This PDF file contains the front matter associated with SPIE Proceedings Volume 8057, including the Title Page, Copyright information, Table of Contents, and the Conference Committee listing.

Extending the methods from our previous work on quantum knots and quantum graphs, we describe a general procedure for quantizing a large class of mathematical structures which includes, for example, knots, graphs, groups, algebraic varieties, categories, topological spaces, geometric spaces, and more. This procedure is different from that normally found in quantum topology. We then demonstrate the power of this method by using it to quantize braids. This general method produces a blueprint of a quantum system which is physically implementable in the same sense that Shor's quantum factoring algorithm is physically implementable. Mathematical invariants become objects that are physically observable.

Spontaneous parametric down-conversion (SPDC) is a reliable and robust source of photons for quantum information applications. For applications that involve operations such as entanglement swapping or single-photon heralding, two-photon states are required to be factorable (uncorrelated) in their spectral and spatial degrees of freedom. We report the design and experimental characterization of an SPDC source that has been optimized for high spectral and spatial purity. The source is pumped by the 776 nm output of a mode-locked Ti:Sapphire laser and consists of a periodically-poled Potassium Titanyl Phosphate (PPKTP) crystal phase-matched for collinear type-II SPDC. The dispersive properties of PPKTP at these wavelengths is such that it is possible to minimize the spectral entanglement by matching the widths of the pump to the spectral phase-matching function. The spatial entanglement is minimized through careful control of the pump focus, yielding nearly single-mode emission. An advantage of this approach is that the emission rate into the collection modes is very high, resulting in a very bright SPDC source. We also report a scheme that employs the output of collinear sources such as these to produce polarization-entangled photon pairs. The scheme, which requires only simple polarization elements, can be scaled to N-photon GHZ states.

Modeling the spontaneous emission of a single-photon emitted from a quantum dot is important for several proposed solid-state quantum computers and quantum networks. In this study we seek to model the fully quantized excitation and spontaneous decay of a quantum dot state through the optical emission of a photon in the non-Markovian limit of the photon bath. We propose the use of discretized central-difference approximations of space and time partial derivatives to describe the interaction between single photon and quantum dot states.

In this work, we discuss a novel compact source that generates six pairs of entangled photons via spontaneous parametric down-conversion from a single pass of a pump beam through a crystal assembly. The experimental demonstrations reported are at 810 nm so as to utilize high quantum efficiency Si-APD detectors, but the design can be readily implemented in other wavelength regimes including the telecom bands near 1550 nm. An immediate application of this source enables particular multi-qubit cluster states to be generated in a highly compact unidirectional configuration. This can significantly simplify the interferometric stability, as well as feed-forward methods required in photon-based quantum logic circuitry.

This article presents a proposal for producing photon pairs in states that are entangled in their spatial modes. The method sends collinear pairs of photons to an unbalanced interferometer with diffractive optical elements, and uses coherence and timing discrimination to create entangled states.

Bell's theorem, and inequalities that stem from it, address the conflict between the explanation of key experimental observations by quantum mechanics (QM) and by models expressing Locally Realistic (LR) properties, regardless of their inclusion or exclusion of hidden variables. To demonstrate the conflict between experimental results described by QM and LR models, a physical realization of the quantum state must be chosen. Entangled photons or electrons provide the most viable choices. In this work we consider a simplified version of a Bell inequality (BI) that focuses entirely on the physical state properties of photons in order to demonstrate the difference between QM and LR correlations. While the experiment we propose is in principle similar in intent to prior Bell inequality experiments, our version requires fewer measurements, and is more advantageous in its conceptual clarity.

A factorization scheme will be introduced that potentially permits any N00N or M&M state to be generated and detected. Explicit results for N00N states are presented for N=3. Internal loss within the entanglement generator and external loss due to atmosphere, detectors and targets are modeled. A method using these approaches for quantum entanglement based imaging is provided that gives N times classical resolution, where N is the number photons entangled with explicit results exhibited for N=3. Closed form expressions for the wave function, normalization, density matrix, reduced density matrix, visibility, and probabilities of detection of one through three photons using detectors with general polarization characteristics are provided. Explicit entanglement generator and detector designs are provided in terms of linear and nonlinear photonics devices. The fundamental role of post-selection measurement for generating entanglement is included. A general factorization scheme for M&M states is provided. Discussions of entanglement devices that will produce general M&M states at near visible frequencies are given. A discussion of a bearing measurement device that exhibits both super sensitivity and resolution is provided. Computational results are provided that compare probabilities of detection for three single photon detectors with -45, 45, and 45 degree linear polarization. Results for detecting one to three photons or the vacuum state are compared. Computational results for detecting three photons with these detectors for various values of internal and atmospheric loss are provided. Resolution improvements born of quantum entanglement are shown not to degrade with loss. Loss degrades probability of detection not resolution.

A novel quantum phase amplification (QPA) technique is proposed to enhance the spatial resolution of passive imaging systems, without the commensurate increase of the aperture size. The key component of this approach is the production of phase-amplified light, a squeezed state of the electromagnetic field that can be generated in phase-sensitive three-wave mixing (PSTWM). We have established the theoretical basis for QPA using PSTWM and theoretically derived the conditions for the operation of the ubiquitous PSTWM realized by using standard laser technology. In our approach sub-Rayleigh imaging can be explored to perform the super-resolution enhancement of remotely sensed data. We applied the PSTWM to improve the distinguishability of two point sources in several scenarios. In addition, detector segmentation was modeled to optimize the signal-to-noise ratio.

We present an experimental demonstration of an all-optical flip-flop based on mode competition in a polarization-coupled two-energy system. The two energy systems are comprised of a semiconductor optical amplifier and a supercontinuum generator. Adjusting the polarization state of the coupling between the two energy systems allows for a stable cycling between the four states of a flip-flop namely: (off, off); (off, on); (on, off); (on, on). So that, in this case, representing qubit by orthogonal polarization states enables qubit storage and logic operations.

Quantum gates and simple quantum algorithms can be designed utilizing the diffraction phenomena of a photon within a multiplexed holographic element. The quantum eigenstates we use are the photon's linear momentum (LM) as measured by the number of waves of tilt across the aperture. Two properties of quantum computing within the circuit model make this approach attractive. First, any conditional measurement can be commuted in time with any unitary quantum gate - the timeless nature of quantum computing. Second, photon entanglement can be encoded as a superposition state of a single photon in a higher-dimensional state space afforded by LM. Our theoretical and numerical results indicate that OptiGrate's photo-thermal refractive (PTR) glass is an enabling technology. We will review our previous design of a quantum projection operator and give credence to this approach on a representative quantum gate grounded on coupled-mode theory and numerical simulations, all with parameters consistent with PTR glass. We discuss the strengths (high efficiencies, robustness to environment) and limitations (scalability, crosstalk) of this technology. While not scalable, the utility and robustness of such optical elements for broader quantum information processing applications can be substantial.

In this work we compare the accuracy of two methods used to construct a logical zero state appropriate for the [7, 1, 3] CSS quantum error correction code in a non-equiprobable Pauli operator error environment. The first method is to apply error correction, via syndrome measurement, on seven physical qubits all in the state zero. To do the syndrome measurements in a fault-tolerant fashion requires the construction of four qubit Shor states. These Shor states are also assumed to be constructed in a non-equiprobable Pauli operator error environment and it is these that are used to implement the syndrome measurement. The second construction method is to implement the [7, 1, 3] encoding gate sequence, also in the nonequiprobable Pauli operator error environment. The fidelity of the output states is calculated for each of these methods. With respect to the Shor state construction we find that the implementation of (noisy) parity based verifications does not necessarily raise the fidelity of the resulting Shor state. We also find that the second logical zero construction method outputs a seven qubit state with a respectfully higher fidelity than the first (fault tolerant) method. However, the fidelity of the single qubit of stored information has almost equivalent fidelity from the two construction methods.

Quantum vortex structures and energy cascades are examined for two dimensional quantum turbulence (2D QT) using a special unitary evolution algorithm. The qubit lattice gas (QLG) algorithm, is employed to simulate the weakly-coupled Bose-Einstein condensate (BEC) governed by the Gross-Pitaevskii (GP) equation. A parameter regime is uncovered in which, as in 3D QT, there is a very short Poincare recurrence time. This short recurrence time is destroyed as the nonlinear interaction energy is increased. Energy cascades for 2D QT are considered to examine whether 2D QT exhibits the inverse cascades of 2D classical turbulence. In the parameter regime considered, the spectra analysis reveals no such dual cascades---dual cascades being a hallmark of 2D classical turbulence.

Research in contemporary physics is emphasizing the development and evolution of computer systems to facilitate the calculations. Quantum computing is a branch of modern physics is believed promising results for the future, Thanks to the ability of qubits to store more information than a bit. The work of this paper focuses on the simulation of certain quantum algorithms such as the prisoner's dilemma in its quantum version using the MATHEMATICA® software and implementing stochastic version of the software MAPLE ® and the Grover search algorithm that simulates finding a needle in a haystack.

Causal connectivity at warp speed is a possible consequence of a limiting proper acceleration relative to the vacuum. Normally this would occur only near the Planck scale of spatial separation between two devices measuring the field but at much larger separations when the relative speed of the two measuring devices is near the standard speed of light in vacuum.

Quantum walks have been studied under several regimes. Motivated by experimental results on quantum weak measurements and weak values as well as by the need to develop new insights for quantum algorithm development, we are extending our knowledge by studying the behavior of quantum walks under the regime of quantum weak measurements and weak values of pre- and postselected measurements (QWWM hereinafter). In particular, we investigate the limiting position probability distribution and several statistical measures (such as standard deviation) of a QWWM on an infinite line, and compare such results with corresponding classical and quantum walks position probability distributions and statistical measures, stressing the differences provided by weak measurements and weak values with respect to results computed by using canonical observables. We start by producing a concise introduction to quantum weak values and quantum weak measurements. We then introduce definitions as well as both analytical and numerical results for a QWWM under Hadamard evolution and extend our analysis to quantum evolution ruled by general unitary operators. Moreover, we propose a definition and focus on the properties of mixing time of QWWM on an infinite line, followed by a comparison of known corresponding results for classical and quantum walks mixing times. We finish this paper by presenting a plausible experimental implementation of a QWWM.

We show that communication of single-photon quantum states in a multi-user environment is improved by using spread spectrum communication techniques. We describe a framework for spreading, transmitting, despreading, and detecting single-photon spectral states that mimics conventional spread spectrum techniques. We show in the cases of inadvertent detection, unintentional interference, and multi-user management, that quantum spread spectrum communications may minimize receiver errors by managing quantum channel access.

A working definition of the term "quantum game" is developed in an attempt to gain insights into aspects of quantum mechanics via game theory.

We describe an optical implementation of a CNOT gate in which the control qubit is the polarization of a single photon and the target qubit is the spatial parity of the same photon. The gate is implemented with a polarizationsensitive spatial light modulator. We characterize the operation of the gate using quantum process tomography and the spatial parity is analyzed with a modified Mach-Zehnder interferometer. We also demonstrate the CNOT-gate operation with arbitrary rotation of the target qubit and discuss the possibility of implementing multi-qubit CNOT gates using the same approach.

We show how to construct quantum cellular automata (QCA) based on the formalism introduced by Watrous but without that formalism's "quiescent states," by using shift-invariant Lebesgue measure on Cantor space. Although QCA's with quiescent states are strictly sufficient for computational purposes, removing quiescent states as a requirement allows global QCA states with infinite support that allows the state space of the QCA to be identified with the class of interpretations of logic-based formalism in a formal methods approach to proving the correctness correctness of QCAs with respect to formal specifications.

In the context of quantum theory, recently we distinguished mathematics for expressing evidence from mathematics for explaining evidence. Here this distinction is made in spacetime physics. We offer a system of mathematical thought-or as termed in geodesy a reference system-for evidence, separated out from additional assumptions of a geometry in terms of which to explain that evidence. The offered reference system for evidence, free of any assumption of a particular explanatory geometry, whether Euclidean or general relativistic, amounts to a (theoretical) "assemblage of histories accumulated in the memories of parties to a synchronous communications network." The assemblage of histories gives voice to the known experimental finding, sometimes forgotten by theorists, that any memory device for recording logical symbols must be insensitive to variations in signals in which those symbols are carried. Out of acknowledging this insensitivity comes an appreciation of rhythms essential to the communication of digital symbols and of the need for analog measurements to maintain these rhythms. The separate reference system for evidence reconciles what otherwise is a conflict between the demand in quantum mechanics for repeatable experiments and the lack in spacetime metrics appropriate to the Global Positioning System of any exact symmetry, a lack that rules out an isometry between two spacetime regions for two occurrences of an experiment.

Forbidden transitions often have moderate energy separations and long lifetimes. They can be induced in elements by external force including laser absorption, magnetic effect, crystal forces and collision with energetic particles. In this paper we discuss the prospects of quantum information processing involving forbidden transitions in the inner shell of Ytterbium. We derive the Schrodinger wave equation for the qubit states with the ³P and 1S as basis states. We also determine a coherent norm calculate the decoherence rate for trapped atoms.

The best design for practical quantum computing is one that emulates the multi-agent quantum logic function of natural biological systems. Such systems are theorized to be based upon a quantum gate formed by a nucleic acid Szilard engine (NASE) that converts Shannon entropy of encountered molecules into useful work of nucleic acid geometric reconfiguration. This theoretical mechanism is logically and thermodynamically reversible in this special case because it is literally constructed out of the (nucleic acid) information necessary for its function, thereby allowing the nucleic acid Szilard engine to function reversibly because, since the information by which it functions exists on both sides of the theoretical mechanism simultaneously, there would be no build-up of information within the theoretical mechanism, and therefore no irreversible thermodynamic energy cost would be necessary to erase information inside the mechanism. This symmetry breaking Szilard engine function is associated with emission and/or absorption of entangled photons that can provide quantum synchronization of other nucleic acid segments within and between cells. In this manner nucleic acids can be considered as a natural model of topological quantum computing in which the nonabelian interaction of genes can be represented within quantum knot/braid theory as anyon crosses determined by entropic loss or gain that leads to changes in nucleic acid covalent bond angles. This naturally occurring biological form of topological quantum computing can serve as a model for workable man-made multi-agent quantum computing systems.

We discuss an approach to encoding qubits in the spatial distribution of single photons and entangled-photon pairs based on utilizing the spatial parity of the photon wave fronts. Using simple optical components, that do not require nonlinearities or moving parts, we discuss implementing rotations in the Hilbert space of spatial parity and measurement of the state of parity. Using entangled photon pairs, we use spatial parity to demonstrate quantum nonlocality by violating a Bell inequality using an EPR state. We also discuss generalizations of this scheme that may allow for a larger number of qubits to be encoded per photon.

Grover's oracle based unstructured search algorithm is often stated as "given a phone number in a directory, find the associated name." More formally, the problem can be stated as "given as input a unitary black box Uf for computing an unknown function f:{0,1}n →{0,1}find x=x0 an element of {0,1}n such that f(x0) =1, (and zero otherwise). The crucial role of the externally supplied oracle Uf (whose inner workings are unknown to the user) is to change the sign of the solution 0 x , while leaving all other states unaltered. Thus, Uf depends on the desired solution x0. This paper examines an amplitude amplification algorithm in which the user encodes the directory (e.g. names and telephone numbers) into an entangled database state, which at a later time can be queried on one supplied component entry (e.g. a given phone number t0) to find the other associated unknown component (e.g. name x0). For N=2n names x with N associated phone numbers t , performing amplitude amplification on a subspace of size N of the total space of size N2 produces the desired state 0 0 x t in √N steps. We discuss how and why sequential (though not concurrent parallel) searches can be performed on multiple database states. Finally, we show how this procedure can be generalized to databases with more than two correlated lists (e.g. x t s r ...).

This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic and combinatorial structures. We include a description of work of the first author on the construction of Hilbert spaces from the states of the bracket polynomial with applications to algorithms for the Jones polynomial and relations with Khovanov homology. The purpose of this paper is to place such constructions in a general context of the quantization of combinatorial structures.

An emergent trend in quantum computation is the topological quantum computation (TQC). Briefly, TQC results from the application of quantum computation with the aim to solve the problems of quantum topology such as topological invariants for knots and links (Jones polynomials, HOMFLY polynomials, Khovanov polynomials); topological invariants for graphs (Tutte polynomial and Bollobás-Riordan polynomial); topological invariants for 3-manifolds (Reshetiskin-Turaev, Turaev-Viro and Turaer-Viro-Ocneanu invariants) and topological invariants for 4-manifolds( Crane-Yetter invariants). In a few words, TQC is concerned with the formulation of quantum algorithms for the computation of these topological invariants in quantum topology. Given that one of the fundamental achievements of quantum topology was the discovery of strong connections between monoidal categories and 3-dimensional manifolds, in TQC is possible and necessary to exploit such connections with the purpose to formulate universal quantum algorithms for topological invariants of 3-manifolds. In the present work we make an exploration of such possibilities. Specifically we search for universal quantum algorithms for generalized Turaev-Viro invariants of 3-manifolds such as the Turaev-Viro-Ocneanu invariants, the Kashaev-Baseilhac-Benedetti invariants of 3-manifolds with links and the Geer-Kashaev-Turaev invariants of 3-manifolds with a link and a principal bundle. We also look for physical systems (three dimensional topological insulators and three-dimensional gravity) over which implement the resulting universal topological quantum algorithms.

One of the current trends in quantum computing is the application of algebraic topological methods in the design of new algorithms and quantum computers, giving rise to topological quantum computing. One of the tools used in it is the Yang-Baxter equation whose solutions are interpreted as universal quantum gates. Lately, more general Yang-Baxter equations have been investigated, making progress as two-spectral equations and Yang-Baxter systems. This paper intends to apply these new findings to the field of topological quantum computation, more specifically, the proposition of the two-spectral Yang-Baxter operators as universal quantum gates for 2 qubits and 2 qutrits systems, obtaining 4x4 and 9x9 matrices respectively, and further elaboration of the corresponding Hamiltonian by the use of computer algebra software Mathematica® and its Qucalc package. In addition, possible physical systems to which the Yang-Baxter operators obtained can be applied are considered. In the present work it is demonstrated the utility of the Yang-Baxter equation to generate universal quantum gates and the power of computer algebra to design them; it is expected that these mathematical studies contribute to the further development of quantum computers

Gauge theories can be described by assigning a vector space ¯V (x) to each space time point x. A common set of complex numbers, ¯ C, is usually assumed to be the set of scalars for all the ¯ V_x. This is expanded here to assign a separate set of scalars, ¯ C_x, to ¯ V_x. The freedom of choice of bases, expressed by the action of a gauge group operator on the ¯V_x, is expanded here to include the freedom of choice of scale factors, _cy, _x, as elements of GL(1, C) that relate ¯ Cy to ¯ C_x. A gauge field representation of cy,x gives two gauge fields, A(x) and iB (x). Inclusion of these fields in the covariant derivatives of Lagrangians results in A(x) appearing as a gauge boson for which mass is optional and B(x) as a massless gauge boson. B(x) appears to be the photon field. The nature of A(x) is not known at present. One does know that the coupling constant of A(x) to matter fields is very small compared to the fine structure constant.