# Optimal control of superconducting gmon qubits using Pontryagin's minimum principle: preparing a maximally entangled state with singular bang-bang protocols

## TL;DR

This paper applies optimal control theory, including Pontryagin's minimum principle, to efficiently generate maximally entangled states in superconducting gmon qubits, revealing the nature of reachable states and optimal protocols.

## Contribution

It introduces a comprehensive optimal control framework for gmon qubits, demonstrating the use of multiple methods to identify and analyze optimal protocols, including singular control cases.

## Key findings

- Optimal protocols are bang-bang, confirmed by three different approaches.
- Some target states are unreachable due to symmetry constraints.
- Analytical solutions provide insight into singular control scenarios.

## Abstract

We apply the theory of optimal control to the dynamics of two "gmon" qubits, with the goal of preparing a desired entangled ground state from an initial unentangled one. Given an initial state, a target state, and a Hamiltonian with a set of permissible controls, can we reach the target state with coherent quantum evolution and, in that case, what is the minimum time required? The adiabatic theorem provides a far from optimal solution in the presence of a spectral gap. Optimal control yields the fastest possible way of reaching the target state and helps identify unreachable states. In the context of a simple quantum system, we provide examples of both reachable and unreachable target ground states and show that the unreachability is due to a symmetry. We find the optimal protocol in the reachable case using three different approaches: (i) a brute-force numerical minimization (ii) an efficient numerical minimization using the bang-bang ansatz expected from the Pontryagin minimum principle, and (iii) direct solution of the Pontryagin boundary value problem, which yields an analytical understanding of the numerically obtained optimal protocols. Interestingly, our system provides an example of singular control, where the Pontryagin theorem does not guarantee bang-bang protocols. Nevertheless, all three approaches give the same bang-bang protocol.