Theoretical Calculation And Simulation Of Continuous Transitions From Quantum Walk To Classical Random Walk

As the quantum mechanical correspondence of classical random walk,quantum walk has been widely used in quantum information processing,quantum simulation and quantum algorithm due to its coherent superposition and quantum interference.The essential difference between quantum walk and classical random walk is that the basic properties of microscopic particles are completely different from those of macroscopic objects.Microscopic particles have wave-particle duality,and can be in completely different wave and particle states at the same time.Many existing research works have explored the mixed state of quantum walk and classical random walk through decoherence,measurement and other operations,and found that a small amount of decoherence operation can effectively improve the performance of quantum algorithms.However,the coherent superposition of wave motion and particle walk has not been discussed so far.In this thesis,the evolution process and properties of wave-particle quantum walk are studied by theoretical calculation and quantum computing simulator simulation.Through quantum control operation,the walker in quantum walk is in wave-particle coherent superposition state.The post-selection operation is used to realize the continuous control of quantum walk from the state of waves with multipath coherence to the state of particles without coherence in two different ways :coherence and mixing.This work lays the foundation for the research of quantum computation and quantum simulation based on quantum walk.The main work is summarized as follows :Firstly,the evolution process and properties of wave-particle quantum walk are studied by theoretical calculation.The introduction of auxiliary bits is related to the coin-walker system to achieve quantum manipulation.The calculation of the state of the entire system shows that the introduction of quantum control can prepare the coin-walker into fluctuations,particles,and intermediate states,and the intermediate state can be coherent superposition and mixing.These different states can be quantitatively described by location distribution and variance.Finally,the position distribution of coherent superposition and mixed wave-particle walk is calculated respectively,and on this basis,the position variance with the change of evolution steps and auxiliary bit parameters is further calculated.The theoretical calculation results show that when the walker is in the wave-particle coherent state,two completely different properties can be observed simultaneously through one measurement.By adjusting the relative phase in the wave-particle coherent state,the diffusion rate of the walker can be controlled.Secondly,the evolution process and properties of wave-particle quantum walk are simulated by means of quantum computing simulator.Furthermore,by introducing quantum control,the continuous regulation between quantum coherence and non-quantum coherence in quantum walk is realized by coherent superposition and mixing.The quantum circuit constructed in the simulation experiment is a quantum circuit composed of coin-walker multi-qubits,auxiliary qubits,quantum logic gates and measurements constructed using python code.Using the probability distribution obtained by quantum computing simulation,the position distribution corresponding to each position is calculated,and the position variance can also be obtained based on this.Finally,the auxiliary bit is placed in the depolarization error noise quantum channel to consider the influence of environmental noise on the state regulation of the auxiliary bit and the coin-walk system when implemented in the actual physical system,and the feasibility analysis of the regulation scheme of the actual physical system is given.The quantum control proposed in this paper is different from the previous ones.It can not only realize the mixed wave-particle quantum walk,but also obtain the wave-particle coherent quantum walk,that is,the continuous control of two states with(without)multipath is realized simultaneously by coherent superposition.