  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 circuit complexity they are
able to achieve within circuit-based quantum computation. We do this by
deriving an upper bound on the average gate fidelity achievable under imperfect
timekeeping for a general class of random circuits. 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.
