  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 (\sigma_{z} error) with
probability p independently at every step in the algorithm, where $0\leq p\leq
1$. 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 $x\equiv 2Mnp(\geq 0)$ 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_{th} or more. (We refer
to this maximum value of x as x_{c}, a critical point of x.) We obtain a curve
of x_{c} as a function of P_{th} by repeating this numerical calculation for
many points of P_{th} and find the following facts: a tangent of the obtained
curve at P_{th}=1 is given by x=(8/5)(1-P_{th}), and we have
x_{c}>-(8/5)\log_{e}P_{th} near P_{th}=0.
