The abundant amount of prior work credited above provides ample evidence that **nonlinear** optics resem- bles fluid dynamics to a certain degree. In order to use **nonlinear** optics as a useful and practical computational tool for fluid dynamics, however, simply drawing analogies between the two kinds of dynamics is not enough.
One must be able to show an exact correspondence or, at the very least, an approaching convergence between a problem in **nonlinear** optics and a problem in fluid dynamics, in order to produce any useful prediction of fluid dynamics via **nonlinear** optics. Moreover, as computers nowadays have enough capabilities to simulate two-dimensional fluids, the mere correspondence between optics and two-dimensional fluid dynamics con- sidered in most of the prior work would not motivate the use of metaphoric **optical** computing in preference to conventional digital computing. A three-dimensional fluid modeling, on the other hand, requires a pro- cessing capability orders of magnitude higher than that available in today’s supercomputers, so metaphoric **optical** computing would need to compute such problems much more efficiently to compete with electronic computers and the Moore’s law.

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allow for a relaxation of the security assumptions typically made in QKD security proofs, opening the way to the so-called device independent security (see e.g., [56, 57, 58].
The two families of QKD schemes described above can be further distinguished into three main categories, which differ in the **information** coding technique: discrete-variables coding (DV-C), continuous-variables coding (CV-C) and distributed-phase-reference co- ding (DPR-C). DV-C is the first technique which has been proposed for QKD and, to date, is probably the most used. According to this approach, **information** is encoded in a discrete **quantum** degree of freedom of photons; in particular, widely used solutions are photon polarization for free-space implementations and phase coding for fiber-based im- plementations. The receiver uses a photon detector and only the events which resulted in a detection are taken into account for key distillation. DV-C protocols have the main advantage that, if the **quantum** channel is error-free and no eavesdropper is tampering with it, the legitimate parties immediately share a perfect secret key. On the other hand, they suffer from a low efficiency of photon detectors, high dark count rates and rather long dead-times, thus resulting in high overall losses [51]. These limitations were the dri- ving reasons for the introduction of CV-C protocols, which are based on the measurement of quadrature components of light by means of homodyne detection (see, e.g., [59, 60]). Despite getting rid of hardware limitations of photon detectors, however, in CV-C imple- mentations losses translate into noise, 1 resulting in a decrease of the signal-to-noise ratio. This entails an overhead in the **information** reconciliation procedure, which is now required to deal with noisier signals. In order to overcome the drawbacks of both the DV-C and the CV-C protocols, some experimental groups proposed a new approach to QKD, where the raw key bits are encoded into discrete **quantum** states (as for DV-C) and the **quantum** channel is monitored by observing the phase coherence of subsequent pulses. In that way, the communication efficiency can be improved with respect to DV-C and CV-C schemes. As previously mentioned, this class of protocols is referred to as DPR-C; examples of such protocols can be found in [61] and in [62].

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The single-qubit rotation by using STIRAP technique [107] (briefly pre- sented in Sec. 4.2) was developed in 2002. Type-I DQD configurations to implement single- and two- qubit rotations by using STIRAP have also been introduced before [19]. Our contribution here is that we show how to overcome the limitations of the scheme in Ref. [19] by using the novel type-II DQD-in- a-nanowire system instead of the conventional type-I configuration. By cal- culating the transition dipole moments, we quantitatively illustrate type-II QDs are significantly better than type-I QDs for **quantum** gating by STI- RAP. By developing a multiband formalism we show that a charged-exciton featuring a mixed-hole part acting as an intermediate state provides essential coupling with three ground-states of a DQD to perform qubit rotations by STIRAP. We then introduce a system consisting of two neighboring DQDs in a nanowire each encoding one qubit. By using the configuration-interaction method, we show that the strong Coulomb interaction between the charges, which causes a significant shift of the STIRAP transition frequencies, can be exploited to efficiently perform high fidelity conditional two-qubit CNOT op- eration on the two qubits. Importantly, we also show that the implementation is robust against the decoherence posed by spin and charge fluctuations in the environment.

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In this article we bridge this gap, presenting a complete analysis of the performance of the main photonic architectures under imperfect operational conditions. The three interferometric layouts have been investigated in the general case, by admitting different levels of losses and noise in unitary transformations of increasing size. Specifically, following the approach commonly adopted for reconfigurable **quantum** circuits 24 , 25 , i.e. by model- ling beam splitters as Mach-Zehnder interferometers with variable phases and two cascaded symmetric beam splitters, noise was added to their reflectivities and to phases in both Mach-Zehnders and outer phase shifters. For our numerical benchmark we employ as figures of merit the fidelity 39 , 52 , 55 , 56 and the total variation distance (TVD) 53 – 56 , as good estimators of the distance between ideal and imperfect implementations in relevant applica- tions. In particular, while depending also on external factors like purity and degree of distinguishability between the input photons, the TVD has been already identified as a key figure of merit for the goodness of multiphoton experiments 53 – 56 (see Supplementary Note 3). The article is structured as follows. First, we briefly discuss the interferometric structure of the triangular, square and fast designs, whose main characteristics are outlined in Fig. 1 . Thereafter, we compare the performances of the first two schemes, which were shown to be universal for unitary evolutions, for the implementation of Haar-random transformations of increasing size. Finally, we com- pare the operation of the three schemes for the implementation of Fourier and Sylvester interferometers, which represent the fundamental building blocks in a significant number of relevant **quantum** **information** protocols. Our analysis highlights the advantages and limitations of each scheme, providing an essential reference point for the design of future larger-scale photonic technologies, whose optimal configurations may well benefit from a joint integration.

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Scanning the monolith ensures that the cavity experiences a wide range of dispersions, and thus does not limit the possible resonances of the signal and idler. The output of the scanning grating is monitored with a silicon photodiode (EG & G FND-100). Fig. 7.2 shows the scanning grating output for the most nondegenerate case, A = 31nm. This is the broadest nondegeneracy reported to date for this system. The large peak in the centre of the plot is the fundamental, at 1064 nm. The peak at the far left of the plot is the signal, that at the far right is the idler. The signal intensity is 4.3% of the 1064nm peak; the idler is much weaker. This may be intrinsic, i.e. idler production is weaker than signal, however it is more probable that this is due to the very steep roll-off in **quantum** efficiency in silicon photodetectors between 1000 & 1100 nm. The signal wavelength of 1033 nm agrees well w ith the calculated maximum of **nonlinear** gain centred at 1031 nm.

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Chapter 7
Conclusion and outlook
**Quantum** and **classical** properties of light propagating in the **nonlinear** opti- cal waveguides spark significant of research driven by fundamental interest and applications. **Nonlinear** waveguide arrays can be used to efficiently generate en- tangled photon pairs and simultaneously shape their spatial correlations through **quantum** walks. Such integrated photon sources can find applications in the development of on-chip **quantum** communication and computation devices. In recent years the **quantum** optics of attenuating media was developed. Firstly, be- cause **quantum** **information** protocols are inevitably affected by noise, secondly photonic structures composed of coupled waveguides with lossy regions offer new possibilities for shaping **optical** beams and pulses. Finally, there has been in- creased interest in **nonlinear** hybrid plasmonic waveguide as surface plasmon po- laritons offer increased field confirmation and minituisation.

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This linear relationship holds for all D and D 0 . In contrast, for the cubic process, the relation is not linear, and the log-log plot on the right in Fig. 2 shows a power-law dependence with the power-law index p. This power-law index p varies between 1.52 and 1.91 and depends on the width (∝ D 1/2 0 ) of initial PDF and stochastic forcing amplitude D, as shown in [18]. This demonstrates that **nonlinear** interaction tends to change geometric structure of a non-equilibrium process from linear to power-law scalings. In either cases here, L ∞ has a smooth variation with y 0 with its minimum value at y 0 = 0 since the equilibrium point 0 is stable. This will be compared with the behaviour in chaotic systems in

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mechanism that is employed. Finally, the feed-forward nature of linear opti- cal **quantum** computing means that we require fast, low-loss **optical** switches.
This is currently a major challenge.
An actual implementation of a linear **optical** **quantum** computer will not use bulk **optical** elements, but rather have a chip-based architecture in which microscopic waveguides are wired into programmable circuits. Beam splitters can then be constructed from evanescently coupled waveguides. By adjusting the distance between the waveguides, the transmission coefficient can in principle be carefully calibrated. Recently, photon sources have been placed in or on top of waveguides, which allows for directional coupling of the photon into the waveguide depending on the spin of the photon source [53, 54, 55]. This new technology can be employed for alternative Bell pair generation methods based on photon which-path erasure and spin readout.

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Part I (Chapters 1–4) explores what is possible if the two parties may use only **classical** communication. A well-known result by M. Nielsen says that this is inti- mately connected to the majorization relation: if x is the vector of eigenvalues of the initial state, then y can be the vector of eigenvalues of the final state if and only if x is majorized by y. It was recently observed that it is possible for x ⊗ z to be majorized by y ⊗ z, even if x is not majorized by y; physically, this means that the presence of a state with eigenvalues z is a catalyst that allows a certain transformation to occur.

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is still sufficiently low.
Conclusions and outlook
In summary, we have proposed the application of NEMS as a universal **quantum** transducer for spin–spin interactions. Com- pared with direct magnetic coupling or probabilistic **optical** en- tanglement schemes, our approach enables the implementation of long-range and deterministic spin-entanglement operations as well as the design and control of multispin interactions by simple electric circuitry. The universality of the basic underlying concept, namely to use the mechanical resonator for a coherent conversion of magnetic into electric dipoles, also opens a wide range of possibilities for the integration of electronic and nuclear spins with other charge-based **quantum** systems. Specifically, it might be interesting to consider hybrid architectures by coupling spins with transmission-line cavities 27 , charge qubits 29 and trapped ions 31,32 or atoms 30 . Moreover, this technique can be applicable for a remote magnetic sensing of ‘dark’ spins in a condensed-matter or biological environment that is incompatible with direct laser illumination. In a broader perspective, the **quantum** transducer ability of NEMS can therefore be seen as one of the fundamental applications of

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3.4.1 ‘S MALL ’ Q UANTUM S IMULATIONS
**Quantum** simulations explore physical systems that are themselves highly **quantum** in nature.
This is a potential early application because a **quantum** simulator is thought to be useful with only a small number of qubits. In addition, the laboratory environment is much more forgiving of technical problems than the commercial environment, so devices could be released by manufacturers early in the technology lifecycle. Below 30 qubits, simulations could, in general, be performed as quickly on a **classical** computer. Above about 30 qubits, a **quantum** computer can outperform any conceivable **classical** computer, for some applications, and in particular for **quantum** simulations of many-bodied problems. An example application is the study of dipole- dipole interactions: **classical** simulations of crystal unit-cells are limited to approximately 200 independent atoms before the **classical** algorithms used become unstable. If a **quantum** computer could improve this performance then it would be of great value to the field of materials research.

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Chapter 4
Reservoir Engineering
4.1 Introduction
The previous two chapters have both been focused on developing the theoretical tools we require for this one. We now turn to the task of applying this to cavity QED systems that can be used to perform useful **quantum** **information** **processing** tasks. The motivation for us here thus comes from limitations of current exper- iments with **optical** cavities. Recent progress in experiments with these **optical** cavities has mainly been motivated by potential applications in **quantum** informa- tion **processing**. These applications often require the simultaneous trapping of at least two atomic qubits inside a single resonator field mode. It has been shown that the common coupling to a quantised mode can be used to implement **quantum** gate operations [110, 100, 111, 101, 112] and the controlled generation of entanglement [113, 94, 114, 51, 115]. However, the practical realisation of these schemes with current technologies is experimentally challenging. The reason is that strong atom- cavity interactions require relatively small mode volumes and high quality mirrors;

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We also would like to admit that many physical realizations of QCM’s has been demonstrated recently. An **optical** implementation of the universal 1 → 2 QCM ( 2.24 ) for an arbitrary input qubit state based on parametric down-conversion has been demonstrated to have fidelity 0.810 ± 0.008 [ 123 ] which is in a good agreement with the theoretical prediction 5/6 = 0.833. Another physical realization of the universal QCM was achieved by using **optical** fibers doped with erbium ions. The universal cloning transformation based on this technique was shown to have fidelity F ≈ 0.82 which is again in good agreement with the theoretical prediction [ 46 ]. Also, several realistic theoretical schemes for the physical realization of a QCM on atoms in a cavity have been recently proposed [ 46 ]. However, to our knowledge no experimental results are available at the moment. Also, some physical realizations of universal and equatorial 1 → N QCMs where N = 2, 3... has been already reported. Apart of the optimal cloning transformations, a single-qubit state independent transformation — universal NOT operation ( 2.43 ) — has been experimentally demon- strated with an **optical** setup to have fidelity 0.630 ± 0.008 which is in a good agree- ment with the theoretical prediction 2/3 ≈ 0.666 [ 123 ]. However, it is worth noticing that while most of efforts has been devoted to physical realizations of 1 → N QCMs where N = 2, 3..., no attention has been paid to realization of universal and equa- torial N → N + K QCMs. Therefore, it is hard to judge whether such cloning machines and corresponding optimal multiqubit state independent transformations can be efficiently realized in practice.

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D/γ y 0 by taking the limit of t → ∞ (y → 0) in Equation (A10).
In contrast, for the cubic process, the relation is not linear, and the log-log plot on the right in Figure 2 shows a power-law dependence with the power-law index p. This power-law index p varies between 1.52 and 1.91 and depends on the width ( ∝ D 1/2 0 ) of initial PDF and stochastic forcing amplitude D, as shown in [16]. This indicates that **nonlinear** force breaks the linear scaling of geometric structure and changes it to power-law scalings. In either cases here, L ∞ has a smooth variation with y 0

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control. Since data is transmitted in the form of **optical** pulses over fiber optic net- works, problems arise when several pulse trains simultaneously request right of the same route. Certain mechanisms will be helpful if they can save the pulse trains of low priorities to yield for the high priority one. Currently, no good **optical** buffer memo- ries exist to deal with this need, and so the **optical** signal must first be converted to an electronic signal, which severely limits the overall network speed. In 1999, Hau et al. reported that the group velocity of light was slowed down to the unusual 17 m/s in an ultra cold atomic gas [7]. This has spurred ultraslow light research for applica- tions in **optical** buffers, delay lines and **quantum** **information** storage. For the sake of commercial competitiveness, ultraslow light has to be realized at room temperature by solid-state devices. However, there is an inherent conflict between the delay time and bandwidth, which limits the device capacities to an unacceptable value. The capacity limit is represented by the delay-time-bandwidth product which is related to the length of an **optical** pulse stream that can be buffered by this technique. Here, we will study the problem of delay-time-bandwidth product in slow light and a discuss a possible solution.

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As we have mentioned in the introduction, some frac- tional transforms arise under consideration of di ﬀ erent problems: description of paraxial diﬀraction in free space and in a quadratic refractive index medium, resolution of the nonstationary Schr¨odinger equation in **quantum** mechanics, phase retrieval, and so forth. Other fractional transforms can be constructed for their own sake, even if their direct ap- plication may not be obvious yet. In particular, in Section 9 we consider a general algorithm for the fractionalization of a given linear cyclic integral transform. The application of a particular fractional transform for **optical** **information** pro- cessing then depends on its properties and on the possibility of its experimental realization in optics.

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By performing the NFT of a given profile q ( z, t ) , we segre- gate two distinct components: the dispersive **nonlinear** radia- tion and the non-dispersive solitons, although either of these
two can be absent for specific profiles. For normal dispersion, the inputs localised in time cannot nucleate solitons. For the dis- persive part of the **nonlinear** spectrum, the NLSE evolution pro- duces exactly the linear phase rotation of spectral components as we have for linear systems. For the anomalous dispersion, the solitons, associated with the complex “**nonlinear** frequen- cies” (eigenvalues), in addition to the rotation of soliton phases can involve either the motion as a whole or a more nontrivial beating dynamics of bound states–the so-called multi-soliton breathers [27, 33], although inside the NFT domain the solitonic degrees of freedom remain decoupled. Note that NFT methods are much richer, more flexible and versatile with respect to the system design and performance compared to just soliton-based techniques, studied previously in many details [26, 27, 33]. In the NFT methods dealing with the discrete part of **nonlinear** spectrum (solitonic eigenvalues), the **information** carriers are not the fundamental solitons themselves but the NFT parameters (**nonlinear** spectral data) attributed to a multisolution pulse. In this sense, the traditional soliton-based transmission emerges as the simplest (and not necessarily optimal) subclass of the NFT methods. The NFT communications are, to some extent, the extension of not only the soliton-based approach but also of the coherent communication idea itself: While for the latter both signal’s amplitude and phase are used for modulation, the NFT approach goes further and employs the **nonlinear** charac- teristics of the signal.

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Palabras clave: Solitones ópticos, Ecuación no lineal de Helmholtz, Gestión de la Dispersión, Multiplexación de información cuántica, Computación cuántica, Materiales no lineales
ABSTRACT:
We outline the main research lines in **Nonlinear** and **Quantum** Optics of the Group of Photonics and **Quantum** **Information** at the University of Valladolid. These works focus on **Optical** Soli- tons, **Quantum** **Information** using Photonic Technologies and the development of new materials for Nonlinar Optics. The investigations on **optical** solitons cover both temporal solitons in disper- sion managed fiber links and nonparaxial spatial solitons as described by the **Nonlinear** Helmholtz Equation. Within the **Quantum** **Information** research lines of the group, the studies address new photonic schemes for **quantum** computation and the multiplexing of **quantum** data. The investiga- tions of the group are, to a large extent, based on intensive and parallel computations. Some asso- ciated numerical techniques for the development of the activities described are briefly sketched.

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1. Stationary Qubit Model
One of the most popular communication models in **classical** communication is the bus-based model. In this model, the bus/register is the primary unit of **information** **processing** and **information** is mediated between buses using flying bits. In the world of **quantum** **information** **processing**, this has traditionally been done by carrier particles such as photons. We can observe short-range and yet effective mediation done by the exchange interaction in spin-systems. In our stationary-qubit model, we have an integrated computing-and-communication system. Each compu- tation unit comprises of an array of spins being driven through channels, such as electrons driven by surface acoustic waves in semiconductor heterostructures, and made to interact at specific locations in the system.

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