We investigate a new research area: we are interested in the ultimate thermodynamic cost of computing from x to y. Other than its fundamental importance, such research has implications for future miniaturiza-tion of VLSI chips reducing the energy dissipation below kT (thermal noise), and the similarity distance problem in pattern recognition. It turns out that the theory of thermodynamic cost o...

The thermodynamic uncertainty relation (TUR) for underdamped dynamics has intriguing problems while its counterpart overdamped recently been derived. Even the case of steady states, a proper way to match and TURs not found. We derive TUR systems subject general time-dependent protocols, that covers by using Cram\'{e}r-Rao inequality. show resultant give rise inequality product variance entropy ...