Pontryagin’s maximum principle (PMP), formulated in the 1950s for engineering problems, gives the necessary conditions that an optimal control must satisfy. It turns an optimization over functions into a boundary-value problem for the state and an auxiliary co-state, together with a pointwise maximization condition on the control. In quantum control it plays two roles. It yields exact optimal solutions for small systems, and it underlies the numerical gradient methods used for large ones. A tutorial written for quantum physicists is [1]; the geometric theory is developed in [2].

Statement

Consider a system \(\dot x = f(x, u)\) with controls \(u(t) \in \mathcal U\), and a cost

$$C[u] = \varphi\big(x(T)\big) + \int_0^T f^0\big(x(t), u(t)\big)\,dt$$

to be minimized. Define the pseudo-Hamiltonian

$$\mathcal H(x, p, p^0, u) = \langle p, f(x, u)\rangle + p^0 f^0(x, u).$$

Maximum principle

If \(u^*\) is optimal with trajectory \(x^*\), there exist a constant \(p^0 \le 0\) and a co-state \(p(t)\), not both zero, such that

  • \(\dot x = \partial\mathcal H/\partial p\) and \(\dot p = -\partial\mathcal H/\partial x\);
  • for almost every \(t\), \(u^*(t)\) maximizes \(\mathcal H\big(x^*(t), p(t), p^0, v\big)\) over \(v \in \mathcal U\);
  • boundary (transversality) conditions hold, e.g. \(p(T) = p^0\,\nabla\varphi\big(x(T)\big)\) when the final state is free; and if the final time is free, \(\mathcal H = 0\) along the optimal trajectory.

Trajectories satisfying these conditions are extremals. They are called normal if \(p^0 \neq 0\) (normalized to \(p^0 = -1\)) and abnormal if \(p^0 = 0\). The principle only gives candidates: an extremal need not be optimal, and optimality must be checked separately.

Quantum form

For a closed system \(|\dot\psi\rangle = -iH(u)|\psi\rangle\) with \(H(u) = H_0 + \sum_k u_k H_k\), the co-state is a wavefunction \(|\chi(t)\rangle\) and the pseudo-Hamiltonian reads

$$\mathcal H = \mathrm{Re}\,\big\langle \chi \big|\, {-i}H(u)\, \big|\psi\big\rangle + p^0 f^0 .$$

Its equation of motion \(\dot p = -\partial\mathcal H/\partial x\) is again the Schrödinger equation, \(|\dot\chi\rangle = -iH(u)|\chi\rangle\), solved backward from a final condition set by the figure of merit. The way the controls enter is summarized by the switching functions

$$\Phi_k(t) = \frac{\partial \mathcal H}{\partial u_k} - p^0\frac{\partial f^0}{\partial u_k} = \mathrm{Im}\,\big\langle \chi(t) \big| H_k \big| \psi(t) \big\rangle .$$

Two cases recur:

  • Energy-penalized, unbounded controls, \(f^0 = \tfrac12\sum_k u_k^2\) with \(p^0 = -1\). Maximization gives \(u_k(t) = \Phi_k(t) = \mathrm{Im}\langle\chi(t)|H_k|\psi(t)\rangle\). This is the structure of the updates of Krotov's method and of the gradient in GRAPE: numerical methods solve these conditions iteratively.
  • Bounded controls, time-optimal problems, \(|u_k| \le u_{\max}\) and \(f^0 = 1\). \(\mathcal H\) is linear in \(u\), so \(u_k(t) = u_{\max}\,\mathrm{sign}\,\Phi_k(t)\) wherever \(\Phi_k \neq 0\) — bang-bang control. If \(\Phi_k\) vanishes on a whole interval, the arc is singular and the control is found by differentiating \(\Phi_k\) until \(u\) appears.

What it has revealed

  • Exact optimal protocols for two- and three-level systems, e.g. for laser-driven population transfer [3] and for time-optimal single-qubit operations [4]; see Geometric & time-optimal control.
  • Singular arcs are physical. In the time-optimal control of a dissipative spin-1/2, the optimal solution contains singular arcs, and it has been realized experimentally in NMR [5].
  • Bang-bang versus smooth. The maximum principle has been used to argue that optimal variational protocols for fixed total time have bang-bang form, supporting the structure of QAOA [6]. A later analysis showed that generic optimal protocols are bang–anneal–bang — pulsed at the beginning and end with a smooth, singular annealing segment in between [7]. See Variational algorithms & QAOA as control.
  • Minimal times. Free-final-time problems, with the condition \(\mathcal H = 0\), give minimal control times — control-based quantum speed limits — which are known exactly for driven two-level systems [8, 4].

Practical notes

  • Shooting. Solving the boundary-value problem by guessing the initial co-state and integrating forward (“shooting”) works for low-dimensional systems. Larger problems are solved by gradient iterations (GRAPE, Krotov's method), which converge towards normal extremals.
  • Existence matters. The principle assumes that an optimum exists. With unbounded controls, or without penalties, it may not — the infimum can be approached without being reached.
  • Sufficiency. Second-order conditions, or a direct comparison of all extremals, are needed to confirm optimality. Landscape analysis studies the same critical points from a global point of view.