Quantum Representation Research Articles

Irreducibility of quantum representations of mapping class groups with boundary

We prove that theWitten–Reshetikhin–Turaev SU(2) quantum representations of mapping class groups are always irreducible in the case of surfaces equipped with colored banded points, provided that at least one banded point is colored by 1. We thus generalize a well-known result due to J. Roberts.

An Optimized Quantum Representation for Color Digital Images

With the continuous development of quantum computation, quantum mechanics has been widely exploited to meet the storage requirement of high definition image. In this paper, an optimized quantum representation for color digital images (OCQR) is proposed, which makes full use of quantum superposition characteristic to store the RGB value of every pixel. Compared with latest novel quantum representation of color digital images (NCQI), OCQR uses nearly one-third times the qubits to store the pixel value. Meanwhile, some image processing operations related to color information can be executed more simultaneously and conveniently based on OCQR. Therefore, the proposed OCQR model is better suited to represent the quantum color image.

Woit’s Way

Peter Woit’s book Quantum Theory, Groups and Representations is based on a series of lectures that he gave at Columbia University. Woit is a fine teacher. Andrew Jordan finds much to admire.

The Kirillov picture for the Wigner particle**In memory of E C G Sudarshan.

We discuss the Kirillov method for massless Wigner particles, usually (mis)named ‘continuous spin’ or ‘infinite spin’ particles. These appear in Wigner’s classification of the unitary representations of the Poincaré group, labelled by elements of the enveloping algebra of the Poincaré Lie algebra. Now, the coadjoint orbit procedure introduced by Kirillov is a prelude to quantization. Here we exhibit for those particles the classical Casimir functions on phase space, in parallel to quantum representation theory. A good set of position coordinates are identified on the coadjoint orbits of the Wigner particles; the stabilizer subgroups and the symplectic structures of these orbits are also described.

Quantum realization of the nearest neighbor value interpolation method for INEQR

This paper presents the nearest neighbor value (NNV) interpolation algorithm for the improved novel enhanced quantum representation of digital images (INEQR). It is necessary to use interpolation in image scaling because there is an increase or a decrease in the number of pixels. The difference between the proposed scheme and nearest neighbor interpolation is that the concept applied, to estimate the missing pixel value, is guided by the nearest value rather than the distance. Firstly, a sequence of quantum operations is predefined, such as cyclic shift transformations and the basic arithmetic operations. Then, the feasibility of the nearest neighbor value interpolation method for quantum image of INEQR is proven using the previously designed quantum operations. Furthermore, quantum image scaling algorithm in the form of circuits of the NNV interpolation for INEQR is constructed for the first time. The merit of the proposed INEQR circuit lies in their low complexity, which is achieved by utilizing the unique properties of quantum superposition and entanglement. Finally, simulation-based experimental results involving different classical images and ratios (i.e., conventional or non-quantum) are simulated based on the classical computer’s MATLAB 2014b software, which demonstrates that the proposed interpolation method has higher performances in terms of high resolution compared to the nearest neighbor and bilinear interpolation.

Fuzzy Matching Based on Gray-scale Difference for Quantum Images

Quantum image processing has recently emerged as an essential problem in practical tasks, e.g. real-time image matching. Previous studies have shown that the superposition and entanglement of quantum can greatly improve the efficiency of complex image processing. In this paper, a fuzzy quantum image matching scheme based on gray-scale difference is proposed to find out the target region in a reference image, which is very similar to the template image. Firstly, we employ the proposed enhanced quantum representation (NEQR) to store digital images. Then some certain quantum operations are used to evaluate the gray-scale difference between two quantum images by thresholding. If all of the obtained gray-scale differences are not greater than the threshold value, it indicates a successful fuzzy matching of quantum images. Theoretical analysis and experiments show that the proposed scheme performs fuzzy matching at a low cost and also enables exponentially significant speedup via quantum parallel computation.

Quantum color image watermarking based on Arnold transformation and LSB steganography

In this paper, a quantum color image watermarking scheme is proposed through twice-scrambling of Arnold transformations and steganography of least significant bit (LSB). Both carrier image and watermark images are represented by the novel quantum representation of color digital images model (NCQI). The image sizes for carrier and watermark are assumed to be [Formula: see text] and [Formula: see text], respectively. At first, the watermark is scrambled into a disordered form through image preprocessing technique of exchanging the image pixel position and altering the color information based on Arnold transforms, simultaneously. Then, the scrambled watermark with [Formula: see text] image size and 24-qubit grayscale is further expanded to an image with size [Formula: see text] and 6-qubit grayscale using the nearest-neighbor interpolation method. Finally, the scrambled and expanded watermark is embedded into the carrier by steganography of LSB scheme, and a key image with [Formula: see text] size and 3-qubit information is generated at the meantime, which only can use the key image to retrieve the original watermark. The extraction of watermark is the reverse process of embedding, which is achieved by applying a sequence of operations in the reverse order. Simulation-based experimental results involving different carrier and watermark images (i.e. conventional or non-quantum) are simulated based on the classical computer’s MATLAB 2014b software, which illustrates that the present method has a good performance in terms of three items: visual quality, robustness and steganography capacity.

QRMW: quantum representation of multi wavelength images

In this study, we propose quantum representation of multi wavelength images (QRMW) which gives preparation and retrieving procedures of quantum images. Proposed QRMW model represents multi-channel and 2^n x 2^m images. Also, we present image comparison and some image operations based on QRMW model. Comparing our model with the models in literature, QRMW model has less time complexity. Also QRMW model uses fewer qubits than existing models in the literature.

Profinite Completions of Burnside-Type Quotients of Surface Groups

Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.

A multilayer multiconfiguration time-dependent Hartree study of the nonequilibrium Anderson impurity model at zero temperature

Quantum transport is studied for the nonequilibrium Anderson impurity model at zero temperature employing the multilayer multiconfiguration time-dependent Hartree theory within the second quantization representation (ML-MCTDH-SQR) of Fock space. To address both linear and nonlinear conductance in the Kondo regime, two new techniques of the ML-MCTDH-SQR simulation methodology are introduced: (i) the use of correlated initial states, which is achieved by imaginary time propagation of the overall Hamiltonian at zero voltage and (ii) the adoption of the logarithmic discretization of the electronic continuum. Employing the improved methodology, the signature of the Kondo effect is analyzed.

A Quantum Image Representation Based on Bitplanes

Quantum image representation plays an important role in quantum image processing. In this paper, a bitplane representation of quantum images (BRQI) is proposed, which uses (n+4) or (n+6) qubits to store a grayscale or RGB color image of ${2^{n}}$ pixels. Compared with a novel enhanced quantum representation (NEQR), and a novel quantum representation of color digital images (NCQI), the storage capacity of BRQI improves 16 times and 218 times, respectively. Next, some quantum operations based on BRQI are proposed, these operations including complement of colors, reverse of bitplanes, and translation of bitplanes. Analyzed the implementation circuits of these operations, the result indicates that these operations based on BRQI have lower quantum cost than NEQR and NCQI. Furthermore, an image scrambling algorithm based on BRQI is presented by combining the above operations. Simulation experiments and performance analysis show that proposed scrambling algorithm is effective and efficient.

Continuum approach to the BF vacuum: The U(1) case

A quantum representation of holonomies and exponentiated fluxes of a U(1) gauge theory that contains the Pullin-Dittrich-Geiller (DG) vacuum is presented and discussed. Our quantization is performed manifestly in a continuum theory, without any discretization. The discreteness emerges on the quantum level as a property of the spectrum of the quantum holonomy operators. The new type of a cylindrical consistency present in the DG approach now follows easily and naturally. A generalization to the non-Abelian case seems possible.

Quantum multiverse optimization algorithm for optimization problems

In this paper, a new hybrid algorithm called quantum multiverse optimization (QMVO) is proposed. The proposed QMVO is based on quantum computing and multiverse optimization (MVO) algorithm. The main features of quantum theory and MVO were applied in a new algorithm to find the optimal trade-off between exploration and exploitation. QMVO algorithm depends on adopting a quantum representation of the search space and the integration of the quantum interference and operators in the multiverse optimization algorithm to obtain the optimal solution of the objective function. The performance of QMVO algorithm is evaluated by using 50 unimodal and multimodal benchmark functions. The experimental results show that the proposed algorithm has comprehensive superiority in solving complex numerical optimization problems. Also, the results show that the proposed QMVO is a promising optimization algorithm compared with other well-known and popular algorithms.

Efficient quantization and fixed-point representation for MIMO turbo-detection and turbo-demapping

In the domain of wireless digital communication, floating-point arithmetic is generally used to conduct performance evaluation studies of algorithms. This is typically limited to theoretical performance evaluation in terms of communication quality and error rates. For a practical implementation perspective, using fixed-point arithmetic instead of floating-point reduces significantly implementation costs in terms of area occupation and energy consumption. However, this implies a complex conversion process, particularly if the considered algorithm includes complex arithmetic operations with high accuracy requirements and if the target system presents many configuration parameters. In this context, the purpose of the paper is to present an efficient quantization and fixed-point representation for turbo-detection and turbo-demapping. The impact of floating-to-fixed-point conversion is illustrated upon the error-rate performance of the receiver for different system configurations. Only a slight degradation in the error-rate performance of the receiver is observed when implementing the detector and demapper modules which utilize the devised quantization and fixed-point arithmetic rather than floating-point arithmetic.

Nonautonomous Hamiltonian quantum systems, operator equations, and representations of the Bender–Dunne Weyl-ordered basis under time-dependent canonical transformations

We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the basis elements.

Asymptotic behavior of the spectrum of combination scattering at Stokes phonons

For a class of polynomial quantum Hamiltonians used in models of combination scattering in quantum optics, we obtain the asymptotic behavior of the spectrum for large occupation numbers in the secondary quantization representation. Hamiltonians of this class can be diagonalized using a special system of polynomials determined by recurrence relations with coefficients depending on a parameter (occupation number). For this system of polynomials, we determine the asymptotic behavior a discrete measure with respect to which they are orthogonal. The obtained limit measures are interpreted as equilibrium measures in extremum problems for a logarithmic potential in an external field and with constraints on the measure. We illustrate the general case with an exactly solvable example where the Hamiltonian can be diagonalized by the canonical Bogoliubov transformation and the special orthogonal polynomials degenerate into the Krawtchouk classical discrete polynomials.

Character varieties as a tensor product

In this short note we show that representation and character varieties of discrete groups can be viewed as tensor products of suitable functors over the PROP of cocommutative Hopf algebras. Such view point has several interesting applications. First, it gives a straightforward way of deriving the functor sending a discrete group to the functions on its representation variety, which leads to representation homology. Second, using a suitable deformation of the functors involved in this construction, one can obtain deformations of the representation and character varieties for the fundamental groups of 3-manifolds, and could lead to better understanding of quantum representations of mapping class groups.

Power Allocation in Mobile Cellular Communication Under Multiplicative Noise and Interference Uncertainty

In this paper, we study the problem of closed-loop power allocation in mobile cellular-communication networks subject to multiplicative uncertainty. Multiple-access interference, nonuniform quantization, and finite wordlength representations are all characterized by multiplicative uncertainty models in the forward and feedback paths of closed-loop power allocation strategies. The quality of service (QoS) in the cellular network is evaluated with respect to signal to interference-noise ratio after the estimation process, where the uplink channel (mobile users to base station) is addressed under arbitrary round-trip delays per user. The multiplicative uncertainty terms are modeled by a stochastic framework, where the mean-square small gain theorem is employed to evaluate robust stability. Two inequalities are then derived to check internal stability with respect to the norms of the multiplicative uncertainty terms, and the sensitivity and its complementary transfer functions of the closed-loop system. The distributed power control scheme that maximizes the stability upper bounds is derived that depends on the round-trip delay per user, and whose structure enables simplifying the inequalities to evaluate robust stability. By establishing a numerical relation between the size of the interference and the multiplicative noise, a unique stability bound is derived for closed-loop power allocation. A numerical evaluation was carried out in a code-division multiple-access cellular network with multiple base stations, and under distinct QoS requirements.

Complex Structures and Quantum Representations for Scalar QFT in Curved Spacetimes

We confirm the equivalence of the Schr\"odinger representation and the holomorphic one, based on previous results of the General Boundary Formulation (GBF) of quantum field theory. On a wide class of curved spacetimes, we consider real Klein-Gordon theory in two types of regions: interval regions (consisting e.g. of a time interval times all of space), and rod regions (a solid ball of space extended over all of time). Using mode expansions, we provide explicit expressions for the Schr\"odinger vacuum (which determines this representation) and for the corresponding complex structure on the space of classical solutions (which determines the holomorphic representation). For both representations we give the corresponding coherent states and calculate the generalized free transition amplitudes of the GBF, which coincide and hence confirm the equivalence of the two representations. We also transcribe the complex structure to phase space and show that it agrees with earlier results.

Quantum Endpoint Detection Based on QRDA

Speech recognition technology is widely used in many applications for man - machine interaction. To face more and more speech data, the computation of speech processing needs new approaches. The quantum computation is one of emerging computation technology and has been seen as useful computation model. So we focus on the basic operation of speech recognition processing, the voice activity detection, to present quantum endpoint detection algorithm. In order to achieve this algorithm, the n-bits quantum comparator circuit is given firstly. Then based on QRDA(Quantum Representation of Digital Audio), a quantum endpoint detection algorithm is presented. These quantum circuits could efficient process the audio data in quantum computer.