Quantum searching’s underlying SU(2) structure and its quantum decoherence effects

The search operation for a marked state by means of Grover’s quantum searching algorithm is shown to be an element of group SU(2) which acts on a 2-dimensional space spanned by the marked state and the unmarked collective state. Based on this underlying structure, those exact bounds of the steps in various quantum search algorithms are obtained in a quite concise way. This reformulation of the quantum searching algorithm also enables a detailed analysis of the decoherence effects caused by its coupling with an environment. It turns out that the environment will not only make the quantum search invalid in case of complete decoherence, where the probability of finding the marked state is unchanged, but also it may make the quantum search algorithm worse than expected: It will decrease this probability when the environment shows its quantum feature. PACS numbers: 03.67.Lx, 03.65.Fd, 89.70.+c Typeset using REVTEX 1 Since Shor [1] convincingly demonstrated in 1994 that quantum mechanics can help in factoring a large number exponentially faster than any known classical algorithms, there were a few algorithms presented to overcome the classical limits on the usual computation process [2,3]. Among them the Grover’s quantum searching algorithm [4,5] has been paid much attention [6–8] to at present as it has been implemented by using nuclear magnetic resonance techniques [9]. As well as in other quantum algorithms in quantum computation, the non-classical feature such as quantum coherence plays a dominant role in Grover’s searching algorithm. But the environment surrounding the qubits may force the quantum computer to become classical by decohering the coherent superposition of quantum states. This decoherence effect certainly makes the quantum algorithms invalid [10,11] and thus the problem of decoherence must be overcome or avoided before the quantum computation can be implemented in practice. Recently, with Shor’s factoring algorithm as an explicit example, we have analyzed the decoherence problem of quantum computation in detail [12] based on the generalization of the quantum dynamic theory of quantum measurement [13-20]. The purpose of this note is to investigate the decoherence influences on Grover’s quantum algorithm. To this end, a concise formulation of the quantum search algorithm is presented using its SU(2) structure at first. And then based on this underlying structure we analyze the decoherence effects on this algorithm in some details. Classically, when there are N objects among which there is an unknown marked one, the best way to find this marked object is to search for it one by one among those N objects. For large N the steps of classical search are therefore of order O(N) in order to find the marked object. By using the Grover’s searching algorithm [4] however, the quantum computer needs only O( √ N) steps of searching to find a marked quantum object (e.g. state) with a probability near one. Suppose that our system has N levels and its Hilbert space is spanned by mutual orthogonal bases |k〉 with k = 1, 2, . . . , N and N very large. Initially, the system is prepared in a homogeneous coherent superposition 2