Implementing A Quantum Search Algorithm With Nonorthogonal States In Linear Optics

Quantum interference,quantum entanglement and superposition states are adopt-ed in quantum computation,which have positive feature of parallel computation com-pared with classical counterparts.Quantum computation has much faster computing speed in simulation of quantum system,the unsorted database search problem,and the prime-factorization problem.For example,American scientist Shor proposed the Shor algorithm in 1994,which can factorize integers in polynomial time and can be used to crack public keys（RSA）;In 1996,Grover proposed a search algorithm,which was orig-inally designed to find an element in an unsorted database with no repeated elements.Compared with the classical algorithm,Grover algorithm has much faster computing speed.More specifically,consider a search task in an unsorted database containing N elements,classical algorithm finds the target element in O（N）steps,while Grover al-gorithm only needs O(（N）^1/2)steps.Various schemes for implementing quantum search algorithm in different quantum systems have been proposed,such as the cavity QED system,superconducting system,and NMR system.So far,only a few experimental works about the search algorithm have been reported.Based on Grover algorithm,we propose a scheme of quantum search using nonorthog-onal states and experimentally implement the entire scheme in a linear optical system.In our linear optical experiment,different target states are successfully found from a database of 9 elements with all fidelities>0.99,the state evolution from the initial state to a target state is studied,and the medium states and the final states are shown in a Bloch sphere.The main advantages of this scheme and the experimental setup are:（1）A single qubit can be used to encode large elements.（2）The theoretical search probability of the target state is 100%,regardless of the size of database.（3）The experimental setup and the operations are extremely simple.（4）The scheme could be realized in a wide range of quantum systems because of its generality.This work may pave an avenue to implement quantum algorithms with nonorthogonal states.It provides a new way for the realization of quantum algorithm and quantum simulation.This paper is divided into the following parts:In chapter 1,we briefly summarize the quantum information、quantum algorithm research background and basic theoretical knowledge.In addition,a brief introduction to the linear optical system is presented.In chapter 2,we introduce the basic theory of the Grover algorithm and the imple-mentation of the algorithm in different quantum systems.In chapter 3,we introduce the theoretical part of the nonorthogonal search algo-rithm and the feasibility of the algorithm.In chapter 4,we introduce the experimental scheme,show the experimental result of nonorthogonal search algorithm with N=9,and study the state evolution.In chapter 4,we summarize the whole paper.