A Novel SO(3) Picture for Quantum Searching

An SO(3) picture of the generalized Grover’s quantum searching algorithm,with arbitrary unitary transformation and with arbitrary phase rotations, is constructed. In this picture, any quantum search operation is a rotation in a 3 dimensional space. Exact formulas for the rotation angle and rotational axis are given. The probability of finding the marked state is just (z +1)/2, where z is the z-component of the state vector. Exact formulas for this probability is easily obtained. The phase matching requirement and the failure of algorithm when phase mismatches are clearly explained. 03.67-a, 03.67.Lx,Quantum searching, Phase matching, SO(3) group Typeset using REVTEX 1 Grover’s quantum search algorithm [1,2] is one of the most celebrated quantum computing algorithms. It has been shown that the algorithm is optimal [3]. The algorithm can be generalized to arbitrary initial amplitude distribution [4]. It has many important applications, for instance, in the Simon problem [5] and quantum counting [6]. In the case where multiple marked state is involved, it can even search the data by just one query [7]. Recently, it has been generalized to an arbitrarily entangled initial states [8]. Since Grover’s algorithm involves only simple gate operations, it has been realized in 2 qubits [9–11], and 3 qubit NMR systems [12]. Grover’s original algorithm has a simple geometric interpretation [2,13,15]. When the Hadmard transformation is substituted by any arbitrary unitary transformation, it has been shown there is an SU(2) group structure in the generalized algorithm [2,15]. However, when generalizing the algorithm to arbitrary phase rotations, phase matching is vital [17,18]. In [18] we have given an approximate formula for the amplitude of the marked state. But it is difficult to understand the phase matching requirement, as it is contrary to what one expects from an continuity argument. In this Letter, we give a novel SO(3) picture of the generalized quantum search algorithm by exploiting the relation between SO(3) and SU(2). In this SO(3) picture the process of quantum search is crystalline transparent. The behavior of the algorithm with phase matching or mismatching are clearly understood. This helps us to understand the various aspects of the algorithm, and to further develop the algorithm. The operator for quantum search [2] can be written as Q = −IγUIτU , where |τ〉 is the marked state, |γ〉 is the prepared state, usually |γ〉= |0〉. For arbitrary phase rotations, Iγ = I − (−eiθ + 1) |γ〉〈γ|, Iτ = I − (−eiφ + 1) |τ〉〈τ |. In the basis where |1〉 = U−1|τ〉, |2〉 = −(|γ〉 − UτγU|τ〉)/ √ 1− |Uτγ|2, Q can be written as