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

## TL;DR

This paper derives bounds relating error and evolution time in the adiabatic approximation, extending beyond typical scaling relations, with practical applications to quantum search and error reduction strategies.

## Contribution

It provides new bounds and explicit expressions for error and time in adiabatic processes, including the first error-run-time trade-off for adiabatic search matching Grover's scaling.

## Key findings

- Derived bounds relating error and evolution time beyond standard scaling.
- Applied results to adiabatic quantum search, reproducing Grover's scaling.
- Obtained tight numerical values for error and time using boundary cancellation.

## Abstract

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. In this validity regime, we can make clear and precise statements about the evolution time needed to reach a given error and vice-versa. As an example of practical importance, we apply these results to the adiabatic search, and obtain for the first time an error-run-time trade-off relation that fully reproduces the discrete-Grover-search scaling. We also pioneer the obtention of tight numerical values for $\varepsilon$ and $T$ under the error-reducing strategy ``boundary cancelation''.