Impact of Imperfect Timekeeping on Quantum Control

In order to unitarily evolve a quantum system, an agent requires knowledge of time, a parameter that no physical clock can ever perfectly characterize. 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.

Figure 1

For a circuit comprising $l_t=1$ CNOT per time step as the one in (b), the impact of timekeeping error on average gate fidelity $\overline{\mathcal{F}}(\mathcal{D})$ is plotted in (a) as the number of gates $L$ increases for different values of clock accuracy $N$. Inset (c) is the clock accuracy in logarithmic scaling against the depth $m$ for different numbers of CNOTs per time step, i.e., $l_t=\{1,5,25,100\}$, given a threshold average gate fidelity of 0.5 [46]. This shows that the temporal control one has access to greatly impacts the circuit complexity one can achieve and that this relationship is not linear. In the right panel (d), we present an illustration that visually conveys the setting of Theorem 1. That is, $l_t$ CNOTs per step of an algorithm individually timed by independent clocks following identical tick distributions leading to independent dephasing at each time step. This results in a global dephasing map whose unitarity we use to bound the achievable average gate fidelity.