In order to unitarily evolve a quantum system, an agent requires knowledge of
time, a parameter which no physical clock can ever perfectly characterise. In
this letter, we study how limitations on acquiring knowledge of time impact
controlled quantum operations in different paradigms. We show that the quality
of timekeeping an agent has access to limits the gate complexity they are able
to achieve within circuit-based quantum computation. It also exponentially
impacts state preparation for measurement-based quantum computation. Another
area where quantum control is relevant is quantum thermodynamics. In that
context, we show that cooling a qubit can be achieved using a timer of
arbitrary quality for control: timekeeping error only impacts the rate of
cooling and not the achievable temperature. Our analysis combines techniques
from the study of autonomous quantum clocks and the theory of quantum channels
to understand the effect of imperfect timekeeping on controlled quantum
dynamics.Comment: 5 + 7 pages, 2 figure