Grover量子搜索算法的一般化多相位匹配

General Multiphase Matching for Grover Quantum Search Algorithm

一个量子系统将不可避免地受到不可预知的微扰影响,据此断定文献中的Grover量子搜索算法的实验实现是在三维复子空间中完成的.同时证明在二维复子空间中,对任意给定的初始态|γ0〉=cosβ0|α〉+sinβ0 eiζ|β〉(β0是较小的正实数,ζ是任意的一个实数),存在解集Fj={(θj,θj-1,…,θ1),(φj,φj-1,…,φ1)}(整数j≥2)使得目标态能以100%的最大成功概率找到,其中相位旋转角θl和φl是不为2k'π的实数(1≤l≤j,k'为任意整数).如果只要求目标态以较高的成功概率找到,那么当一个无序数据库中目标态和非目标态的总个数足够大时,对于相对较小的正整数j,解集Fj可表示为Σji=1θl=Σji=1φl的形式.

Since a quantum system is inevitably influenced by some unpredictable perturbations, we thereby conclude that all the experimental realizations of Grover quantum search algorithm reported were, in fact, achieved in a three-dimensional complex subspace. We also prove that in a two-dimensional complex subspace, for any given initial superposition of basis states ｜γ0〉=cosβ0｜α〉＋sinβ0e｜β）（β0 is a small positive real number, ζ is an arbitrary real number） , there exists a set of solutions Fj={（θj,θj-1,…,θ1）,（φj,φj-1,…,φ1）} such that a desired state can be found with certainty for some positive integerj ~ 2, where the phase rotation angles θl andθt are real numbers but not equal to 2k′π ,1≤1≤j,k′is an arbitrary integer. If it is only required that a desired state can be found with high success probability, then as the total number of the desired and undesired states in an unsorted database is J i sufficiently large the above set of solutions Fj can be written in the form j∑t=1θt=j∑t=1φ1 for a relatively small positive integer j.