Optimal control of Gmon qubits using Pontyagin's minimum principle: preparing a maximally entangled state with singular bang-bang protocols

摘要

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 initialunentangled one. Given an initial state, a target state, and a Hamiltonian witha set of permissible controls, can we reach the target state with coherentquantum evolution and, in that case, what is the minimum time required? Theadiabatic theorem provides a far from optimal solution in the presence of aspectral gap. Optimal control yields the fastest possible way of reaching thetarget state and helps identify unreachable states. In the context of a simplequantum system, we provide examples of both reachable and unreachable targetground states, and show that the unreachability is due to a symmetry. We findthe optimal protocol in the reachable case using three different approaches:(i) a brute-force numerical minimization (ii) an efficient numericalminimization using the bang-bang ansatz expected from the Pontryagin minimumprinciple, and (iii) direct solution of the Pontryagin boundary value problem,which yields an analytical understanding of the numerically obtained optimalprotocols. 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.