Before a pulse can be optimized it must be described by finitely many numbers. The choice of that description — the parametrization — and the way experimental limits are imposed on it often decide whether an optimized pulse works in the laboratory at all. A pulse that reaches 99.999 % fidelity in simulation but needs infinite bandwidth, or exceeds the amplifier’s range, is of little use.

Common parametrizations

Parametrization Typical method Parameters Character
Piecewise constant GRAPE one per time slice and control (hundreds–thousands) maximally flexible; no built-in smoothness
Truncated randomized Fourier basis CRAB / dCRAB a few to tens of coefficients smooth, bandwidth-limited by construction
Slepian (prolate spheroidal) functions GRAFS tens optimal concentration in a given bandwidth [1]
B-splines (with carrier waves) Juqbox tens–hundreds smooth, local support, efficient for multi-level systems [2]
Analytic shapes (Gaussian, DRAG, …) GOAT a handful physically interpretable, hardware-native [3, 4]
Polynomial / power series QCPOP a few makes the problem polynomial for global optimization

Analytic corrections such as DRAG [5] are themselves parametrizations: a few well-chosen terms that remove a specific error.

How many parameters are actually needed? An information-theoretic argument relates the minimal number of independent control parameters to the dimension of the set of states the dynamics can reach in the available time [6]. Bandwidth and duration are linked: for a fixed bandwidth there is a minimal time to reach a target [1]. See Control complexity & resources and Quantum speed limits.

Imposing experimental constraints

  • Amplitude. Hard bounds through bound-constrained optimizers (L-BFGS-B), through a smooth mapping such as \(u = u_{\max}\tanh(v)\) that makes the problem unconstrained, or through penalties. Plain clipping after each step is simple but can stall the optimizer.
  • Smoothness and bandwidth. Penalize derivatives, e.g. \(\lambda \int_0^T \dot u(t)^2\,dt\); optimize on a coarse grid and interpolate; or use a smooth basis with limited frequency content (CRAB, splines, Slepians).
  • Transfer functions. Cables, filters and amplifiers distort the programmed pulse. If the distortion is a linear filter with impulse response \(h(t)\), the field seen by the system is the convolution \(u_{\rm act}(t) = \int h(t-\tau)\,u(\tau)\,d\tau\). Optimizing through this map — by the chain rule — yields pulses that are correct after distortion [7]. The same works for non-linear hardware responses [8].
  • Boundary conditions. Pulses usually must start and end at zero. Shape functions that vanish at \(t=0\) and \(t=T\) do this automatically — Krotov’s update shape \(S(t)\) (Krotov's method) or CRAB’s boundary factor \(\lambda(t)\) (CRAB & dCRAB).
  • Energy and power. A fluence penalty \(\lambda \int u(t)^2\,dt\) limits heating and drive-induced errors.
  • Sampling. Arbitrary waveform generators output samples at a fixed rate. Matching the control grid to it, and using a finer grid for simulation, avoids aliasing between model and hardware; see Control electronics & software stacks.

Choosing a parametrization

  • Exploration and benchmarks: piecewise constant with GRAPE — maximal freedom shows what is possible in principle.
  • Experiments: a compact, smooth basis with the hardware constraints built in, so the result can be transferred to the device and recalibrated in closed loop (Closed-loop & in-situ optimization).
  • Robustness: fewer parameters are easier to calibrate, while extra freedom can be spent on making the pulse robust (Robust & ensemble optimal control).