GOAT — Gradient Optimization of Analytic conTrols — computes exact gradients of the fidelity with respect to the parameters of any smooth, analytically defined pulse shape. Machnes, Assémat, Tannor and Wilhelm introduced it to optimize controls for practical qubits [1]: Gaussian or DRAG-like envelopes, sums of Fourier components, pulses passed through a filter — whatever the hardware naturally produces. It avoids the piecewise-constant approximation of GRAPE entirely.
The idea
Let the controls depend on a set of parameters \(\boldsymbol{\alpha}\) through known functions,
$$H(\boldsymbol{\alpha}, t) = H_0 + \sum_k c_k(\boldsymbol{\alpha}, t)\, H_k ,$$
for example \(c(\boldsymbol\alpha, t) = \alpha_1 \exp[-(t-\alpha_2)^2/\alpha_3^2]\cos(\alpha_4 t)\). Differentiating the Schrödinger equation \(\partial_t U = -iHU\) with respect to a parameter \(\alpha_j\) gives an equation for the derivative of the propagator,
$$\partial_t\big(\partial_{\alpha_j} U\big) = -i\,\big(\partial_{\alpha_j} H\big)\,U - i\,H\,\big(\partial_{\alpha_j} U\big), \qquad \partial_{\alpha_j} H = \sum_k \frac{\partial c_k}{\partial \alpha_j}\,H_k .$$
GOAT integrates this together with the evolution as one coupled linear system,
$$\frac{\partial}{\partial t}\begin{pmatrix} U \\ \partial_{\alpha_j} U \end{pmatrix} = -i \begin{pmatrix} H & 0 \\ \partial_{\alpha_j} H & H \end{pmatrix} \begin{pmatrix} U \\ \partial_{\alpha_j} U \end{pmatrix}, \qquad U(0) = \mathbb{1},\;\; \partial_{\alpha_j} U(0) = 0,$$
using a standard high-accuracy ODE solver with error control. At the final time, \(U(T)\) and all \(\partial_{\alpha_j} U(T)\) are known, so the gradient of any figure of merit follows directly. For the gate fidelity \(g = \frac{1}{d}\,\big|\mathrm{Tr}(U_{\rm tgt}^\dagger U(T))\big|\), for example,
$$\frac{\partial g}{\partial \alpha_j} = \frac{1}{d}\,\mathrm{Re}\!\left[ \frac{\mathrm{Tr}\big(U_{\rm tgt}^\dagger U(T)\big)^{*}}{\big|\mathrm{Tr}\big(U_{\rm tgt}^\dagger U(T)\big)\big|}\, \mathrm{Tr}\big(U_{\rm tgt}^\dagger\, \partial_{\alpha_j} U(T)\big)\right].$$
The gradient is then passed to a standard optimizer such as L-BFGS (Quasi-Newton & second-order methods).
Why it is useful
- Hardware-native pulses. The optimizer works directly in the parameter space of the pulses the electronics can produce. Transfer functions, filters and frequency limits are included analytically in \(c_k(\boldsymbol\alpha, t)\) instead of being imposed afterwards; see Pulse parametrization & constraints.
- Few, meaningful parameters. A dozen parameters with a physical meaning (amplitudes, widths, frequencies, phases) are easier to calibrate and to transfer to an experiment than hundreds of piecewise-constant values; see Closed-loop & in-situ optimization.
- Accurate. The ODE solver controls the error of the evolution and of the gradient, with no time-slicing approximation.
How it relates to other methods
GOAT is forward-mode differentiation: one extra copy of the evolution is propagated for every parameter. GRAPE’s backward sweep is reverse-mode differentiation (the adjoint method), which gets all derivatives for the cost of about one extra propagation. So GOAT is the natural choice for a few to a few dozen parameters and smooth pulses, while reverse-mode methods win when there are thousands of parameters. Automatic-differentiation frameworks implement both modes generically; see Automatic differentiation & differentiable programming.
Related approaches optimize smooth pulses in a fixed analytic basis. GRAFS uses discrete prolate spheroidal (Slepian) functions, which concentrate energy in a given bandwidth [2]. CRAB uses randomized Fourier components CRAB & dCRAB.
Strengths and limitations
Strengths
- Exact gradients for arbitrary differentiable pulse shapes, including filters and transfer functions.
- Few, interpretable parameters, and pulses that hardware can reproduce.
- Accuracy controlled by the ODE solver.
Limitations
- Cost grows linearly with the number of parameters (the coupled system has \(1 + n_\alpha\) blocks).
- A local method; the result depends on the chosen pulse family, which may not contain the optimum.
- Evaluating \(\partial c_k / \partial \alpha_j\) by hand for complicated parametrizations is tedious; automatic differentiation is often used for it.