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Landauer's principle imposes a fundamental limit on the energy cost toperfectly initialize a classical bit, which is only reached under the idealoperation with infinite-long time. The question on the cost in the practicaloperation for a quantum bit (qubit) has been posted under the constraint by thefiniteness 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 afinite-time $\tau$ initialization with an error probability $\epsilon$. Thethermodynamic length $\mathcal{L}(\epsilon)$ between the states before andafter initializing in the parametric space increases monotonously as the errordecreases. 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 isproposed for the qubit initialization realized via a finite-time isothermalprocess.

Minimal Energy Cost to Initialize a Quantum Bit with Tolerable Error