A Bayesian view of Single-Qubit Clocks, and an Energy versus Accuracy tradeoff

We bring a Bayesian approach to the analysis of clocks. Using exponential distributions as priors for clocks, we analyze how well one can keep time with a single qubit freely precessing under a magnetic field. We find that, at least with a single qubit, quantum mechanics does not allow exact timekeeping, in contrast to classical mechanics which does. We find the design of the single-qubit clock that leads to maximum accuracy. Further, we find an energy versus accuracy tradeoff --- the energy cost is at least $k_BT$ times the improvement in accuracy as measured by the entropy reduction in going from the prior distribution to the posterior distribution. We propose a physical realization of the single qubit clock using charge transport across a capacitively-coupled quantum dot.