$\eta_d$. Due to the symmetry of the system, the sequential order of coefficients is irrelevant. With the re-defined operator the effect of decoherence reads $$\mD(\varrho) = \varrho' + (1 - {\mathop{\mathrm{Tr}}\nolimits}\varrho')\vrho,
\label{eq:inhomo-dop}$$ where $\varrho={\left|\psi\right>}{\left<\psi\right|}$ is the initial state, and the non-vacuum part of the output state is $\varrho'={\left|\psi'\right>}{\left<\psi'\right|}$, with ${\left|\psi'\right>}= D{\left|\psi\right>}$. Therefore, we can again reduce our problem to calculating the evolution of unnormalized pure states, just as in the uniform case, and use the non-unitary step operator $U''=DU'$ with the more general noise operator.

Telling how well the algorithm performs under these conditions is a complex task. First we give a lower bound on the probability of measuring the target node, based on generic assumptions. To begin, we separate the noise operator into two parts $$D = \eta + D',
\label{eq:D-sep}$$ where, for the moment, we leave $0\le\eta\le1$ undefined. As a consequence of Eq. (\[eq:D-inhomo\]) the diagonal elements of $D'$ are $[D']_{dd}=\delta_d=\eta_d-\eta$, and the off-diagonal elements are zero. From Eq. (\[eq:inhomo-dop\]) it follows that starting from a pure state ${\left|\psi_0\right>}$, after $t$ non-ideal steps the state of the system can be characterized by the unnormalized vector ${\left|\psi'(t)\right>}$, which is related to the state obtained from the same initial state 