  We study a model with a down-type SU(2) singlet vector-like quark (VLQ) as a
minimal extension of the standard model (SM). In this model, flavor changing
neutral currents (FCNCs) arise at tree level and the unitarity of the $3\times
3$ Cabibbo-Kobayashi-Maskawa (CKM) matrix does not hold. In this paper, we
constrain the FCNC coupling from $b\rightarrow s$ transitions, especially
$B_s\rightarrow \mu^+\mu^-$ and $\bar{B}\rightarrow X_s\gamma$ processes. In
order to analyze these processes, we derive an effective Lagrangian which is
valid below the electroweak symmetry breaking scale. For this purpose, we first
integrate out the VLQ field and derive an effective theory by matching Wilson
coefficients up to one-loop level. Using the effective theory, we construct the
effective Lagrangian for $b\rightarrow s\gamma^{(*)}$. It includes the effects
of the SM quarks and the violation of the CKM unitarity. We show the
constraints on the magnitude of the FCNC coupling and its phase by taking
account of the current experimental data on $\Delta M_{B_s}$,
$\mathrm{Br}[B_s\rightarrow\mu^+\mu^-]$, $\mathrm{Br}[\bar{B}\rightarrow
X_s\gamma]$ and CKM matrix elements as well as theoretical uncertainties. We
find that the constraint from the $\mathrm{Br}[B_s\rightarrow\mu^+\mu^-]$ is
more stringent than that from the $\mathrm{Br}[\bar{B}\rightarrow X_s\gamma$].
We also obtain the bound for the mass of the VLQ and the strength of the Yukawa
couplings related to the FCNC coupling of $b\rightarrow s$ transition. Using
the CKM elements which satisfy above constraints, we show how the unitarity is
violated on the complex plane.
