Decoherence in Grover’s quantum algorithm: Perturbative approach

Abstract

In this paper, we study decoherence in Grover's quantum searching algorithm using a perturbative method. We assume that each two-state system (qubit) suffers ${\ensuremath{\sigma}}_{z}$ error with probability $p(0<~p<~1)$ independently at every step in the algorithm. Considering an n-qubit density operator to which Grover's operation is applied M times, we expand it in powers of $2Mnp$ and derive its matrix element order by order under the $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty}$ limit. (In this large-n limit, we assume p is small enough, so that $2Mnp(>~0)$ can take any real positive value or $0.)$ This approach gives us an interpretation about creation of new modes caused by ${\ensuremath{\sigma}}_{z}$ error and an asymptotic form of an arbitrary-order correction. Calculating the matrix element up to the fifth-order term numerically, we investigate a region of $2Mnp$ (perturbative parameter), where the algorithm finds the correct item with a threshold of probability ${P}_{\mathrm{th}}$ or more. It satisfies $2Mnp<(8/5)(1\ensuremath{-}{P}_{\mathrm{th}})$ around $2Mnp\ensuremath{\simeq}0$ and ${P}_{\mathrm{th}}\ensuremath{\simeq}1,$ and this linear relation is applied to a wide range of ${P}_{\mathrm{th}}$ approximately. This observation is similar to a result obtained by Bernstein and Vazirani concerning accuracy of quantum gates for general algorithms. We cannot investigate a quantum-to-classical phase transition of the algorithm, because it is outside the reliable domain of our perturbation theory.