89 research outputs found
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 TecnologÃaPROCIENCI
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 TecnologÃaPROCIENCI
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