Error-run-time trade-off in the adiabatic approximation beyond scaling relations

M. R. Passos, M. M. Taddei, R. L. de Matos Filho

Published 2019-07-08Version 1

The use of the adiabatic approximation in practical applications, as in adiabatic quantum computation, demands an assessment of the errors made in finite-time evolutions. Aiming at such scenarios, we derive bounds relating error and evolution time in the adiabatic approximation that go beyond typical scaling relations. Using the Adiabatic Perturbation Theory, we obtain leading-order expressions valid for long evolution time $T$, while explicitly determining the shortest time $T$ and the largest error $\varepsilon$ for which they are valid. Restricting our considerations to this validity regime, we can make clear and precise statements about the evolution time necessary to reach a given error and vice-versa. As an example of practical importance, we apply these results to the adiabatic version of Grover's quantum search algorithm and obtain highly accurate trade-off relations between run time and error for several evolution schedules. Their examination indicates that different strategies are required to optimize for either shorter time or minimal error.