Most numerical quantum-control methods produce a gradient. What they do with it matters as much as how they compute it. Plain gradient ascent takes small steps in the direction of steepest increase and slows down in long, curved valleys of the landscape. Second-order methods also use the curvature of the figure of merit. Newton methods use the exact Hessian, quasi-Newton methods an approximation built up from successive gradients. They converge much faster, especially near the optimum. Today they are the default engine behind GRAPE, GOAT and most gradient-based control codes.

Newton and quasi-Newton steps

Near an optimum the figure of merit \(J(\mathbf u)\) is well approximated by a quadratic, and Newton’s method jumps to its stationary point:

$$\Delta\mathbf u = -\big[\nabla^2 J(\mathbf u)\big]^{-1}\, \nabla J(\mathbf u).$$

Close to a non-degenerate optimum this converges quadratically. But the Hessian \(\nabla^2 J\) of a problem with \(n\) control parameters is an \(n \times n\) matrix that can be expensive to compute and is not positive-definite away from the optimum.

Quasi-Newton methods avoid computing it. BFGS updates an approximation \(B_k\) of the Hessian from the change in the gradient along each step, so that it satisfies the secant condition

$$B_{k+1}\, \mathbf s_k = \mathbf y_k, \qquad \mathbf s_k = \mathbf u_{k+1} - \mathbf u_k,\quad \mathbf y_k = \nabla J(\mathbf u_{k+1}) - \nabla J(\mathbf u_k).$$

L-BFGS keeps only the last few pairs \((\mathbf s_k, \mathbf y_k)\), so memory grows linearly with \(n\); its bound-constrained variant L-BFGS-B handles amplitude limits directly. Combined with a line search or a trust region that guarantees sufficient progress, these are robust, general-purpose methods [1].

In quantum control

  • GRAPE with exact gradients and BFGS converges much faster than the original gradient ascent. Its accuracy depends on using the exact rather than the first-order gradient, because quasi-Newton methods are sensitive to gradient errors [2]. Frameworks that compare update strategies systematically include such hybrids [3].
  • Newton GRAPE. The Hessian of the GRAPE fidelity turns out to be unusually cheap, with the same asymptotic cost as the fidelity itself. Regularized so that it is positive-definite far from the optimum (rational function optimization), Newton–Raphson GRAPE needs fewer trajectory evaluations than other GRAPE variants [4]. Newton methods for spin systems have also been analysed from the numerical-analysis side [5].
  • Krotov with quasi-Newton updates combines Krotov’s sequential update with BFGS-style acceleration [6]; see Krotov's method.
  • Constraints. Hard limits on amplitudes, slopes, energy or populations of unwanted levels are handled by constrained solvers such as sequential quadratic programming and interior-point methods. For constrained gate synthesis, exact second derivatives have been shown to outperform the BFGS approximation [7].

Direct collocation

A different way to use second-order solvers treats not only the controls but also the states at every time step as optimization variables. The Schrödinger equation is imposed as a set of equality constraints, one per time step, and the resulting large but very sparse problem is solved with an interior-point method. This direct collocation makes it easy to constrain the state trajectory, the control and its derivatives, and it gives the optimizer more freedom to move through infeasible intermediate points [8].

Practical notes

  • A good default is L-BFGS(-B) with exact gradients. It needs nothing beyond the gradient, and little tuning.
  • Full Newton methods pay off when very high fidelities are needed and the number of parameters is moderate.
  • Quasi-Newton methods assume a smooth, deterministic objective. With noisy objectives (experimental fidelities, stochastic simulations) they can fail, and Gradient-free & global optimizers methods or stochastic optimizers (Adam-type, as in Automatic differentiation & differentiable programming) are more appropriate.
  • Second-order information speeds up convergence to a local optimum. It does not help find a better one — that needs restarts, continuation in the control time, or a global method such as QCPOP.