Higher-dimensional Quantum Walk In Environment Of Quantum Bernoulli Noises And Its Applications To Quantum Parameter Estimation

Quantum walks,also known as quantum random walks,are quantum analogs of the classical random walks.They play an important role in quantum information theory(especially in quantum computing).Generally a model of quantum walks is described by the state space.The state space contains the walker's internal degrees of freedom as well as the location information.A space is called the coin space if it describes internal degrees of freedom while the operators on the coin space that determine the moving directions of the walker are known as the coin operators.Most of the research has focused on the finite dimensional coin space but little on the infinite dimensional case.Introduced in the recent decade,quantum Bernoulli noises refer to Bernoulli noise functionals and the annihilation and creation operators acting on them,which can be viewed as the operators acting on infinite-dimensional space.In this thesis,we investigate further application of quantum Bernoulli noises to quantum walks and other related problems.Our main work is as follows.(1)Constructing a general framework of coin operators.We define an abstract coin operator couple in the framework of a general Hilbert space,and establish two characterization theorems for coin operator couples,which show links between coin operator couples and unitary operators together with projection operators(self-adjoint unitary operators).We also define an abstract coin operator system in the framework of a general Hilbert space,and two characterization theorems are obtained accordingly for coin operator systems.(2)Constructing models of unitary quantum walks.We generalize the model of1-dimensional unitary quantum walks based on quantum Bernoulli noises and show that the commutativity of the coin operator couple(sequence)can have a strong impact on the model's probability distribution and its evolution behavior.Our results either generalize or make complete the existing ones in the literature,correspondingly.By using a coin operator system,we introduce a general model of high-dimensional unitary quantum walk based on quantum Bernoulli noises.Giving a tensor representation as well as a Fourier representation of the model,we establish a formula calculating its probability distributions.(3)Constructing models of open quantum walks.We generalize the model of 1-dimensional open quantum walks based on quantum Bernoulli noises and show its links with its unitary counterpart.We introduce high-dimensional nucleuses on the space of square integrable Bernoulli functionals,and then use them to define a general model of highdimensional open quantum walks based on quantum Bernoulli noises.Offering a quantum channel representation of the model,we examine its separability,probability distribution property and other properties.(4)Applications in Quantum Parameter Estimation.Finally,as an application of our high-dimensional quantum walk models,we consider quantum parameter estimation in 1-dimensional quantum walks.Novel ideas are proposed for constructing estimation schemes via our high-dimensional quantum walk models,and example are also shown.