Minimal Energy Cost to Initialize a Bit with Tolerable Error

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 infinitely long time. The question on the cost in the practical operation for a bit has been raised under the constraint by the finiteness of operation time. We discover a raise-up of energy cost by $\mathscr{L}^2/\tau$ from the Landaeur’s limit ($k_B T \ln 2$) for a finite-time $T$ initialization of a bit with an error probability $E$. The thermodynamic length $\mathscr{L}(\varepsilon)$ 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 $\varepsilon = 1%$ and $1.288⁢ k_B T / \gamma_0 \tau$ for $\varepsilon=0.1%$. Furthermore, the optimal protocol to reach the bound of minimal energy cost is proposed for the bit initialization realized via a finite-time isothermal process.

Physical Review E