Noise effects in the quantum search algorithm from the computational complexity point of view

It is often said that the strength of quantum computation lies in the phenomena of quantum superposition and quantum entanglement. These features of quantum computation allow to perform the computation on all possible inputs that fit to the quantum register. One of the greatest achievements in the theory of quantum algorithms is quantum search algorithm introduced by Grover. The detailed description of this algorithm can be found in [1], [2], [3], [4]. Any physical implementation of a quantum computer will be error-prone because of the interaction of the computing device with the environment. In this paper we investigate the resilience of Grover’s algorithm in the presence of quantum noise. We use the language of density matrices and quantum channels. Our goal is to find the maximal amount of noise, for which the quantum algorithm is better, in terms of mean number of operations, than classical algorithm. We aim to achieve this objective by considering some classes of quantum channels modelling environmentally induced noise. The paper is organised as follows: in Section II we make a short review of the subject. In Section III we describe the formalism of quantum information theory. In Section IV we present the quantum search algorithm. In Section V we introduce the noise model we have applied to the system. In Section VI we analyse the results and finally in Section VII we present some conclusions.