A variational quantum algorithm (VQA) runs a parametrized evolution \(U(\boldsymbol\theta)\) on a quantum device, measures a cost such as an energy \(C(\boldsymbol\theta) = \langle\psi_0|U^\dagger(\boldsymbol\theta)\,H\, U(\boldsymbol\theta)|\psi_0\rangle\), and updates the parameters with a classical optimizer [1, 2]. The best-known examples are the variational quantum eigensolver (VQE) [3] and the quantum approximate optimization algorithm (QAOA) [4]. Read as control problems, the parameters are controls, the cost is a figure of merit, and the hybrid loop is closed-loop optimization on hardware. VQAs and optimal control are variational optimization at two levels of abstraction, circuits and pulses. Results move between them in both directions [5].

QAOA is bang-bang control

QAOA applies \(p\) alternating layers,

$$ |\psi(\boldsymbol\gamma, \boldsymbol\beta)\rangle = e^{-i\beta_p H_M} e^{-i\gamma_p H_C} \cdots e^{-i\beta_1 H_M} e^{-i\gamma_1 H_C}\, |\psi_0\rangle , $$

where \(H_C\) encodes the cost function and \(H_M\) is a “mixer” [4]. This is the evolution under \(H(t) = u(t) H_C + [1-u(t)] H_M\) with a control \(u(t) \in \{0, 1\}\) that alternates between its two values in \(2p\) segments, whose durations \(\gamma_k, \beta_k\) are the parameters. In other words, it is bang-bang control. Three control-theoretic results frame it:

  • Optimal protocols. Pontryagin’s principle has been used to argue that the optimal protocols at fixed total time are bang-bang [6]. A later analysis found that generic optimal protocols are bang–anneal–bang, with a smooth annealing segment between pulsed ends [7]. See Pontryagin's maximum principle.
  • Annealing and beyond. A smooth \(u(t)\) that varies slowly is quantum annealing [8] and adiabatic quantum computation [9], which QAOA can approximate at large depth. Optimized QAOA can also learn non-adiabatic mechanisms that get around the small spectral gaps where annealing fails [10].
  • Universality. Whether alternating two fixed Hamiltonians generates every unitary is a Quantum controllability question. For one-dimensional cost Hamiltonians with nearest-neighbour \(ZZ\) terms, the precise conditions for universality have been proven [11].

Landscapes and trainability

  • Barren plateaus. For a wide class of randomly initialized circuits, the gradient is exponentially small in the number of qubits [12]. The dynamical Lie algebra of the circuit’s generators, the same object that decides controllability, controls the variance of the cost [13]. A general Lie-algebraic theory gives this variance exactly for deep circuits and unifies the known sources of barren plateaus: expressiveness, entangled inputs, local observables and noise [14]. Full controllability and trainability therefore pull in opposite directions.
  • Overparametrization. Above a critical number of parameters, bounded by the dimension of that Lie algebra, spurious local minima start to disappear and training becomes much easier [15]. Problem-inspired ansätze such as the Hamiltonian variational ansatz show mild barren plateaus and become almost trap free when overparametrized [16]. This is the circuit counterpart of the trap-free Control landscapes of optimal control.

From circuits to pulses, and back

  • Gradients from measurements. The parameter-shift rule obtains exact gradients from the same circuit run at shifted parameters [17, 18]. A stochastic version extends it to general parametrized evolutions, such as pulse amplitudes [19]. These are hardware-ready gradient estimators of the kind Closed-loop & in-situ optimization control needs.
  • Optimizing pulses directly. In ctrl-VQE, the pulse shapes of the device, not gate angles, are the variational parameters. In simulations of molecules on transmons, this shortened the state-preparation time by roughly three orders of magnitude [20]. See Pulse-level compilation.
  • Ansätze from control. Hardware-efficient ansätze use the device’s native interactions [21]. Ansätze inspired by optimal control add symmetry-breaking terms that improve convergence, for example for the Fermi–Hubbard model [22]. Counterdiabatic terms enrich QAOA layers [23] (see Shortcuts to adiabaticity). Digital–analog schemes combine single-qubit gates with an always-on analog Ising interaction [24].
  • No optimizer at all. Lyapunov control gives FALQON, where each layer’s parameters are fixed by measurements on the previous layers [25, 26]. See Lyapunov, tracking & model-predictive control.

What each field brings

Optimal control brings controllability and Lie-algebra tools, landscape theory, Quantum speed limits and robustness, and the lesson that enough control resources (parameters, time) make optimization easy [5]. Variational algorithms bring gradient estimation from measurements, optimizers that cope with shot noise, and a precise theory of when the cost concentrates. Both use the same mathematics for problems on many-body Hamiltonians.

Limitations

  • Every cost evaluation is a finite-shot estimate. Gradients below the shot-noise level cannot be resolved, which turns barren plateaus into a practical limit.
  • Hardware noise flattens landscapes as well and biases the cost.
  • Pulse-level ansätze optimized in simulation need an accurate device model. Without one, they must be optimized on the hardware, with every evaluation costing measurements.