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We present an experimental realization of the first iteration in Grover's quantum algorithm using classical light and linear optical elements. The algorithm serves to find an entry marked by an oracle in an unstructured database. In our scheme, the quantum states encoding the database are represented by helical modes generated by means of a Spatial Light Modulator, while the marking corresponds to a π phase shift of the hidden mode. The optical implementation of Grover's algorithm then selectively amplifies the intensity of the marked mode such that it can be revealed by a modal decomposition. The core of the algorithm – a geometrical reflection of the point representing all database entries – is implemented in a single step independent of the size of the database. Moreover, we demonstrate experimentally that one iteration of the algorithm is enough to identify the marked entry, as a consequence of using classical states of light.

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... Controlling various degrees of freedom of light, i.e., time, frequency, space, and momentum, has become an emerging and promising tool for numerous information processing tasks in classical and quantum domains, ranging from novel imaging methods [1][2][3][4][5][6][7] to communications [8][9][10][11][12][13] to computations, [14][15][16][17] all harnessing the high dimensional nature of structured light fields. [18][19][20][21][22] This is because spatial modes enable information encoding in qudit spaces for dimensions d > 2 per particle as opposed to the d = 2 (qubits) encoding levels offered by traditional qubit encoding, such as with polarization states. ...

... On the other hand, harnessing classical optical fields for quantum computing has also been an active area of research. 14,[36][37][38][39][40] The drive in this field has been inspired by the observation that some of the essential resources of quantum computing can be found in classical coherent fields. Similar to quantum states, classical waves can be prepared in superpositions and can be interfered with, allowing for parallel information processing. ...

Optical computing harnesses the speed of light to perform vector-matrix operations efficiently. It leverages interference, a cornerstone of quantum computing algorithms, to enable parallel computations. In this work, we interweave quantum computing with classical structured light by formulating the process of photonic matrix multiplication using quantum mechanical principles such as state superposition and subsequently demonstrate a well-known algorithm, namely, Deutsch–Jozsa’s algorithm. This is accomplished by elucidating the inherent tensor product structure within the Cartesian transverse degrees of freedom of light, which is the main resource for optical vector-matrix multiplication. To this end, we establish a discrete basis using localized Gaussian modes arranged in a lattice formation and demonstrate the operation of a Hadamard gate. Leveraging the reprogrammable and digital capabilities of spatial light modulators, coupled with Fourier transforms by lenses, our approach proves adaptable to various algorithms. Therefore, our work advances the use of structured light for quantum information processing.

... Controlling various degrees of freedom of light, i.e. time, frequency, spatial and momentum, has become an emerging and promising tool for numerous information processing tasks in classical and quantum domains, ranging from novel imaging methods [1][2][3][4][5][6][7], communications [8][9][10][11][12][13] and computation [14][15][16][17], all harnessing the high dimensional nature of structured light fields [18][19][20][21][22]. This is because spatial modes enable for information encoding in qudit spaces for dimensions d > 2 per particle as opposed to the d = 2 (qubits) encoding levels offered by traditional qubit encoding such as with polarisation states. ...

... On the other hand, harnessing classical optical fields for quantum computing has also been an active area of research [14,[35][36][37][38][39]. The drive in this field has been inspired by the observation that some of the essential resources of quantum computing can be found in classical coherent fields. ...

Optical computing harnesses the speed of light to perform matrix-vector operations efficiently. It leverages interference, a cornerstone of quantum computing algorithms, to enable parallel computations. In this work, we interweave quantum computing with classical structured light. This is achieved by formulating the process of photonic matrix multiplication using quantum mechanical principles such as state superposition and subsequently demonstrate a well known algorithm, namely the Deutsch-Jozsa's algorithm. This is accomplished by elucidating the inherent tensor product structure within the Cartesian transverse degrees of freedom of light. To this end, we establish a discrete basis using localized Gaussian modes arranged in a lattice formation and demonstrate the operation of a Hadamard Gate. Leveraging the reprogrammable and digital capabilities of spatial light modulators, coupled with Fourier transforms by lenses, our approach proves adaptable to various algorithms. Therefore our work advances the use of structured light for quantum information processing.

... Structured light is a valuable resource for information processing [1,2] as it offers the opportunity for increasing the encoding capacity of many protocols. In particular, structured light encoded in the transverse spatial degree of freedom, such as its orbital angular momentum [3], position [4] and pixel/position [5,6], is finding applications in communications [7,8], cryptography [9][10][11][12][13], imaging [14][15][16], and computing [17][18][19]. However, a crucial task is to characterize the channels that the photons/laser fields are propagated through, and undoing any perturbations that may have acted on the field [5]. ...

Structured light, light tailored in its internal degrees of freedom, has become topical in numerous quantum and classical information processing protocols. In this work, we harness the high dimensional nature of structured light modulated in the transverse spatial degree of freedom to realize an adaptable scheme for learning unitary operations. Our approach borrows from concepts in variational quantum computing, where a search or optimization problem is mapped onto the task of finding a minimum ground state energy for a given energy/goal function. We achieve this by a pseudo-random walk procedure over the parameter space of the unitary operation, implemented with optical matrix-vector multiplication enacted on arrays of Gaussian modes by exploiting the partial Fourier transforming capabilities of a cylindrical lens in the transverse degree of freedom for the measurement. We outline the concept theoretically, and experimentally demonstrate that we are able to learn optical unitary matrices for dimensions d = 2, 4, 8, and 16 with average fidelities of $>90{\%}$ > 90 % . Our work advances high dimensional information processing and can be adapted to both process and quantum state tomography of unknown states and channels.

... Structured light is a valuable resource for information processing [1,2] as it offers the opportunity for increasing the encoding capacity of many protocols. In particular, structured light encoded in the transverse spatial degree of freedom, such as its orbital angular momentum [3], position [4] and pixel/position [5,6], is finding applications in communications [7,8], cryptography [9][10][11][12][13], imaging [14][15][16], and computing [17][18][19]. However, a crucial task is to characterise the channels that the photons/laser fields are propagated through, and undoing any perturbations that may have acted on the field [5]. ...

Structured light, light tailored in its internal degrees of freedom, has become topical in numerous quantum and classical information processing protocols. In this work, we harness the high dimensional nature of structured light modulated in the transverse spatial degree of freedom to realise an adaptable scheme for learning unitary operations. Our approach borrows from concepts in variational quantum computing, where a search or optimisation problem is mapped onto the task of finding a minimum ground state energy for a given energy/goal function. We achieve this by a pseudo-random walk procedure over the parameter space of the unitary operation, implemented with optical matrix-vector multiplication enacted on arrays of Gaussian modes by exploiting the partial Fourier transforming capabilities of a cylindrical lens in the transverse degree of freedom for the measurement. We outline the concept theoretically, and experimentally demonstrate that we are able to learn optical unitary matrices for dimensions d = 2, 4, 8 and 16 with average fidelities of >90%. Our work advances high dimensional information processing and can be adapted to both process and quantum state tomography of unknown states and channels.

... For instance, quantum algorithms involving up to two-qubit operations can be constructed using common optical components, such as beam splitters, polarization beam splitters, lenses, and Dove prisms, by leveraging the two paths and polarization degrees of freedom [12], [13]. Recent experiments have explored the use of vortex beams generated or manipulated by a spatial light modulator (SLM) to achieve quantum algorithms with up to 4 qubits, utilizing 16 orbital angular momenta (OAM) [15], [16]. Additionally, more efficient and programmable utilization of the SLM has demonstrated the implementation of the Deutsch-Jozsa algorithm with an effective complexity of 20 qubits [17], [18]. ...

Metasurfaces have recently opened up applications in the quantum regime, including quantum tomography and the generation of quantum entangled states. With their capability to store a vast amount of information by utilizing the various geometric degrees of freedom of nanostructures, metasurfaces are expected to be useful for processing quantum information. Here, we propose and experimentally demonstrate a programmable metasurface capable of performing quantum algorithms using both classical and quantum light with single photons. Our approach encodes multiple programmable quantum algorithms and operations, such as Grover’s search algorithm and the quantum Fourier transform, onto the same metalens array on a metasurface. A spatial light modulator selectively excites different sets of metalenses to carry out the quantum algorithms, while the interference patterns captured by a single-photon camera are used to extract information about the output state at the selected output directions. Our programmable quantum metasurface approach holds promising potential as a cost-effective means of miniaturizing components for quantum computing and information processing.

... While this classical notion of entanglement is an inherently local phenomenon, some works have suggested that non-locality is not required for many quantum information tasks [25]. In fact, this "classical entanglement" has been observed [22,[26][27][28][29][30][31][32][33] and used to implement classical analogues of quantum information algorithms [34][35][36][37][38][39]. ...

Quantum discord has been shown to be a resource for quantum advantage in addition to quantum entanglement. While many experiments have demonstrated classical analogies of entanglement, none have done so for discord. We present a proof-of-concept demonstration for creating a classical analogue of quantum discord using classical light that takes advantage of the analogy between the state of two qubits and the spatial modes of a Laguerre-Gauss beam. We demonstrate the validity of this approach by comparing the intensity profiles of theoretical simulations to experimental results for different values of discord. Such a classical analogue of quantum discord may provide further insight in understanding and development of quantum information technologies that make use of discord.

... The healing of defects more generally shows that the system can respond and adapt to aperiodic features of the phase mask. This opens up the possibility of iteratively manipulating information encoded as topological charges and could be used to develop photonic simulators 46 using OAM as a synthetic dimension, including an optical simulator of synthetic gauge fields 47 , and implementing quantum search algorithms in a laser solver with a marked array 48 . For instance, our laser arrays could be used to model the population dynamics of other physical systems, such as interacting galaxies or hydrodynamic vortices, mapping their angular momenta to different topological charges of the laser. ...

Geometric arrays of vortices found in various systems owe their regular structure to mutual interactions within a confined system. In optics, such vortex crystals may form spontaneously within a resonator. Their crystallization is relevant in many areas of physics, although their usefulness is limited by the lack of control over their topology. On the other hand, programmable devices like spatial light modulators allow the design of nearly arbitrary vortex distributions but without any intrinsic evolution. By combining non-Hermitian optics with on-demand topological transformations enabled by metasurfaces, we report a solid-state laser that generates 10 × 10 vortex laser arrays with actively tunable topologies and non-local coupling dictated by the array’s topology. The vortex arrays exhibit sharp Bragg diffraction peaks, witnessing their coherence and topological charge purity, which we spatially resolve over the whole lattice by introducing a parallelized analysis technique. By structuring light at the source, we enable complex transformations that allow to arbitrarily partition orbital angular momentum within the cavity and to heal topological charge defects, thus realizing robust and versatile resonators for applications in topological optics. A solid-state laser that generates 10 × 10 vortex laser arrays is demonstrated. The topologies are actively tunable.

... In a parallel approach, there have been many theoretical proposals and successful attempts to simulate (or perform) quantum computation [17], quantum search algorithm [18], quantum information processing [19] and quantum random walk [20] with classical waves. The employment of classical waves for their implementation overcomes the problem of sustaining quantum coherence for a long duration. ...

Information is encoded in a qubit in the form of its Bloch vector. In this paper, we propose protocols for remote transfers of information in a known and an unknown qubit to qudits using SU(2)-invariant $2 \times N$-level separable discordant states as quantum channels. These states have been identified as separable equivalents of the two-qubit entangled Werner states in Bharath and Ravishankar (Phys Rev A 89:062110, 2014). Due to $SU(2) \times SU(2)$ invariance of these states, the remote qudit can be changed by performing appropriate measurements on the qubit. We also propose a protocol for transferring information of a family of unknown qudits to remote qudits using $2 \times N$-level states as channels. Finally, we propose a protocol for swapping of quantum discord from $2\times N$-level systems to $N \times N$-level systems. All the protocols proposed in this paper involve separable states as quantum channels.

... The proposed experimental setups accomplish the same task without the need for a single photon source, which is difficult to prepare [36]. There are many more quantum phenomena and algorithms that have been simulated with classical light but not necessarily with OAM modes [37][38][39][40][41][42][43][44][45]. For a detailed review of classical entanglement, one may refer to [29] and references therein. ...

Classical electromagnetic fields and quantum mechanics obey the principle of superposition alike. This opens up many avenues for simulation of a large variety of phenomena and algorithms, which have hitherto been considered quantum mechanical. In this paper, we propose two such applications. In the first, we introduce a new, to the best of our knowledge, class of beams, called “equivalent optical beams,” in parallel with equivalent states introduced in Phys. Rev. A 89, 062110 (2014)PLRAAN1050-294710.1103/PhysRevA.89.062110. These beams have the same information content for all practical purposes. Employing them, we show how to transfer information from one degree of freedom of classical light to another, without need for classically entangled beams. Next, we show that quantum machine learning can be performed with OAM beams through the implementation of a quantum classifier circuit. We provide explicit protocols and explore the possibility of their experimental realization.

... The proposed experimental setups accomplish the same task without the need for a single photon source, which is hard to prepare [36]. There are many more quantum phenomena and algorithms that have been simulated with classical light, but not necessarily with OAM modes [37][38][39][40][41][42][43][44][45]. For a detailed review of classical entanglement, one may refer to [29] and references therein. ...

Classical electromagnetic fields and quantum mechanics -- both obey the principle of superposition alike. This opens up many avenues for simulation of a large variety of phenomena and algorithms, which have hitherto been considered quantum mechanical. In this paper, we propose two such applications. In the first, we introduce a new class of beams, called equivalent optical beams, in parallel with equivalent states introduced in [Bharath \& Ravishankar, \href{https://doi.org/10.1103/PhysRevA.89.062110}{Phys. Rev. A 89, 062110}]. These beams have the same information content for all practical purposes. Employing them, we show how to transfer information from one degree of freedom of classical light to another, without any need for classically entangled beams. Next, we show that quantum machine learning can be performed with OAM beams through the implementation of a quantum classifier circuit. We provide explicit protocols and experimental setups for both the applicaions.

... Nous ne nous attarderons pas sur l'implémentation effective de l'Oracle (nous considérerons l'opérateur d'Oracle comme une boite noire et un opérateur unitaire). Plusieursétudes théoriques et expérimentales ontété néanmoins menées sur le sujet [200,218]. Le lecteur prendra tout de même soin de distinguer le fait de connaître la solution, et le fait de pouvoir reconnaître la solution. On peutêtre capable de reconnaître la bonne solution lorsqu'elle se présenteà nous, sans pour autantêtre en mesure de la déterminer [200]. ...

L’intrication quantique est un des phénomènes les plus intéressants et intriguant en Mécanique Quantique, et de surcroît en Théorie de l’Information Quantique. Ressource fondamentale pour le calcul quantique, son rôle dans l’efficacité et la fiabilité des protocoles ou algorithmes quantiques n’est toujours pas totalement compris. Dans cette thèse, nous étudions l’intrication quantique des états multipartites, et notamment la nature de sa présence dans les algorithmes quantiques. L’étude de l’intrication se fait d’un point de vue théorique, en utilisant principalement des outils issus de la géométrie algébrique.Nous nous intéressons alors aux algorithmes de Grover et de Shor et déterminons quelles sont les classes d’intrication présentes (ou non) dans ces algorithmes, et ceci constitue donc une étude qualitative de l’intrication. De plus, nous mesurerons l’intrication quantitativement, à l’aide de mesures algébriques et géométriques, et étudions son évolution tout au long des différentes étapes de ces algorithmes. Nous proposons également des interprétations géométriques originales de ces résultats numériques.D’autre part, nous cherchons également à développer et exploiter de nouveaux outils pour mesurer, caractériser et classifier l’intrication quantique. Ceci se fait dans un premier temps d’un point de vue mathématique en étudiant les singularités des hypersurfaces liées aux systèmes quantiques pour caractériser différentes classes d’intrication. Dans un second temps, nous proposons des candidats pour les états maximalement intriqués, notamment pour les états symétriques et fermioniques, en utilisant des polynômes invariants et une mesure géométrique de l’intrication pour quantifier l’intrication. Enfin, nous avons également adopté une approche de type Machine Learning, notamment en entraînant des réseaux de neurones artificiels de manière supervisée, afin de reconnaitre certaines variétés algébriques modélisant certains types d’intrication précis.

... Since the inherent wave nature shared by both classical optics and quantum mechanics, such as superposition principle and interference effect, it is possible to simulate certain quantum algorithms with classical light. For instance, by encoding quantum bits into different degrees of freedom for the electromagnetic field (frequency, polarization, orbital angular momentum, space and time bins), many quantum computations can be efficiently simulated in optics, such as DJ algorithm [15][16][17][18], Grover's search algorithm [19][20][21][22] and quantum walks [23,24]. ...

During the past few years, a lot of efforts have been devoted in studying optical analog computing with artificial structures. Up to now, much of them are primarily focused on classical mathematical operations. How to use artificial structures to simulate quantum algorithm is still to be explored. In this work, an all-dielectric metamaterial-based model is proposed and realized to demonstrate the quantum Deutsch-Jozsa algorithm. The model is comprised of two cascaded functional metamaterial subblocks. The oracle subblock encodes the detecting functions (constant or balanced), onto the phase distribution of the incident wave. Then, the original Hadamard transformation is performed with a graded-index subblock. Both the numerical and experimental results indicate that the proposed metamaterials are able to simulate the Deutsch-Jozsa problem with one round operation and a single measurement of the output eletric field, where the zero (maximum) intensity at the central position results from the destructive (constructive) interference accompanying with the balance (constant) function marked by the oracle subblock. The proposed computational metamaterial is miniaturized and easy-integration for potential applications in communication, wave-based analog computing, and signal processing systems.

... A HD single quantum system, sometimes referred to as a qudit, widely existing in single-atom [11], photon [9,10], and superconducting quantum circuits [12], has been widely applied in quantum computations and communications, for example, the Grover search algorithm [11,13]. In a photonic system, one of the widely used qudits is a photon state depicted by a spatial degree of freedom (DOF), i.e., orbital angular momentum (OAM) 14], where | refers to a state with a topological charge of ; c ( c 2 = 1) represents amplitude occupations on each eigenstate. ...

In high-dimensional quantum communication networks, the quantum frequency converter (QFC) is indispensable as an interface in the frequency domain. For example, many QFCs have been built to link atomic memories and fiber channels. However, almost all QFCs work in a two-dimensional space. It is still a pivotal challenge to construct a high-quality QFC for some complex quantum states, e.g., a high-dimensional single-photon state that refers to a qudit. Here, we firstly propose a high-dimensional QFC for an orbital-angular-momentum qudit via sum-frequency conversion with a flat-top beam pump. As a proof-of-principle demonstration, we realize quantum frequency conversions for a qudit from infrared to visible range. Based on the qudit quantum state tomography, the fidelities of a converted state are 98.29(95.02)%, 97.42(91.74)%, and 86.75(67.04)% for a qudit without (with) accidental counts in 2, 3, and 5 dimensions, respectively. The demonstration is very promising for constructing a high-capacity quantum communication network.

... Altogether, arbitrary shaping of the OAM spectrum of light remains an open challenge. Yet, many applications require this, for example, the quantum Fourier transform as well as crucial steps in quantum computational algorithms such as that of Shor [14] and Grover [15] and general quantum information processing simulated with classical light [16][17][18]. In general, the task is to take an arbitrary initial OAM spectrum (specified by the coefficients c ) and modify it according to c | → c | . ...

Control of orbital angular momentum (OAM) in optical fields has seen tremendous growth of late, with a myriad of tools existing for their creation and detection. What has been lacking is the ability to arbitrarily modify the OAM spectrum of a superposition in amplitude and phase, especially if a priori knowledge of the initial OAM spectrum is absent. Motivated by a quasi-mapping that exists between the position and OAM of Laguerre-Gaussian modes, we propose an approach for a single-step modulation of a field’s OAM spectrum. We outline the concept and implement it through the use of binary ring apertures encoded on spatial light modulators. We show that complete control of the OAM spectrum is achievable in a single step, fostering applications in classical and quantum information processing that utilise the OAM basis.

... 'Nonlocality' is the feature distinguishing non-local entanglement from local entanglements. Recently, some significant advances suggest that the 'Nonlocality' is not a necessary condition for implementing many quantum computing tasks [21], for example, quantum walk [22,23] and several parallel-search algorithms [24,25], where local non-separable state can increase computation resources while keeping the physical number of particles constant. Also, because of the equivalent mathematical form between them, the non-separable state supports an effective platform to simulate the behaviors of quantum non-local entanglements [14,16,26], i.e., researches of the Bell measurement [21,27] and quantum contextuality [20,28]. ...

An analogous model system for high-dimensional quantum entanglement is proposed, based on the angular and radial degrees of freedom of the improved Laguerre Gaussian mode. Experimentally, we observed strong violations of the Bell-CGLMP inequality for maximally non-separable states of dimension 2 through 10. The results for violations in classical non-separable state are in very good agreement with quantum instance, which illustrates that our scheme can be a useful platform to simulate high-dimensional non-local entanglement. Additionally, we found that the Bell measurements provide sufficient criteria for identifying mode separability in a high-dimensional space. Similar to the two-dimensional spin-orbit non-separable state, the proposed high-dimensional angular-radial non-separable state may provide promising applications for classical and quantum information processing.

... In our classical theory of the physical world we have developed both particle and wave mechanics, and so it is not surprising that the classical wave mechanics holds many of the notions of the wave mechanics as described by our quantum theory, being as it is a theory that does not distinguish between particles and waves. Separable classical light fields have long been used to mimic quantum experiments, including path correlated photons in waveguide [24], quantum computing algorithms [25][26][27][28], entangled photons in spontaneous parametric downconversion (SPDC) experiments [29] and in quantum walks [30], while non-separable classically entangled light have been used to perform teleportation [31,32], quantum walks [33], dense coding [34], real-time quantum channel tomography and error correction [35] and testing local entanglement through violations of Hardy and Bell-type inequalities [36]. Rather than cover all of these topics scantly, we will illustrate the power of classical light for quantum tasks using a few selected examples. ...

The similarities between quantum mechanics and paraxial optics were already well-known to the founding fathers of quantum mechanics; indeed knowledge of paraxial optics partly informed quantum mechanics as a wave theory. Likewise quantum mechanical methods have been employed to better understand optics, for example, to determine which optical transformations are in principle realisable and which not. Recently the notion of classical entanglement has been mooted, ushering in a new avenue to explore, perhaps bridging the classical-quantum divide. These developments have raised questions as to which quantum tasks could be implemented with classical light, taking advantage of the wealth of four centuries of experience in optics. In this article we review the similarities as well as differences between optics and quantum mechanics, providing a quantum notation for classical light. We review the evidence for cross-fertilisation between quantum mechanics and classical optics, in particular considering the issue of classical entanglement and its exploitation for quantum tasks. Our work provides a concise theoretical framework punctuated with relevant examples, and critically assesses the current state of the field and its limits.

... Recent work has shown that many quantum processes can be mimicked with classical light, from teleportation (da Silva, Leal, Guzman-Silva et al., 2016) to quantum computing algorithms (Perez-Garcia, Hernandez-Aranda, Forbes, & Konrad, 2018;Perez-Garcia et al., 2016), higher-dimensional systems (Balthazar et al., 2016) and used to demonstrate entanglement beating in free-space (Otte, Rosales-Guzmán, Ndagano, Denz, & Forbes, 2018). ...

The concept of entanglement is so synonymous with quantum mechanics that the prefix “quantum” is often deemed unnecessary; there is after all only quantum entanglement. But the hallmark of entangled quantum states is nonseparability, a property that is not unique to the quantum world. On the contrary, nonseparability appears in many physical systems, and pertinently, in classical vector states of light: classical entanglement? Here we outline the concept of classical entanglement, highlight where it may be found, how to control and exploit it, and discuss the similarities and differences between quantum and classical entangled systems. Intriguingly, we show that quantum tools may be applied to classical systems, and likewise that classical light may be used in quantum processes. While we mostly use vectorial structured light throughout the text as our example of choice, we make it clear that the concepts outlined here may be extended beyond this with little effort, which we showcase with a few selected case studies.

Non-separable states of structured light have the analogous mathematical forms with quantum entanglement, which offer an effective way to simulate quantum process. However, the classical multi-partite non-separable states analogue to multi-particle entanglements can only be controlled by bulky free-space modulation of light through coupling multiple degrees of freedom (DoFs) with orbital angular momentum (OAM) to achieve high dimensionality and other DoFs to emulate multi-parties. In this paper, a scheme is proposed to directly emit multi-partite non-separable states from a simple laser cavity to mimic multi-particle quantum entanglement. Through manipulating three DoFs as OAM, polarization, and wavevector inside a laser cavity, the eight-dimensional (8D) tripartite states and all Greenberger-Horne-Zeilinger (GHZ)-like states can be generated and controlled on demand. In addition, an effective method is proposed to perform state tomography employing convolutional neural network (CNN), for measuring the generated GHZ-like states with highest fidelity up to 95.11%. This work reveals a feasibility of intra-cavity manipulation of high-dimensional multipartite non-separable states, opening a compact device for quantum-classical analogy and paving the path for advanced quantum scenarios.

Quantum Mechanics and classical optics feature similar phenomena such as superposition, interference and even entanglement. Hence, techniques from optics can be used in quantum mechanics and vice versa. In this article I address the question: What can we learn from formulating optics in the language of quantum mechanics? It is argued that the solutions of the wave equations for the electromagnetic field form a tensor product of Hilbert spaces corresponding to the degrees of freedom of classical light. Therefore, it comprises non-separable solutions reminiscent of entanglement. Moreover, the two spatial degrees of freedom each carry non-commuting position and momentum variables forming a Heisenberg algebra like quantum particles moving in a two dimensional space. An analogy between the dynamics of a quantum harmonic oscillator and paraxial light propagating through a converging lens is drawn. This article presents a modern formulation of optics in the language of state vectors and operators (based on Dirac’s notation) along the lines of an earlier contribution [1], specifying and explaining its results.

We generate a spatial comb comprising a superposition of a record 51 orbital angular momentum modes with a flat spectrum. This is a first step towards versatile digital control of the spatial shape of light.

We present the implementation of two quantum-like algorithms which exploit classical entanglement (i.e., nonseparability) of elastic waves. The Deutsch–Jozsa-like algorithm can distinguish between the even or odd character of all possible binary functions for two elastic bits inputs. The Deutsch-like algorithm distinguishes between constant and balance binary functions also using two elastic bits. These algorithms can be implemented on simple elastic systems composed of a planar array of two finite one-dimensional discrete elastic waveguides mutually coupled periodically along their length and driven externally.

Research submitted as a dissertation, a requirement for obtaining a Bachelor's Degree in Computer Science. Quantitative systematic review of articles on algorithms for quantum computing published during the last 3 years.

Recently, the study of analog optical computing raised renewed interest due to its natural advantages of parallel, high speed and low energy consumption over conventional digital counterpart, particularly in applications of big data and high-throughput image processing. The emergence of metamaterials or metasurfaces in the last decades offered unprecedented opportunities to arbitrarily manipulate the light waves within subwavelength scale. Metamaterials and metasurfaces with freely controlled optical properties have accelerated the progress of wave-based analog computing and are emerging as a practical, easy-integration platform for optical analog computing. In this review, the recent progress of metamaterial-based spatial analog optical computing is briefly reviewed. We first survey the implementation of classical mathematical operations followed by two fundamental approaches (metasurface approach and Green’s function approach). Then, we discuss recent developments based on different physical mechanisms and the classical optical simulating of quantum algorithms are investigated, which may lead to a new way for high-efficiency signal processing by exploiting quantum behaviors. The challenges and future opportunities in the booming research field are discussed.

Modal decomposition of light has been known for a long time, applied mostly to pattern recognition. With the commercialization of liquid-crystal devices, digital holography as an enabling tool has become accessible to all, and with it all-digital tools for the decomposition of light have finally come of age. We review recent advances in unravelling the properties of light, from the modal structure of laser beams to decoding the information stored in orbital angular momentum (OAM)-carrying fields. We show application of these tools to fiber lasers, solid-state lasers, and structured light created in the laboratory by holographic laser beam shaping. We show by experimental implementation how digital holograms may be used to infer the intensity, phase, wavefront, Poynting vector, polarization, and OAM density of some unknown optical field. In particular, we outline how virtually all the previous ISO-standard beam diagnostic techniques may be readily replaced with all-digital equivalents, thus paving the way for unravelling of light in real time. Such tools are highly relevant to the in situ analysis of laser systems, to mode division multiplexing as an emerging tool in optical communication, and for quantum information processing with entangled photons.

Light beams can carry a discrete, in principle unbounded amount of angular
momentum. Examples of such beams, the Laguerre-Gauss modes, are frequently
expressed as solutions of the paraxial wave equation. There, they are
eigenstates of the orbital angular momentum (OAM) operator. The paraxial
solutions predict that beams with large OAM could be used to resolve
arbitrarily small distances - a dubious situation. Here we show how to solve
that situation by calculating the properties of beams free from the paraxial
approximation. We find the surprising result that indeed one can resolve
smaller distances with larger OAM, although with decreased visibility. If the
visibility is kept constant (for instance at the Rayleigh criterion, the limit
where two points are reasonably distinguishable), larger OAM does not provide
an advantage. The drop in visibility is due to a field in the direction of
propagation, which is neglected within the paraxial limit.

Vector beams have the defining property of nonseparable spatial and polarization degrees of freedom and are now routinely generated in the laboratory and used in a myriad of applications. Here we exploit the nonseparability of such beams, akin to entanglement of quantum states, to apply tools traditionally associated with quantum measurements to these classical fields. We find that the entanglement entropy is a proxy for the average degree of polarization and thus provides a single number for the vector nature of such beams. In addition to providing tools for the analysis of vector beams, we also explore the concept of classical entanglement to explain why these tools are appropriate in the first place.

Quantum computers are able to outperform classical algorithms. This was long
recognized by the visionary Richard Feynman who pointed out in the 1980s that
quantum mechanical problems were better solved with quantum machines. It was
only in 1994 that Peter Shor came up with an algorithm that is able to
calculate the prime factors of a large number vastly more efficiently than
known possible with a classical computer. This paradigmatic algorithm
stimulated the flourishing research in quantum information processing and the
quest for an actual implementation of a quantum computer. Over the last fifteen
years, using skillful optimizations, several instances of a Shor algorithm have
been implemented on various platforms and clearly proved the feasibility of
quantum factoring. For general scalability, though, a different approach has to
be pursued. Here, we report the realization of a fully scalable Shor algorithm
as proposed by Kitaev. For this, we demonstrate factoring the number fifteen by
effectively employing and controlling seven qubits and four "cache-qubits",
together with the implementation of generalized arithmetic operations, known as
modular multipliers. The scalable algorithm has been realized with an ion-trap
quantum computer exhibiting success probabilities in excess of 90%.

Classical optics can be used to efficiently implement certain quantum
information processing tasks with a high degree of control, for example,
one-dimensional quantum walks through the space of orbital angular momentum of
light directed by its polarization. To explore the potential of quantum
information processing with classical light we here suggest a method to realize
d-dimensional quantum walks with classical optics -- an important step towards
robust implementation of certain quantum algorithms. In this scheme different
degrees of freedom of light, such as frequency, orbital angular momentum and
time bins represent different directions for the walker while the coin to
decide which direction the walker takes is realized employing the polarization
combined with different light paths.

There is recent interest in the use of light beams carrying orbital angular
momentum (OAM) for creating multiple channels within free-space optical
communication systems. One limiting issue is that, for a given beam size at the
transmitter, the beam divergence angle increases with increasing OAM, thus
requiring a larger aperture at the receiving optical system if the efficiency
of detection is to be maintained. Confusion exists as to whether this
divergence scales linarly with, or with the square root of, the beam's OAM. We
clarify how both these scaling laws are valid, depending upon whether it is the
radius of the Gaussian beam waist or the rms intensity which is kept constant
while varying the OAM.

The book is now in its fifth printing. Note that the second, fourth and fifth printings contain corrections based on the errata below. The third printing does not contain any corrections beyond those in the second printing. First, Second, Third, Fourth & Fifth Printing (July, 2002) pp 4 On page 4, in the first full sentence on the page, there is a missing comma after "qubits". pp 6 In the errata for the First through Fourth printings there is an error related to page 6, and the Solovay-Strassen test. Namely, "no deter-ministic test" should be "no efficient deterministic test".

Quantum approaches relying on entangled photons have been recently proposed
to increase the efficiency of optical measurements. We demonstrate here that,
surprisingly, the use of classical light with entangled degrees of freedom can
also bring outstanding advantages over conventional measurements in
polarization metrology. Specifically, we show that radially polarized beams of
light allow to perform real-time single-shot Mueller matrix polarimetry. Our
results also indicate that quantum optical procedures requiring entanglement
without nonlocality can be actually achieved in classical optics regime.

We present an experimental study of higher-dimensional quantum key distribution protocols based on mutually unbiased bases, implemented by means of photons carrying orbital angular momentum.We perform (d + 1) mutually unbiased measurements in a classically simulated prepare-and-measure scheme and on a pair of entangled photons for dimensions ranging from d = 2 to 5. In our analysis, we pay attention to the detection efficiency and photon pair creation probability. As security measures,we determine from experimental data the average error rate, the mutual information shared between the sender and receiver, and the secret key generation rate per photon. We demonstrate that increasing the dimension leads to an increased information capacity as well as higher key generation rates per photon. However, we find that the benefit of increasing the dimension is limited by practical implementation considerations, which in our case results in deleterious effects observed beyond a dimension of d = 4.

In analogy with Bell's inequality for two-qubit quantum states, we propose an inequality criterion for the nonseparability of the spin-orbit degrees of freedom of a laser beam. A definition of separable and nonseparable spin-orbit modes is used in consonance with the one presented in Phys. Rev. Lett. 99, 160401 (2007). As the usual Bell's inequality can be violated for entangled two-qubit quantum states, we show both theoretically and experimentally that the proposed spin-orbit inequality criterion can be violated for nonseparable modes. The inequality is discussed in both the classical and quantum domains.

We present an implementation scheme for a quantum walk in the orbital angular momentum space of a laser beam. The scheme makes use of a ring interferometer, containing a quarter-wave plate and a q plate. This setup enables one to perform an arbitrary number of quantum walk steps. In addition, the classical nature of the implementation scheme makes it possible to observe the quantum walk evolution in real time. We use nonquantum entanglement of the laser beam's polarization with its orbital angular momentum to implement the quantum walk.

As they travel through space, some light beams rotate. Such light beams have angular momentum. There are two particularly important ways in which a light beam can rotate: if every polarization vector rotates, the light has spin; if the phase structure rotates, the light has orbital angular momentum (OAM), which can be many times greater than the spin. Only in the past 20 years has it been realized that beams carrying OAM, which have an optical vortex along the axis, can be easily made in the laboratory. These light beams are able to spin microscopic objects, give rise to rotational frequency shifts, create new forms of imaging systems, and behave within nonlinear material to give new insights into quantum optics.

We demonstrate a simple approach, using digital holograms, to perform a complete azimuthal decomposition of an optical field. Importantly, we use a set of basis functions that are not scale dependent so that unlike other methods, no knowledge of the initial field is required for the decomposition. We illustrate the power of the method by decomposing two examples: superpositions of Bessel beams and Hermite-Gaussian beams (off-axis vortex). From the measured decomposition we show reconstruction of the amplitude, phase and orbital angular momentum density of the field with a high degree of accuracy.

We recently demonstrated a new method to efficiently analyse the orbital angular
momentum (OAM) states of light by application of an optical geometric transformation
(Berkhout et al 2010 Phys. Rev. Lett. 105 153601). Here we study the performance of
such a system to measure the change in the observed OAM spectrum, as the
input beam is misaligned with respect to the analyser. We present modelled and
experimental results which show that our reformatting approach does correctly
measure the OAM spectrum for lateral and tilt misalignment of the input beam.

We study light propagation in a photonic system that shows stepwise evolution in a discretized environment. It resembles a discrete-time version of photonic waveguide arrays or quantum walks. By introducing controlled photon losses to our experimental setup, we observe unexpected effects like subexponential energy decay and formation of complex fractal patterns. This demonstrates that the interplay of linear losses, discreteness and energy gradients leads to genuinely new coherent phenomena in classical and quantum optical experiments. Moreover, the influence of decoherence is investigated.

We present a method to efficiently sort orbital angular momentum (OAM) states of light using two static optical elements. The optical elements perform a Cartesian to log-polar coordinate transformation, converting the helically phased light beam corresponding to OAM states into a beam with a transverse phase gradient. A subsequent lens then focuses each input OAM state to a different lateral position. We demonstrate the concept experimentally by using two spatial light modulators to create the desired optical elements, applying it to the separation of eleven OAM states.

Laser light with a Laguerre-Gaussian amplitude distribution is found to have a well-defined orbital angular momentum. An astigmatic optical system may be used to transform a high-order Laguerre-Gaussian mode into a high-order Hermite-Gaussian mode reversibly. An experiment is proposed to measure the mechanical torque induced by the transfer of orbital angular momentum associated with such a transformation.

We report on an experiment on Grover's quantum search algorithm showing that classical waves can search a N-item database as efficiently as quantum mechanics can. The transverse beam profile of a short laser pulse is processed iteratively as the pulse bounces back and forth between two mirrors. We directly observe the sought item being found in approximately square root[N] iterations, in the form of a growing intensity peak on this profile. Although the lack of quantum entanglement limits the size of our database, our results show that entanglement is neither necessary for the algorithm itself, nor for its efficiency.

Determining classically whether a coin is fair (head on one side, tail on the other) or fake (heads or tails on both sides) requires an examination of each side. However, the analogous quantum procedure (the Deutsch-Jozsa algorithm) requires just one examination step. The Deutsch-Jozsa algorithm has been realized experimentally using bulk nuclear magnetic resonance techniques, employing nuclear spins as quantum bits (qubits). In contrast, the ion trap processor utilises motional and electronic quantum states of individual atoms as qubits, and in principle is easier to scale to many qubits. Experimental advances in the latter area include the realization of a two-qubit quantum gate, the entanglement of four ions, quantum state engineering and entanglement-enhanced phase estimation. Here we exploit techniques developed for nuclear magnetic resonance to implement the Deutsch-Jozsa algorithm on an ion-trap quantum processor, using as qubits the electronic and motional states of a single calcium ion. Our ion-based implementation of a full quantum algorithm serves to demonstrate experimental procedures with the quality and precision required for complex computations, confirming the potential of trapped ions for quantum computation.

Standard quantum computation is based on sequences of unitary quantum logic gates that process qubits. The one-way quantum computer proposed by Raussendorf and Briegel is entirely different. It has changed our understanding of the requirements for quantum computation and more generally how we think about quantum physics. This new model requires qubits to be initialized in a highly entangled cluster state. From this point, the quantum computation proceeds by a sequence of single-qubit measurements with classical feedforward of their outcomes. Because of the essential role of measurement, a one-way quantum computer is irreversible. In the one-way quantum computer, the order and choices of measurements determine the algorithm computed. We have experimentally realized four-qubit cluster states encoded into the polarization state of four photons. We characterize the quantum state fully by implementing experimental four-qubit quantum state tomography. Using this cluster state, we demonstrate the feasibility of one-way quantum computing through a universal set of one- and two-qubit operations. Finally, our implementation of Grover's search algorithm demonstrates that one-way quantum computation is ideally suited for such tasks.

We discuss a class of phase computer-generated holograms for the encoding of arbitrary scalar complex fields. We describe two holograms of this class that allow high quality reconstruction of the encoded field, even if they are implemented with a low-resolution pixelated phase modulator. In addition, we show that one of these holograms can be appropriately implemented with a phase modulator limited by a reduced phase depth.

The essential operations of a quantum computer can be accomplished using
solely optical elements, with different polarization or spatial modes
representing the individual qubits. We present a simple all-optical
implementation of Grover's algorithm for efficient searching, in which a
database of four elements is searched with a single query. By `compiling' the
actual setup, we have reduced the required number of optical elements from 24
to only 12. We discuss the extension to larger databases, and the limitations
of these techniques.

We show that the coined quantum walk on a line can be understood as an interference phenomenon, can be classically implemented, and indeed already has been. The walk is essentially two independent walks associated with the different coin sides, coupled only at initiation. There is a simple analogy between the evolution of walker positions and the propagation of light in a dispersive optical fiber. Comment: 8 pages, 2 figures; corrected following referees comments

High-dimensional entanglement with spatial modes of light promises increased security and information capacity over quantum channels. Unfortunately, entanglement decays due to perturbations, corrupting quantum links that cannot be repaired without performing quantum tomography on the channel. Paradoxically, the channel tomography itself is not possible without a working link. Here we overcome this problem with a robust approach to characterize quantum channels by means of classical light. Using free-space communication in a turbulent atmosphere as an example, we show that the state evolution of classically entangled degrees of freedom is equivalent to that of quantum entangled photons, thus providing new physical insights into the notion of classical entanglement. The analysis of quantum channels by means of classical light in real time unravels stochastic dynamics in terms of pure state trajectories, and thus enables precise quantum error correction in short- and long-haul optical communication, in both free space and fibre.

Inducing strong coupling
Quantum dots, or artificial atoms, are being pursued as prospective building blocks for quantum information processing architectures. Communication with other, distant quantum dots requires strong coupling between photons and the electronic states of the dots. Mi et al. used double quantum dots defined in silicon and embedded in a superconducting cavity to achieve such coupling. This demonstration in an industry-relevant material bodes well for the large-scale development of semiconductor-based quantum processors.
Science , this issue p. 156

Using an experimental setup that simulates a turbulent atmosphere, we study the secret key rate for quantum key distribution (QKD) protocols in orbital angular momentum based free space quantum communication. The QKD protocols under consideration include the Ekert 91 protocol for different choices of mutually unbiased bases and the six-state protocol. We find that the secret key rate of these protocols decay to zero, roughly at the same scale where the entanglement of formation decays to zero.

Coherent dressing of a quantum two-level system provides access to a new quantum system with improved properties - a different and easily tuneable level splitting, faster control, and longer coherence times. In our work we investigate the properties of the dressed, donor-bound electron spin in silicon, and probe its potential for the use as quantum bit in scalable architectures. The two dressed spin-polariton levels constitute a quantum bit that can be coherently driven with an oscillating magnetic field, an oscillating electric field, by frequency modulating the driving field, or by a simple detuning pulse. We measure coherence times of $T_{2\rho}^*=2.4$ ms and $T_{2\rho}^{\rm Hahn}=9$ ms, one order of magnitude longer than those of the undressed qubit. Furthermore, the use of the dressed states enables coherent coupling of the solid-state spins to electric fields and mechanical oscillations.
Free full-text access to a view-only version: http://rdcu.be/lp6C

Increasing the information capacity per unit bandwidth has been a perennial goal of scientists and engineers. Multiplexing of independent degrees of freedom, such as wavelength, polarization and more recently space, has been a preferred method to increase capacity in both radiofrequency and optical communication. Orbital angular momentum, a physical property of electromagnetic waves discovered recently, has been proposed as a new degree of freedom for multiplexing to achieve capacity beyond conventional multiplexing techniques, and has generated widespread and significant interest in the scientific community. However, the capacity of orbital angular momentum multiplexing has not been established or compared to other multiplexing techniques. Here, we show that orbital angular momentum multiplexing is not an optimal technique for realizing the capacity limits of a free-space communication channel and is outperformed by both conventional line-of-sight multi-input multi-output transmission and spatial-mode multiplexing.

We propose an optical implementation of the Deutsch-Jozsa Algorithm using
classical light in a binary decision-tree scheme. Our approach uses a ring
cavity and linear optical devices in order to efficiently quarry the oracle
functional values. In addition, we take advantage of the intrinsic Fourier
transforming properties of a lens to read out whether the function given by the
oracle is balanced or constant.

We present an implementation of the Deutsch Algorithm using linear optical elements and laser light. We encoded two quantum bits in form of superpositions of electromagnetic fields in two degrees of freedom of the beam: its polarisation and orbital angular momentum. Our approach, based on a Sagnac interferometer, offers outstanding stability and demonstrates that optical quantum computation is possible using classical states of light.

A classical analogy of quantum mechanical entanglement is presented, using classical light beams. The analogy can be pushed a long way, only to reach its limits when we try to represent multiparticle, or nonlocal, entanglement. This demonstrates that the latter is of exclusive quantum nature. On the other hand, the entanglement of different degrees of freedom of the same particle might be considered classical. The classical analog cannot replace Einstein-Podolsky-Rosen type experiments, nor can it be used to build a quantum computer. Nevertheless, it does provide a reliable guide to the intuition and a tool for visualizing abstract concepts in low-dimensional Hilbert spaces.

We examine a classical version of entanglement between spatial and polarization degrees of freedom for classical light. We examine the relation between classical entanglement, polarization, and several recently introduced measures of coherence for vectorial waves. We show that there is no definite relation between quantum and classical entanglement.

The use of programmable liquid crystal displays for optical simulation of quantum algorithms was developed. The Deutsch-Jozsa algorithms and Grover search algorithms were implemented in optical simulation. The Deutsch-Jozsa's algorithms were found advantageous during the manipulation of spatial frequencies of the input in the fourier domain, and the homogeneous portion of the wave front defined an initial state with equal amplitude for every item. The Grover's search algorithm helped to find marked item in a database with N registers. It was found that the both algorithms could be constructed as a sequence of operators diagonal in position (U v), and allows optical simulation of all such unitary operators.

Quantum computing holds the promise of solving problems that would be intractable with conventional computers by implementing principles from quantum physics in the development of computer hardware, software and communications equipment. Quantum-assisted computing will be the first step towards full quantum systems, and will cause immense disruption of our traditional networks. The world's biggest manufacturers are investing large amounts of resources to develop crucial quantum-assisted circuits and devices. Quantum Computing and Communications: Gives an overview of basic quantum computing algorithms and their enhanced versions such as efficient database searching, counting and phase estimation. Introduces quantum-assisted solutions for telecom problems including multi-user detection in mobile systems, routing in IP based networks, and secure ciphering key distribution. Includes an accompanying website featuring exercises (with solution manual) and sample algorithms from the classical telecom world, corresponding quantum-based solutions, bridging the gap between pure theory and engineering practice. This book provides telecommunications engineers, as well as graduate students and researchers in the fields of computer science and telecommunications, with a wide overview of quantum computing & communications and a wealth of essential, practical information.

The microscopic image of illuminated objects results from a twofold diffraction, at the object and at the lens-aperture. The theories of Rayleigh and of Abbe differ only as to the order in which they consider these diffractions. The Abbe method is here used to calculate the image of a coarse transparent grating with shallow grooves of arbitrary form (phase grating). In the ideal case the image is invisible. The formulae are successively applied to the following practical methods to make the image visible: the schlierenmethod, where the diffraction spectra of the grating are intercepted on one side, the ordinary oblique dark ground illumination, where the central image is intercepted also, the central dark ground illumination, with intercepts the central image only, and the bright ground observation with illumination by a narrow pencil, where the visibility is caused by out-of-focus observation. It is found that none of these methods can show the real groove form. This is possible by the new method of phase contrast, where a path difference of λ/4 is introduced between the spectra and the central image by passing the last through a slightly thicker or thinner part (phase-strip) of a glass plate.In Part II the new method is treated in another way which applies to objects of arbitrary irregular structure. The general result is that by the phase-contrast method transparent details of the object which differ in thickness or in refractive index appear as differences of intensity in the image. An important increase of sensitivity can further be obtained by the use of an absorbing phase strip. The effect of the diffraction by the phase strip is then considered and practical methods discussed to make the resulting diffraction-halo as faint as possible. Various reasons are found why the strip should preferably be of circular form, with a corresponding annular diafragm in the condenser. Finally the methods of preparing phase strips and of placing and adjusting them in the microscope are discussed.

It is found that the rotation of the Poynting vector of a Laguerre-Gaussian laser mode is proportional to the Gouy phase and for most cases of interest, rotates through less than one revolution in reaching the far field. For a p = 0 mode, we show the expression for the rotation is particularly simple and independent of l. At the radius corresponding to maximum intensity, it yields a rotation of only as z→±∞.

Some of the potential applications of optical vortices may require a certain density of topological charges within a given area. However, topological charge densities cannot extend over areas of arbitrary size. It is shown here that the net topological charge in an area cannot exceed the circumference of that area divided by the wavelength. This observation is used to explain the depletion of light in different situations, which are illustrated with numerical examples. One such phenomenon is the depletion of light from the core of an optical vortex that is produced by a phase singularity in the transmission function of a kinoform. The result is an effective launch core function for such a vortex. The shape of this core function is derived here.

Reversible logic has received much attention in recent years when calculation with minimum energy consumption is considered. Especially, interest is sparked in reversible logic by its applications in some technologies, such as quantum computing, low-power CMOS design, optical information processing and nanotechnology. This article proposes two new reversible logic gates, ZRQ and NC. The first gate ZRQ not only implements all Boolean functions but also can be used to design optimised adder/subtraction architectures. One of the prominent functionalities of the proposed ZRQ gate is that it can work by itself as a reversible full adder/subtraction unit. The second gate NC can complete overflow detection logic of Binary Coded Decimal (BCD) adder. This article proposes two approaches to design novel reversible BCD adder using new reversible gates. A comparative result which is presented shows that the proposed designs are more optimised in terms of number of gates, garbage outputs, quantum costs and unit delays than the existing designs.

Part I. Fundamental Concepts: 1. Introduction and overview; 2.
Introduction to quantum mechanics; 3. Introduction to computer science;
Part II. Quantum Computation: 4. Quantum circuits; 5. The quantum
Fourier transform and its application; 6. Quantum search algorithms; 7.
Quantum computers: physical realization; Part III. Quantum Information:
8. Quantum noise and quantum operations; 9. Distance measures for
quantum information; 10. Quantum error-correction; 11. Entropy and
information; 12. Quantum information theory; Appendices; References;
Index.

The issue raised in this Letter is classical, not only in the sense of being nonquantum, but also in the sense of being quite ancient: which subset of 4\ifmmode\times\else\texttimes\fi{}4 real matrices should be accepted as physical Mueller matrices in polarization optics? Nonquantum entanglement or inseparability between the polarization and spatial degrees of freedom of an electromagnetic beam whose polarization is not homogeneous is shown to provide the physical basis to resolve this issue in a definitive manner.

The number of steps any classical computer requires in order to find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying the security of widely used cryptographic codes. Quantum computers, however, could factor integers in only polynomial time, using Shor's quantum factoring algorithm. Although important for the study of quantum computers, experimental demonstration of this algorithm has proved elusive. Here we report an implementation of the simplest instance of Shor's algorithm: factorization of N = 15 (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a molecule as quantum bits, which can be manipulated with room temperature liquid-state nuclear magnetic resonance techniques. This method of using nuclei to store quantum information is in principle scalable to systems containing many quantum bits, but such scalability is not implied by the present work. The significance of our work lies in the demonstration of experimental and theoretical techniques for precise control and modelling of complex quantum computers. In particular, we present a simple, parameter-free but predictive model of decoherence effects in our system.

Quantum computing shows great promise for the solution of many difficult problems, such as the simulation of quantum systems and the factorization of large numbers. While the theory of quantum computing is fairly well understood, it has proved difficult to implement quantum computers in real physical systems. It has recently been shown that nuclear magnetic resonance (NMR) can be used to implement small quantum computers using the spin states of nuclei in carefully chosen small molecules. Here we demonstrate the use of a NMR quantum computer based on the pyrimidine base cytosine, and the implementation of a quantum algorithm to solve Deutsch's problem (distinguishing between constant and balanced functions). This is the first successful implementation of a quantum algorithm on any physical system. (C) 1998 American Institute of Physics.

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