Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete-time quantum walks.

Quantum walks: a comprehensive review

The Journal of the Acoustical Society of America

A few years ago the possibility of coupling and inter-converting the spin and orbital angular momentum (SAM and OAM) of paraxial light beams in inhomogeneous anisotropic media was demonstrated. An important case is provided by waveplates having a singular transverse pattern of the birefringent optical axis, with a topological singularity of charge q at the plate center, hence named 'q-plates'. The introduction of q-plates has given rise in recent years to a number of new results and to significant progress in the field of orbital angular momentum of light. Particularly promising are the quantum photonic applications, because the polarization control of OAM allows the transfer of quantum information from the SAM qubit space to an OAM subspace of a photon and vice versa. In this paper, we review the development of the q-plate idea and some of the most significant results that have originated from it, and we will briefly touch on many other related findings concerning the interaction of the SAM and OAM of light.

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Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications

Twisted photons can be used as alphabets to encode information beyond one bit per single photon. This ability offers great potential for quantum information tasks, as well as for the investigation of fundamental questions. In this review article, we give a brief overview of the theoretical differences between qubits and higher dimensional systems, qudits, in different quantum information scenarios. We then describe recent experimental developments in this field over the past three years. Finally, we summarize some important experimental and theoretical questions that might be beneficial to understand better in the near future. Photons possessing orbital angular momentum are promising for systems for realizing new quantum information applications. Quantum computing and communications are set to revolutionize information technology, but most systems studied to date are based on qubits —quantum analogs of classical bits that can take one of only two states. Manuel Erhard at the University of Vienna, Austria, and co-workers review progress in higher dimensional systems that use photons with orbital angular momentum, or twisted photons, as ‘qudits’, which can have any number of levels. They look at the advantages of such higher-dimensional systems, which include higher information capacity and greater protection from eavesdropping. The researchers then examine exciting developments in the field in the past two to three years, such as the creation of high-dimensional entanglement and optimal quantum cloning. Finally, they consider future challenges.

/pdf/twisted-photons-new-quantum-perspectives-in-high-dimensions-41fpvm7use.pdf

Twisted photons: new quantum perspectives in high dimensions

Photonic quantum technologies represent a promising platform for several applications, ranging from long-distance communications to the simulation of complex phenomena. Indeed, the advantages offered by single photons do make them the candidate of choice for carrying quantum information in a broad variety of areas with a versatile approach. Furthermore, recent technological advances are now enabling first concrete applications of photonic quantum information processing. The goal of this manuscript is to provide the reader with a comprehensive review of the state of the art in this active field, with a due balance between theoretical, experimental and technological results. When more convenient, we will present significant achievements in tables or in schematic figures, in order to convey a global perspective of the several horizons that fall under the name of photonic quantum information.

https://iopscience.iop.org/article/10.1088/1361-6633/aad5b2/pdf

Photonic quantum information processing: a review

Photons can carry spin angular momentum (SAM) and orbital angular momentum (OAM), which can be used to realize a qubit system and a high-dimension system, respectively. This spin-orbital system is very suitable for implementing one-dimensional discrete-time quantum random walks. We propose a simple scheme of quantum walks on the spin-orbital angular momentum space of photons, where photons walk on the infinite OAM space controlled by their SAM. By employing the recent invention of an optical device, the so-called ``$q$-plate,'' our scheme is more simple and efficient than others because there is no Mach-Zehnder interferometer in the scheme.

Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons

When a vortex beam with the spiral phase structure passes through a dynamic angular double slits (ADS), the interference pattern changes alternatively between destructive and constructive at the angular bisector direction of the ADS due to their phase difference. Based on this property, we experimentally demonstrate a simple method, which can precisely and efficiently determine the topological charge of vortex beams. What is more, this scheme allows determining both the modulus and sign of the topological charge of vortex beams simultaneously.

Probing the topological charge of a vortex beam with dynamic angular double slits

When a vortex beam with a spiral phase structure passes through dynamic angular double slits (ADS), the interference pattern changes alternatively between destructive and constructive at the angular bisector of the ADS. This change is due to their phase difference. Based on this property, we experimentally demonstrate a simple method to precisely and efficiently determine the topological charge of vortex beams. Furthermore, this scheme allows for the simultaneous determination of the modulus and the sign of the topological charge of the vortex beams.

The spatial second-order interference of two independent pseudothermal light beams in a Hong-Ou-Mandel interferometer is studied experimentally and theoretically. The similar cosine modulation in the second-order coherence function as the one with entangled-photon pairs in a Hong-Ou-Mandel interferometer is observed. Two-photon interference based on Feynman's path integral theory is employed to interpret the results. The experimental results and theoretical simulations agree with each other very well.

Spatial second-order interference of pseudothermal light in a Hong-Ou-Mandel interferometer.

Spatial shaping of light beams has led to numerous new applications in fields such as imaging, optical communication, and micromanipulation. However, structured radiation is less well explored beyond visible optics, where methods for shaping fields are more limited. Binary amplitude filters are often used in these regimes and one such example is a photon sieve consisting of an arrangement of pinholes, the positioning of which can tightly focus incident radiation. Here, we describe a method to design generalized photon sieves: arrays of pinholes that generate arbitrary structured complex fields at their foci. We experimentally demonstrate this approach by the production of Airy and Bessel beams, and Laguerre–Gaussian and Hermite–Gaussian modes. We quantify the beam fidelity and photon sieve efficiency, and also demonstrate control over additional unwanted diffraction orders and the incorporation of aberration correction. The fact that these photon sieves are robust and simple to construct will be useful for the shaping of short- or long-wavelength radiation and eases the fabrication challenges set by more intricately patterned binary amplitude masks.

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Ruifeng Liu

Papers

Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons

Probing the topological charge of a vortex beam with dynamic angular double slits

Probing the topological charge of a vortex beam with dynamic angular double slits

Spatial second-order interference of pseudothermal light in a Hong-Ou-Mandel interferometer.

Generalized photon sieves: fine control of complex fields with simple pinhole arrays