  Landauer's principle imposes a fundamental limit on the energy cost to
perfectly initialize a classical bit, which is only reached under the ideal
operation with infinite-long time. The question on the cost in the practical
operation for a quantum bit (qubit) has been posted under the constraint by the
finiteness of operation time. We discover a raise-up of energy cost by
$\mathcal{L}^{2}(\epsilon)/\tau$ from the Landaeur's limit ($k_{B}T\ln2$) for a
finite-time $\tau$ initialization with an error probability $\epsilon$. The
thermodynamic length $\mathcal{L}(\epsilon)$ between the states before and
after initializing in the parametric space increases monotonously as the error
decreases. For example, in the constant dissipation coefficient ($\gamma_{0}$)
case, the minimal additional cost is $0.997k_{B}T/(\gamma_{0}\tau)$ for
$\epsilon=1\%$ and $1.288k_{B}T/(\gamma_{0}\tau)$ for $\epsilon=0.1\%$.
Furthermore, the optimal protocol to reach the bound of minimal energy cost is
proposed for the qubit initialization realized via a finite-time isothermal
process.
