Obtaining the Quantum Fourier Transform from the Classical FFT with QR Decomposition

We present the detailed process of converting the classical Fourier Transform algorithm into the quantum one by using QR decomposition. This provides an example of a technique for building quantum algorithms using classical ones. The Quantum Fourier Transform is one of the most important quantum subroutines known at present, used in most algorithms that have exponential speed up compared to the classical ones. We briefly review Fast Fourier Transform and then make explicit all the steps that led to the quantum formulation of the algorithm, generalizing Coppersmith's work.Comment: 12 pages, 1 figure (generated within LaTeX). To appear in Journal of Computational and Applied Mathematic

Image enhancement for a Bose-Einstein condensate interferometer

The atom, thanks to its wave behaviour, can manifest phenomena which are, usually, associated to light: interference is one of them. The possibility of cooling atomic clouds and manipulating the states of the atoms contained in them opened many new opportunities to exploit these states in many ways; one of them is measuring various kinds of physical observables with high precision, thanks to the aforementioned interference phenomena: this is atom interferometry. Since the first Bose-Einstein condensates in atomic gases were obtained, there has been a keen interest in interference between them, as it would mean to observe coherent quantum phenomena between macroscopic objects. Nevertheless, the high atomic density of condensates with respect to non condensed, thermal atomic clouds makes it difficult to ignore the effects of interactions within them. For the applications, understanding the role of interactions in the formation of interference figures is crucial. In this thesis, an algorithm for the enhancement of absorption images of a condensate has been developed. This algorithm computes an image basis for the noise and then remove the projection of the starting image from this basis, thus obtaining a clean image. This algorithm has then been applied to the enhancement of images obtained from atom interferometry. These images have then been analyzed using two techniques, and the obtained results have been compared to those for an ideal condensate. The results have been found not compatible with the ideal case, and are then due to atom-atom interactions

Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems

Solving linear systems is of great importance in numerous fields. In particular, circulant systems are especially valuable for efficiently finding numerical solutions to physics-related differential equations. Current quantum algorithms like HHL or variational methods are either resource-intensive or may fail to find a solution. We present an efficient algorithm based on convex optimization of combinations of quantum states to solve for banded circulant linear systems whose non-zero terms are within distance $K$ of the main diagonal. By decomposing banded circulant matrices into cyclic permutations, our approach produces approximate solutions to such systems with a combination of quantum states linear to $K$, significantly improving over previous convergence guarantees, which require quantum states exponential to $K$. We propose a hybrid quantum-classical algorithm using the Hadamard test and the quantum Fourier transform as subroutines and show its PromiseBQP-hardness. Additionally, we introduce a quantum-inspired algorithm with similar performance given sample and query access. We validate our methods with classical simulations and actual IBM quantum computer implementation, showcasing their applicability for solving physical problems such as heat transfer.Comment: 21 pages, 12 figure

Intelligent OFDM telecommunication system. Part 2. Examples of complex and quaternion many-parameter transforms

In this paper, we propose unified mathematical forms of many-parametric complex and quaternion Fourier transforms for novel Intelligent OFDM-telecommunication systems (OFDM-TCS). Each many-parametric transform (MPT) depends on many free angle parameters. When parameters are changed in some way, the type and form of transform are changed as well. For example, MPT may be the Fourier transform for one set of parameters, wavelet transform for other parameters and other transforms for other values of parameters. The new Intelligent-OFDM-TCS uses inverse MPT for modulation at the transmitter and direct MPT for demodulation at the receiver. © 2019 IOP Publishing Ltd. All rights reserved