Every optimal-control calculation starts by turning a physical wish — “flip this qubit”, “prepare this entangled state”, “make the reaction go this way” — into a mathematical optimization problem. That means choosing the model of the dynamics, the controls, the target, a figure of merit that measures success, and the constraints and penalties that encode what the hardware can do. These choices often matter more than the choice of algorithm: an optimizer can only find what the functional asks for.
The ingredients
Dynamics. For closed systems, the Schrödinger equation for a state or a propagator, \(\dot U = -i\big(H_0 + \sum_k u_k(t) H_k\big)U\). For open systems, a master equation for the density matrix, most often of Lindblad form; see Control of open quantum systems. For robustness, an ensemble of such equations with different parameters; see Robust & ensemble optimal control.
Controls. Functions \(u_k(t)\) on \([0, T]\), in practice parametrized by finitely many numbers; see Pulse parametrization & constraints.
Functional. The quantity to minimize, usually written as
$$J[u] = J_T\big(\text{state at } T\big) \;+\; \int_0^T g_a\big(u(t)\big)\,dt \;+\; \int_0^T g_b\big(\text{state at } t\big)\,dt .$$
The final-time term \(J_T\) measures how close the system ends up to the goal. The running cost \(g_a\) penalizes the control, for example its energy \(\lambda \sum_k u_k^2\). The running cost \(g_b\) penalizes undesired dynamics, such as population in a leaky level during the pulse. In the language of control theory these are the Mayer, Lagrange and Bolza forms of the problem.
Constraints. Fixed or free final time \(T\), bounds on amplitudes, limited bandwidth, fixed values at the start and end of the pulse.
The idea of choosing a field to steer molecular dynamics, and of formulating it as a variational problem, took shape in chemistry in the 1980s [1, 2, 3]. In the same period, optimal control theory was applied to selective molecular excitation [4], and the quantum optimal-control problem was given a rigorous formulation, including the existence of optimal solutions [5].
Common figures of merit
State-to-state transfer. The probability of reaching the target,
$$F = \big|\langle \psi_{\rm tgt} | \psi(T) \rangle\big|^2 ,$$
or, for mixed states and general targets, the expectation value of an observable, \(\langle O \rangle_T = \mathrm{Tr}\big(O\,\rho(T)\big)\).
Gates. For a unitary \(U(T)\) on a \(d\)-dimensional space and a target \(U_{\rm tgt}\), two standard choices are
$$F_{\rm sm} = \frac{1}{d^2}\Big|\mathrm{Tr}\big(U_{\rm tgt}^\dagger U(T)\big)\Big|^2 , \qquad F_{\rm re} = \frac{1}{d}\,\mathrm{Re}\,\mathrm{Tr}\big(U_{\rm tgt}^\dagger U(T)\big).$$
\(F_{\rm sm}\) ignores the physically irrelevant global phase; \(F_{\rm re}\) fixes it, which makes the landscape simpler but also stricter. Experiments usually report the average gate fidelity — the fidelity averaged over all input states — which for a unitary error is a simple function of \(F_{\rm sm}\) [6]:
$$F_{\rm avg} = \frac{d\,F_{\rm sm} + 1}{d + 1}.$$
Gates with leakage. When the qubits live in a larger space (transmons, atoms, molecules), let \(P\) project onto the \(n\)-dimensional computational subspace and \(M = P\,U_{\rm tgt}^\dagger U(T)\,P\). The average fidelity over the subspace is [7]
$$F = \frac{1}{n(n+1)}\Big[\mathrm{Tr}\big(M M^\dagger\big) + \big|\mathrm{Tr}\,M\big|^2\Big],$$
which automatically penalizes population lost from the subspace. See DRAG & leakage suppression and Quantum gate synthesis.
Gates in open systems. Propagating a full basis of Liouville space (all \(d^2\) density matrices) is expensive. But when the target is a unitary, the dynamics of only three suitably chosen states suffices to judge it, whatever the dimension [8].
Classes of targets. Often any gate of a class will do — for instance any perfect entangler, from which every two-qubit gate can be built with local operations. Functionals built from the local invariants of a gate target the whole class at once, which leaves the optimizer more freedom than fixing one specific gate [9].
Other tasks. Sensing problems optimize the quantum Fisher information (Control for sensing & metrology), thermodynamic problems optimize work or power (Control in quantum thermodynamics), and robust problems optimize averages or worst cases over parameter ensembles (Robust & ensemble optimal control).
Choosing well
- Match the functional to the experiment. A gate that maps one basis state correctly but scrambles phases scores well on state transfer and badly on gate fidelity. Optimize what will actually be measured.
- Phase conventions. Decide whether global, or even local single-qubit, phases matter. Removing unnecessary requirements makes the problem easier.
- Penalties shape the solution. Energy penalties produce smooth, low-amplitude pulses but bias the optimum. In Krotov’s method the penalty is placed on the change of the pulse between iterations, so it vanishes at convergence; see Krotov's method.
- Final time. With \(T\) fixed, the optimizer maximizes fidelity. With \(T\) free, minimizing time at fixed fidelity is a different problem with its own structure; see Geometric & time-optimal control and Quantum speed limits.
- The landscape depends on the functional. Constraints and the form of \(J\) change whether local optima appear; see Control landscapes.
The necessary conditions that an optimal solution of any such problem must satisfy are given by Pontryagin's maximum principle. Numerical methods such as GRAPE and Krotov are efficient ways of solving them.