Investigation Of The Quantum Searching In A Three-dimensional Complex Subspace And The Multiphase Matching

The original Grover's quantum search algorithm goes quadratically faster than any possible classical counterpart for searching a single desired state in a large unsorted database and it was shown to be optimal. So far, several generalizations of the original Grover's algorithm have been developed from different aspects by some modifications mainly resulted from(1) dealing with the case of more than one desired state;(2) substituting almost any unitary transformation for the Walsh-Hadamard transformation, which is used in the original setting;(3) introduction of the concept of amplitude amplification;(4) further speedup for repeated quantum search by means of quantum computations in parallel;(5) the phase inversion being replaced by arbitrary phase rotations;(6) allowing for an arbitrary complex initial amplitude distribution, instead of the uniform initial amplitude distribution;(7) investigating the case of an arbitrarily entangled initial state. To guarantee that a desired state can be found with certainty, many researchers gave their different forms of accurate phase formulae. It is natural to ask whether a superposition of desired states or a unique desired state can be found with certainty only under the circumstances that the search space is confined to the two-dimensional complex subspace. To render the manipulation of complicated computations substantially simple and to obtain mathematically tractable results for any3×3unitary matrix, we consider the following properties and techniques.1) The eigenvalues of a Hermitian matrix are all real and the eigenvectors of a Hermitian matrix corresponding to different eigenvalues must be orthogonal to each other;2) Due to commutativity, a complete set of common orthonormal eigenvectors shared by two Hermitian matrices possibly exists;3) Suppose that there exists a complete set of common orthonormal eigenvectors shared by two Hermitian matrices. Then, if some eigenvalue belonging to one Hermitian matrix is degenerate, the degeneracy of this eigenvalue should be lifted through combining the other Hermitian matrix. Using the above properties, we prove that no matter what the initial superposition may be, neither a superposition of desired states nor a unique desired state can be found with certainty in a three-dimensional complex subspace, provided that a deflection angle is not exactly equal to zero. By using two different methods of dividing a3×3unitary matrix into two Hermitian matrices, which are commutative one another, and of the properties of the matrix exponential, we further demonstrate that if the total number of the desired and undesired states in an unsorted database is sufficiently large, then corresponding to the case of identical rotation angles, the maximum success probability of finding a unique desired state is approximately equal to the square of the cosine function of a deflection angle.On the other hand, due to the fact that a quantum system will inevitably be influenced by some unpredictable perturbations, we thereby conclude that the experimental realizations of Grover quantum search algorithm previously reported in the literature were, in fact, achieved in a three-dimensional complex subspace. Meanwhile, by utilizing the properties of the matrix exponential, we also showed that in a two-dimensional complex subspace, a unique desired state can be found with high success probability provided the multiphase matching equation is satisfied for any given initial superposition.The above results were rigorously verified by concrete numerical examples in terms of arbitrary initial superposition, arbitrary unitary transformation and arbitrary phase rotation angles in this paper.