Quantum phase gate for photonic qubits using only beam splitters and postselection

The main difficulty in realizing quantum computation and quantum-information processing for single photon qubits is the optical implementation of controlled interactions between individual qubits. A deterministic interaction between separate photons would require the ability to apply strong nonlinearities during well-defined time intervals @1#. At present, there are still many technological difficulties preventing the realization of such reliable quantum gates for photonic qubits. Recently, however, it has been shown that optical nonlinearities are not necessary for photonic qubit operation if reliable single photon sources and single photon detectors are available @2#. The desired interaction between photonic qubits can also be realized by postselection. In particular, quantum phase gates and quantum control-not gates have been proposed, using additional input ports for single photons and additional output ports for postselection @2,3#. In particular, the postselection condition for a successful gate operation requires that the number of photons detected in the postselection ports is equal to the number previously added to the system. It is then possible to predict the success of the operation without further measurements on the output ports. In principle, quantum feedback and teleportation could then be used to realize a scalable optical quantum computer. However, the technological difficulties associated with the requirements of single photon sources and fast feedback are such that a realization of networks using these quantum gates is unlikely in the near future. In this Brief Report we show that a quantum phase gate operation can also be achieved using only beam splitters if postselection in the output port is permitted. Additional single photon inputs are not necessary, since the effective nonlinearity can be realized by the direct interaction of the photonic qubits at the beam splitter. While the operation of such a random gate cannot be confirmed without measuring the output, the fact that single photon sources are not necessary and that higher efficiency can be achieved with fewer optical elements should make this proposal an attractive alternative in the experimental realization of networks processing multiple photonic qubits. The starting point for all linear optics manipulations of single photon qubits is the unitary operation ÛR of a beam splitter of reflectivity R on the two mode input. In the following, we consider only the four-dimensional Hilbert space associated with zero or one photon in each input mode. The unitary operation ÛR is then characterized by