Random numbers are a fundamental resource in science and engineering with important applications in simulation and cryptography. The inherent randomness at the core of quantum mechanics makes quantum systems a perfect source of entropy. Quantum random number generation is one of the most mature quantum technologies with many alternative generation methods. We discuss the different technologies in quantum random number generation from the early devices based on radioactive decay to the multiple ways to use the quantum states of light to gather entropy from a quantum origin. We also discuss randomness extraction and amplification and the notable possibility of generating trusted random numbers even with untrusted hardware using device independent generation protocols.

We show that the Hong-Ou-Mandel effect from quantum optics is equivalent to the SWAP test, a quantum information primitive which compares two arbitrary states. We first derive a destructive SWAP test that does not need the ancillary qubit that appears in the usual quantum circuit. Then we study the Hong-Ou-Mandel effect for two photons meeting at a beam splitter and prove it is, in fact, an optical implementation of the destructive SWAP test. This result offers both an interesting simple realization of a powerful quantum information primitive and an alternative way to understand and analyze the Hong-Ou-Mandel effect.

The Quantum Fourier Transform offers an interesting way to perform arithmetic operations on a quantum computer. We review existing Quantum Fourier Transform adders and multipliers and comment some simple variations that extend their capabilities. These modified circuits can perform modular and non-modular arithmetic operations and work with signed integers. Among the operations, we discuss a quantum method to compute the weighted average of a series of inputs in the transform domain. One of the circuits, the controlled weighted sum, can be interpreted as a circuit to compute the inner product of two data vectors.

The orbital angular momentum ͑OAM͒ of photons offers a suitable support to carry the quantum data of multiple users. We present two optical setups that send the information of n quantum communication parties through the same free-space optical link. Those qubits can be sent simultaneously and share path, wavelength, and polarization without interference, increasing the communication capacity of the system. The first solution, a qubit combiner, merges n channels into the same link, which transmits n independent photons. The second solution, the OAM multiplexer, uses controlled-not ͑CNOT͒ gates to transfer the information of n optical channels to a single photon. Additional applications of the multiplexer circuits, such as quantum arithmetic, as well as connections to OAM sorting are discussed.Light fields, both classical and quantum, can carry angular momentum. The total angular momentum of light has two components, polarization and orbital angular momentum, or OAM. Polarization can be associated to spin and OAM to the azimuthal phase ͓9͔. Both contributions can be studied independently under the paraxial approximation. The separate analysis and manipulation of the OAM allow for a wide range of classical and quantum applications, such as new free-space communication systems, high-precision atomic control, and higher-dimensional quantum information processing ͓10͔.Beams with different OAM values are defined by phase terms e iᐉ , where is the azimuthal phase and ᐉ is the OAM index, usually referred to as the winding number or topological charge. The properties of OAM fields can be taken down to the single-photon level and open the door for new singlephoton quantum states.OAM is a particularly promising candidate to implement optical d-dimensional quantum information units, the qudits. * juagar@tel.uva.es PHYSICAL REVIEW A 78, 062320 ͑2008͒

We prove that a single photon with quantum data encoded in its orbital angular momentum can be manipulated with simple optical elements to provide any desired quantum computation. We will show how to build any quantum unitary operator using beamsplitters, phase shifters, holograms and an extraction gate based on quantum interrogation. The advantages and challenges of these approach are then discussed, in particular the problem of the readout of the results. OPTICAL IMPLEMENTATIONS OF QUANTUM COMPUTINGQuantum information processing offers a new model of computation and communications. Certain tasks which can only be performed with a limited efficiency in a classical computer can be carried out efficiently in the quantum case [1]. For that reason, there is a great interest in finding a suitable physical implementation of a quantum computer.The DiVincenzo criteria give an important guide to the conditions a physical realization of a quantum computer should meet [2]. A practical quantum computer should be implemented over a system which is scalable with the input size and can be easily initialized and read. Additionally, the system must be able to carry out any desired quantum operation. This means that we must be able to implement any logic function and that the coherence time of the system (the time in which the quantum properties of the system are maintained) is long enough to finish the computation.Optical implementations seem particularly attractive. Photons can be initialized and read (generated and detected) with relative ease and have possibly the longest coherence time from all the quantum information candidates. They are also very well suited for communications. However, the interaction between different photons is complicated and needs to be mediated by non-linear processes. One proposed solution has been the use of measurement, like in the Linear Optics Quantum Computer of Knill, Laflamme and Milburn [3].Most notably, if we only have a single photon, there is always a way to perform any desired quantum computation with linear optics. A linear optics multiport can be described by a scattering matrix that gives the relationship between the amplitudes of the fields at the different input and output modes. For a single photon in 2 n input spatial modes, the scattering matrix corresponds to the unitary operator that gives the quantum evolution of a system of n qubits (n quantum information units). Any desired unitary operator of this kind can be implement using only beamplitters and phase shifters [4]. Consequently, any desired quantum computation can be implemented on a single photon at the cost of having a number of paths which grows exponentially with the number of qubits of the input [5]. In this case, we can obtain universal logic but cannot meet the scalability requisite.In this Letter, we propose a compact variation of single photon quantum computation with orbital angular momentum encoding. We will show how using only a limited set of optical elements it is possible to provide any unitary operator wit...

We present a quantum SWAP gate valid for quantum systems of an arbitrary dimension. The gate generalizes the CNOT implementation of the SWAP gate for qubits and keeps its most important properties, like symmetry and simplicity. We only use three copies of the same controlled qudit gate. This gate can be built with two standard higher-dimensional operations, the Quantum Fourier Transform and the d-dimensional version of the CZ gate

Inside computer networks, different information processing tasks are necessary to deliver the user data efficiently. This processing can also be done in the quantum domain. We present simple optical quantum networks where the orbital angular momentum of a single photon is used as an ancillary degree of freedom which controls decisions at the network level. Linear optical elements are enough to provide important network primitives like multiplexing and routing. First we show how to build a simple multiplexer and demultiplexer which combine photonic qubits and separate them again at the receiver. We also give two different self-routing networks where the OAM of an input photon is enough to make it find its desired destination.Comment: 7 pages, 3 figures, comments welcom

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