| Quantum walk is the generalization of classical random walk.By utilizing the superposition of quantum states,the walkers can walk in different positions at the same time,which will result that the efficiency could be improved compared with the classical walk.Quantum walk has been used to implement universal quantum computing,design search algorithms,and so on.Considering the discrete quantum walk model,the introduction of the coin state might produce many unique properties,which has been widely used in quantum information processing.In addition,various kinds of models of discrete quantum walks,such as quantum walk with different topologies,multiple walkers and multiple coins,have been proposed and studied.A quantum network is an interconnected network based on quantum states where every party could carry out quantum information processing and fulfill some quantum computing tasks.In order to realize the construction of quantum network,a key problem is to realize the transmission of quantum states between different nodes.A natural way is to first distribute the quantum entangled pairs in advance,and then the secure transmission of quantum states can be realized,by utilizing various protocols such as quantum teleportation,quantum state sharing,and quantum teleportation of shared secret state.Another approach is to achieve the transmission of quantum states directly with the fidelity of 1 by selecting the physical system as the quantum channel.In this thesis,we consider combining quantum state transmission and discrete quantum walk,and devote ourselves to achieving the state transmission on the quantum walk structure.The solution to these two problems can not only be an attempt in the application of quantum walk in quantum information science,but also play a certain role in promoting the development of quantum networks.This thesis studies and analyzes various types of models of quantum walks.By selecting a suitable walk model,the quantum state to be transmitted is encoded into the coin state,and the state transmission on the quantum-walk structure is realized.The task of quantum state transfer based on quantum entanglement state is firstly studied.By using discrete quantum walks with multiple walkers,multiple coins,and multiple walkers as well multiple coins,selecting the appropriate coin unitary operators and constructing suitable measurement bases,quantum teleportation,quantum state sharing,and quantum teleportation of shared secret state on the walking structure are realized respectively.In the proposed schemes,the preparation of quantum entangled states in advance is not required,which might make it easier to implement experimentally.In addition,it is also versatile and flexible,meaning that by changing the number of walkers or coiners,a variety of state transfer protocols can be implemented.Secondly,perfect quantum state transfer between different network nodes is studied.By using the model of lazy quantum walks on the line,a class of perfect state transfer and entanglement distribution protocols of arbitrary high-dimensional quantum states is proposed.The setup of the coin operator is relatively simple,which might make the proposed scheme possibly more feasible to implement.The research results and innovations of this thesis are as follows:(1)By introducing quantum walks with N walkers,a class of quantum teleportation scheme of N-qubit on the quantum-walk structure is proposed.Based on three different topological structures:the line,the cycle,and the two-point complete graph with self-loops,three corresponding teleportation protocols are proposed while selecting the appropriate coin unitary operators and constructing an appropriate measurement basis.By the analysis,2N single-particle measurements are required,instead of the joint measurement,and during the recovery processing,only a single one-qubit gate is needed,instead of a two-qubit gate.This effectively improves the feasibility of the experiment.(2)By introducing quantum walks with N+1 coins,a class of schemes of quantum state sharing is proposed,where an arbitrary unknown quantum state is shared among N agents.First,the line is utilized to realize the quantum state sharing of an arbitrary qubit among N agents.In addition,two kinds of different protocols are proposed respectively,based on homogeneous and inhomogeneous coin unitary operators.Secondly,to achieve the state sharing for an arbitrary qudit among N agents,d-dimensional complete graphs with loops and ddimensional regular graphs are respectively utilized.Through analysis and comparison,in order to share an unknown qubit or qudit,one can flexibly choose different structures,and in order to achieve state sharing among different numbers of agents,what we need to do is to choose the different numbers of coins.The proposed protocol is universal and flexible,thus enriching the study of quantum state sharing.(3)By introducing quantum walks with multiple coins and multiple walkers,a class of teleportation schemes of quantum shared secret states on the quantum-walk structure is proposed,which can achieve the teleportation between n senders and m receivers.Firstly,the teleportation of a shared secret state from n senders to 2 receivers,is presented,and two kinds of different coin unitary operators,homogeneous and inhomogeneous coin operators,are used,respectively.Secondly,the proposed protocol above can be generalized to the case that the teleportation of a shared quantum secret state from n senders to m receivers.The proposed protocol is not only limited on quantum walks to the line,but also can be extended to the structure cycle.In addition,by selecting a variety of coin operators,one can construct different entangled states and then realize state transmission This is the first teleportation of a shared quantum secret state on a quantum-walk structure,thereby extending the transport protocol for shared quantum states.(4)By introducing lazy discrete quantum walk,perfect state transfer and entanglement distribution protocol of an arbitrary high-dimensional quantum state are proposed.By encoding the high-dimensional quantum state into the coin state in the walker-coin system,and selecting the appropriate high-dimensional quantum gates at different nodes and times,perfect state transfer of any unknown qudit between two arbitrary nodes is realized.This approach can be generalized to high-dimensional lattices,which can enable quantum entanglement distribution between arbitrary inputs and outputs.In our proposed scheme,the setting of the coin operators is relatively simple and distributing effectively m coin states requires m(2d±1)special coin operators,meaning that the number of quantum gates required increases linearly with the increase of the dimension of the high-dimensional quantum state.Moreover,the time for distributing the entangled state between any nodes is linear to the distance,which could make the proposed scheme more feasible in implementation. |
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