Minimal Energy Cost to Initialize a Quantum Bit with Tolerable Error

Abstract

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 ($k_{B}T\ln 2$) 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.

arXiv comment5+5 pages, 3+3 figure, Comments are welcome

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