  We theoretically and experimentally study the precision of a quantum clock
near zero temperature, explicitly accounting for the effect of continuous
measurement. The clock is created by a superconducting transmon qubit
dispersively coupled to an open coplanar resonator. The cavity and qubit are
driven by coherent fields, and the cavity output is monitored with a
quantum-noise-limited amplifier. When the continuous measurement is weak, it
induces persistent coherent oscillations (with fluctuating periods) in the
conditional moments of the qubit's energy probability distribution, which are
manifest in the output of the resonator. On the other hand, strong continuous
measurement leads to an incoherent cycle of quantum jumps. We theoretically
find an equality for the precision of the clock in each regime. Independently
from the equalities, we derive a kinetic uncertainty relation for the
precision, and find that both equalities satisfy this uncertainty relation.
Finally, we experimentally verify that our quantum clock obeys the kinetic
uncertainty relation for the precision, thus making an explicit link between
the (kinetic) thermodynamic behavior of the clock and its precision, and
achieving an experimental test of a kinetic uncertainty relation in the quantum
domain.
