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 raiseup of energy cost by L2(e)/t from the Landaeur's limit (kBT ln 2) for a finite-time t initialization of a bit with an error probability e. The thermodynamic length L(e) 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 0.997kBT/(gamma 0t ) for e = 1% and 1.288kBT/(gamma 0t ) for e = 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.

Automated summary

Landauer's principle imposes a fundamental limit on the energy cost of perfectly initializing a classical bit, but in practical operations, the finite operation time leads to an increase in energy cost. Specifically, when initializing the bit, the smaller the error, the higher the energy cost. A finite-time isothermal process can be used for bit initialization, and an optimal protocol to minimize the energy cost is proposed.