Imperfect quantum searching algorithm

We point out that the criterion, derived by Long et al.[Phys. Rev A 61, 042305 (2000)], to judging the degree of inaccuracy under systematic errors in phase inversions in Grover’s quantum search algorithm is overestimated. Although this criterion is properly deduced, it lacks a more accurate characterization to the relation between systematic errors and the size of a quantum database. The criterion is improvable for considering the practical equilibrium between the actual gate imperfection and the size of the quantum database. Directing at this subject, we show that a nearly exact ctriterion exist. In their paper[1], Long et al. have found that the degree of inaccuracy of quantum search algorithm due to systematic errors in phase inversions is about 4/(Nδ), where N is the size of the database and δ is the angle difference between two phase rotations. This result is based on the approximate Grover kernel and an assumption: large N and small δ et al. However, we found that the main inaccurancy comes from the approximate Grover kernel. Since all parameters in Grover kernel conect with each other exquisitely, any reduction to the structure of Grover’s kernel would destory this penetrative relation, so accumulative errors emerge from the iterations to a quantum searching. Although this assumption lead their study to a proper result, it is hard to apply this criterion in a more practical situation, e.g. phase tunning technique[2][3]. In what follows, we will get rid of the approximation for the Grover kernel, then derive a nearly exact criterion from the general Grover kernel. The Grover kernel is composed of two unitary operators Gτ and Gη, given by Gτ = I + (e iφ − 1) |τ〉 〈τ | , (1) Gη = I + (e iθ − 1)W |η〉 〈η|W , where W is Walsh-Hadamard transformation, |τ〉 is the marked state, |η〉 is the initial state, and φ and θ are two phase angles. It can also be expressed in a matrix form as long as an orthonormal set of basis vectors is chosen. The orthonormal set is[2] |I〉 = |τ〉 and |II〉 = (W |η〉 −Wτη |τ〉)/l , (2)