## Implementing Grover's search algorithm using the one-way quantum computing model and photonic orbital angular momentum.

##### Abstract

Standard quantum computation proceeds via the unitary evolution of physical qubits (two-level
systems) that carry the information. A remarkably different model is one-way quantum
computing where a quantum algorithm is implemented by a set of irreversible measurements
on a large array of entangled qubits,, known as the cluster state. The order and sequence of
these measurements allow for different algorithms to be implemented. With a large enough
cluster state and a method in which to perform single-qubit measurements the desired computation
can be realised.
We propose a potential implementation of one-way quantum computing using qubits encoded
in the orbital angular momentum degree of freedom of single photons. Photons are good carriers
of quantum information because of their weak interaction with the environment and the
orbital angular momentum of single photons offers access to an infinite-dimensional Hilbert
space for encoding information. Spontaneous parametric down-conversion is combined with
a series of optical elements to generate a four-photon orbital angular momentum entangled
cluster state and single-qubit measurements are carried out by means of digital holography.
The proposed set-up, which is based on an experiment that utilised polarised photons, can be
used to realise Grover’s search algorithm which performs a search through an unstructured
database of four elements. Our application is restricted to a two-dimensional subspace of a
multi-dimensional system, but this research facilitates the use of orbital angular momentum
qubits for quantum information processing and points towards the usage of photonic qudits
(multi-level systems).
We also review the application of Dirac notation to paraxial light beams on a classical and
quantum level. This formalism is generally employed in quantum mechanics but the analogy
with paraxial optics allows us to represent the classical states of light by means of Dirac
kets. An analysis of the analogy between the classical and quantum states of light using this
formalism, is presented.