Higher-order perturbation theory for decoherence in Grover’s algorithm

Abstract

In this paper, we study decoherence in Grover's quantum search algorithm using a perturbative method. We assume that each two-state system (qubit) that belongs to a register suffers a phase-flip error (${\ensuremath{\sigma}}_{z}$ error) with probability $p$ independently at every step in the algorithm, where $0\ensuremath{\leqslant}p\ensuremath{\leqslant}1$. Considering an $n$-qubit density operator to which Grover's iterative operation is applied $M$ times, we expand it in powers of $2Mnp$ and derive its matrix element order by order under the large-$n$ limit. [In this large-$n$ limit, we assume $p$ is small enough, so that $2Mnp$ can take any real positive value or zero. We regard $x\ensuremath{\equiv}2Mnp$ $(\ensuremath{\geqslant}0)$ as a perturbative parameter.] We obtain recurrence relations between terms in the perturbative expansion. By these relations, we compute higher orders of the perturbation efficiently, so that we extend the range of the perturbative parameter that provides a reliable analysis. Calculating the matrix element numerically by this method, we derive the maximum value of the perturbative parameter $x$ at which the algorithm finds a correct item with a given threshold of probability ${P}_{\mathrm{th}}$ or more. (We refer to this maximum value of $x$ as ${x}_{\mathrm{c}}$, a critical point of $x$.) We obtain a curve of ${x}_{\mathrm{c}}$ as a function of ${P}_{\mathrm{th}}$ by repeating this numerical calculation for many points of ${P}_{\mathrm{th}}$ and find the following facts: a tangent of the obtained curve at ${P}_{\mathrm{th}}=1$ is given by $x=(8∕5)(1\ensuremath{-}{P}_{\mathrm{th}})$, and we have ${x}_{\mathrm{c}}g\ensuremath{-}(8∕5){\mathrm{log}}_{e}\phantom{\rule{0.2em}{0ex}}{P}_{\mathrm{th}}$ near ${P}_{\mathrm{th}}=0$.