# The roles of drift and control field constraints upon quantum control speed limits

## TL;DR

This paper derives bounds on the minimum time for implementing quantum gates in controlled quantum systems, analyzing how control constraints affect speed limits and the reachable set of operations, especially for single qubits.

## Contribution

It introduces a lower bound on quantum control speed limits considering control field constraints and analyzes its properties for single qubits and fully controllable systems.

## Key findings

- Analytical bounds accurately describe single qubit control time limits.
- Numerical optimization confirms the tightness of the bounds for single qubits.
- Discussion on extending bounds to higher-dimensional quantum systems.

## Abstract

In this work we derive a lower bound for the minimum time required to implement a target unitary transformation through a classical time-dependent field in a closed quantum system. The bound depends on the target gate, the strength of the internal Hamiltonian and the highest permitted control field amplitude. These findings reveal some properties of the reachable set of operations, explicitly analyzed for a single qubit. Moreover, for fully controllable systems, we identify a lower bound for the time at which all unitary gates become reachable. We use numerical gate optimization in order to study the tightness of the obtained bounds. It is shown that in the single qubit case our analytical findings describe the relationship between the highest control field amplitude and the minimum evolution time remarkably well. Finally, we discuss both challenges and ways forward for obtaining tighter bounds for higher dimensional systems, offering a discussion about the mathematical form and the physical meaning of the bound.