# Robust controllability of two-qubit Hamiltonian dynamics

## TL;DR

This paper investigates the ability to robustly control two-qubit quantum gates despite unknown Hamiltonian parameters, using analytical Lie algebraic methods and numerical approaches to extend controllability results.

## Contribution

It provides a comprehensive analysis of robust controllability in two-qubit systems with unknown parameters, including cases with limited control access, combining analytical and numerical techniques.

## Key findings

- Analytical conditions for robust controllability in two-qubit systems.
- Numerical verification of controllability when analytical methods are insufficient.
- Extension of single-qubit robust control results to two-qubit systems.

## Abstract

Quantum gates (unitary gates) on physical systems are usually implemented by controlling the Hamiltonian dynamics. When full descriptions of the Hamiltonians parameters is available, the set of implementable quantum gates is easily characterised by quantum control theory. In many real systems, however, the Hamiltonians may include unknown parameters due to the difficulty of precise measurements or instability of the system. In this paper, we consider the situation that some parameters of the Hamiltonian are unknown, but we still want to perform a robust control of a quantum gate irrespectively to the unknown parameters. The existence of such control was previously shown in single-qubit systems, and a constructive method was developed for two-qubit systems provided full single-qubit controls are available. We analytically investigate the robust controllability of two-qubit systems, and apply Lie algebraic approaches to handle the cases where only controlling one of the two qubits is allowed. We also use numerical approaches for these problems since our analytical approaches does not work in some systems.