors are outlined.