# An upper bound on the time required to implement unitary operations

## TL;DR

This paper establishes an upper limit on the time required to implement any unitary operation in a quantum system with controllable Hamiltonian elements, depending on system size and coupling strength.

## Contribution

It derives a new upper bound on the implementation time for unitaries in quantum systems with control fields, considering system size and coupling constants.

## Key findings

- Derived an explicit upper bound on implementation time.
- Numerically investigated the tightness of the bound.
- Analyzed a specific energy level system with tight-binding interactions.

## Abstract

We derive an upper bound for the time needed to implement a generic unitary transformation in a $d$ dimensional quantum system using $d$ control fields. We show that given the ability to control the diagonal elements of the Hamiltonian, which allows for implementing any unitary transformation under the premise of controllability, the time $T$ needed is upper bounded by $T\leq \frac{\pi d^{2}(d-1)}{2g_{\text{min}}}$ where $g_{\text{min}}$ is the smallest coupling constant present in the system. We study the tightness of the bound by numerically investigating randomly generated systems, with specific focus on a system consisting of $d$ energy levels that interact in a tight-binding like manner.