Quantum Walks on the Line with Phase Parameters

In this paper, a study on discrete-time coined quantum walks on the line is presented. Clear mathematical foundations are still lacking for this quantum walk model. As a step towards this objective, the following question is being addressed: {\it Given a graph, what is the probability that a quantum walk arrives at a given vertex after some number of steps?} This is a very natural question, and for random walks it can be answered by several different combinatorial arguments. For quantum walks this is a highly non-trivial task. Furthermore, this was only achieved before for one specific coin operator (Hadamard operator) for walks on the line. Even considering only walks on lines, generalizing these computations to a general SU(2) coin operator is a complex task. The main contribution is a closed-form formula for the amplitudes of the state of the walk (which includes the question above) for a general symmetric SU(2) operator for walks on the line. To this end, a coin operator with parameters that alters the phase of the state of the walk is defined. Then, closed-form solutions are computed by means of Fourier analysis and asymptotic approximation methods. We also present some basic properties of the walk which can be deducted using weak convergence theorems for quantum walks. In particular, the support of the induced probability distribution of the walk is calculated. Then, it is shown how changing the parameters in the coin operator affects the resulting probability distribution.Comment: In v2 a small typo was fixed. The exponent in the definition of N_j in Theorem 3 was changed from -1/2 to 1. 20 pages, 3 figures. Presented at 10th Asian Conference on Quantum Information Science (AQIS'10). Tokyo, Japan. August 27-31, 201

A study of the optimality of PCA under spectral sparsification

Principal component analisys (PCA) is a data analysis technique for mapping points in Rn to a two or three dimensional space. This dimensionality reduction preserves the natural grouping of points and information of data.CONACYT – Consejo Nacional de Ciencia y TecnologíaPROCIENCI

Classically Time-Controlled Quantum Automata: Definition and Properties

In this paper we introduce classically time-controlled quantum automata or CTQA, which is a reasonable modification of Moore-Crutchfield quantum finite automata that uses time-dependent evolution and a "scheduler" defining how long each Hamiltonian will run. Surprisingly enough, time-dependent evolution provides a significant change in the computational power of quantum automata with respect to a discrete quantum model. Indeed, we show that if a scheduler is not computationally restricted, then a CTQA can decide the Halting problem. In order to unearth the computational capabilities of CTQAs we study the case of a computationally restricted scheduler. In particular we showed that depending on the type of restriction imposed on the scheduler, a CTQA can (i) recognize non-regular languages with cut-point, even in the presence of Karp-Lipton advice, and (ii) recognize non-regular languages with bounded-error. Furthermore, we study the closure of concatenation and union of languages by introducing a new model of Moore-Crutchfield quantum finite automata with a rotating tape head. CTQA presents itself as a new model of computation that provides a different approach to a formal study of "classical control, quantum data" schemes in quantum computing.Comment: Long revisited version of LNCS 11324:266-278, 2018 (TPNC 2018

Distributed spectral clustering on the coordinator model

Clustering is a popular subject in non-supervised learning. Spectral clustering is a method for clustering that reduces dimensionality of data and guarantees a faster convergence to almost optimal clusters.CONACYT – Consejo Nacional de Ciencia y TecnologíaPROCIENCI