Not every control law has to come from an optimization. Lyapunov control, tracking control and model-predictive control borrow constructive ideas from classical control engineering. Lyapunov control drives the system downhill on an “energy-like” function. Tracking control forces an observable to follow a prescribed path by inverting the equations of motion. Model-predictive control repeatedly re-plans over a short time window. These methods give interpretable, often iteration-free designs. They have also inspired recent quantum algorithms.

Lyapunov control

Choose a function \(V(\psi) \ge 0\) that is minimal at the target — for example \(V = \langle\psi|O|\psi\rangle\) for an observable \(O\) whose lowest eigenstate is the target — and choose the controls so that \(V\) can only decrease. For \(H = H_0 + \sum_k u_k(t) H_k\) and \([O, H_0] = 0\),

$$\frac{dV}{dt} = \sum_k u_k(t)\, f_k(t), \qquad f_k(t) = -i\,\langle\psi(t)|[O, H_k]|\psi(t)\rangle ,$$

so the feedback law \(u_k(t) = -K_k\, f_k(t)\) with gains \(K_k > 0\) gives \(\dot V = -\sum_k K_k f_k^2 \le 0\) [1]. Because measuring the state would disturb it, the “feedback” is computed on a simulation. Lyapunov control is therefore usually a recipe for designing open-loop pulses, with no iterations.

Whether \(V\) decreases all the way to the target, or stalls at another invariant set, is the central question. It depends on spectral and coupling conditions and on the choice of \(V\), and it has been analysed in detail [2, 3, 4]. The same ideas apply to tracking a moving reference trajectory. When the linearized system is not controllable, adiabatic modifications can restore convergence [2]. With real measurements and quantum filtering, Lyapunov methods also stabilize states by genuine feedback [5]; see Measurement-based feedback.

Lyapunov control as a quantum algorithm

On a quantum computer, the quantities \(f_k(t)\) can be measured and used to set the next parameters of a circuit. FALQON (feedback-based quantum algorithm) does exactly this for combinatorial optimization. The circuit is a discretized Lyapunov-controlled evolution, its layer parameters come from measurements of the previous layers, and the solution estimate improves monotonically with depth, with no classical optimization loop. It has been demonstrated on a superconducting processor for MaxCut [6], and extended in several Lyapunov-inspired variants [7]. See Variational algorithms & QAOA as control.

Tracking control

Tracking control prescribes the trajectory \(y(t)\) of an observable’s expectation value and solves for the field that produces it. From \(\frac{d}{dt}\langle O\rangle = -i\langle[O,H_0]\rangle - i\,u(t)\,\langle[O,H_1]\rangle\), setting \(\langle O\rangle(t) = y(t)\) gives

$$u(t) = \frac{\dot y(t) + i\,\langle[O,H_0]\rangle}{-\,i\,\langle[O,H_1]\rangle},$$

computed along the evolution without iterations [8]. The catch is the denominator: where it vanishes, the field becomes singular. Choosing tracks and formulations that avoid singularities is the main design problem, for example for orienting molecular rotors [9] and symmetric-top molecules [10]; see Molecules & chemical control. Tracking also works for correlated many-body systems. Laser fields can make the current of a Fermi–Hubbard system follow an arbitrary predetermined evolution, even making a metal and a Mott insulator mimic each other’s optical response (“driven imposters”) [11]. See Many-body optimal control.

Model-predictive control

Model-predictive control (MPC), a standard of industrial process control, solves a short-horizon optimal-control problem, applies the first part of the solution, updates the model or state estimate with new information, and repeats. This closed loop is naturally robust to model discrepancies, coherent disturbances and noise. MPC has been introduced for quantum state preparation, notably for superconducting qubits, whose Hamiltonians are harder to characterize than those of cold atoms [12]. See Closed-loop & in-situ optimization and System identification for control.

Strengths and limitations

Strengths

  • Constructive and often iteration-free: one forward simulation gives the control.
  • Interpretable, with guarantees (monotonic decrease of \(V\), exact tracking where non-singular).
  • Natural bridges to quantum algorithms (FALQON) and to feedback with real measurements.

Limitations

  • Convergence is not guaranteed: Lyapunov control can stall, and tracking fields can become singular or very large.
  • Not optimal in time or energy. Results are often refined with GRAPE or Krotov.