Discrete event systems (DES) have been established and deeply developed in the framework of probabilistic and fuzzy computing models due to the necessity of practical applications in fuzzy and probabilistic systems. With the development of quantum computing and quantum control, a natural problem is to simulate DES by means of quantum computing models and to establish {\it quantum DES} (QDES). The motivation is twofold: on the one hand, QDES have potential applications when DES are simulated and processed by quantum computers, where quantum systems are employed to simulate the evolution of states driven by discrete events, and on the other hand, QDES may have essential advantages over DES concerning state complexity for imitating some practical problems. The goal of this paper is to establish a basic framework of QDES by using {\it quantum finite automata} (QFA) as the modelling formalisms, and the supervisory control theorems of QDES are established and proved. Then we present a polynomial-time algorithm to decide whether or not the controllability condition holds. In particular, we construct a number of new examples of QFA to illustrate the supervisory control of QDES and to verify the essential advantages of QDES over DES in state complexity.

We suggest a nonparametric framework for unsupervised learning of projection models in terms of density estimation on quantized sample spaces. The objective is not to optimally reconstruct the data but instead thequantizer is chosen to optimally reconstruct the density of the data. For the resulting quantizing density estimator (QDE) we present a general method for parameter estimation and model selection. We show how projection sets which correspond to traditional unsupervised methods likevector quantization or PCA appear in the new framework. For a principal component quantizer we present results on synthetic and realworld data,which show that the QDE can improve the generalization of the kernel density estimator although its estimate is based on significantly lower-dimensional projection indices of the data.