Optimal Control

Optimal control designs a control protocol by optimizing a stated objective subject to physical and numerical constraints. In quantum mechanics the controls may be microwave amplitudes, laser phases, magnetic fields, tunneling amplitudes, detunings, dissipative couplings, or measurement-based feedback laws. In open systems, the target must be balanced against decoherence, leakage, bandwidth, calibration uncertainty, and model error.

The objective should reflect the task: population transfer, preservation of arbitrary qubit inputs, cooling, and state stabilization are different problems. The allowed waveform and the noise model determine which solutions are physically possible.

For a small calculation you can reproduce, see Optimal Control Toy Problems. For algorithms and implementation on processors, see Optimal Control for Quantum Processors.

Controls, Dynamics, and a Target

A typical open-system control model has

$$\dot\rho(t) = \mathcal L_t[u](\rho(t)),$$

where $u(t)$ denotes one or more control waveforms. For a driven Markovian model,

$$\mathcal L_t[u](\rho) = - \frac{i}{\hbar} [H_0+\sum_j u_j(t)H_j,\rho] + \sum_\alpha \gamma_\alpha(t)\mathcal D[L_\alpha(t)]\rho.$$

The optimization chooses $u_j(t)$ from an allowed class. The class matters: a piecewise-constant waveform, a Fourier series, a spline, a Gaussian pulse family, and an arbitrary point-by-point waveform are different optimization problems.

The control problem is incomplete until it specifies:

• initial states or input channels;
• target state, unitary, channel, observable, or steady state;
• final time or allowed duration;
• amplitude, bandwidth, smoothness, phase, and energy constraints;
• known noise and uncertainty parameters;
• objective function and penalty terms;
• solver tolerances and validation tests.

Objectives

An objective function turns the physics goal into a number. Since many optimizers minimize, it is common to write

$$J[u] = 1-F[u] + \lambda P[u] + \mu R[u],$$

where $F[u]$ is a physical fidelity or success score, $P[u]$ penalizes control cost or roughness, and $R[u]$ penalizes lack of robustness. The weights $\lambda$ and $\mu$ are modeling choices, not constants of nature.

Common targets include:

Target	Example score
state preparation	$F_\psi=\langle\psi_\star\rvert\rho(T)\lvert\psi_\star\rangle$
population transfer	$F=\operatorname{Tr}[\Pi_{\mathrm{target}}\rho(T)]$
unitary gate	average gate fidelity or process fidelity
channel synthesis	distance between Choi matrices or action on a test set
cooling	final occupation $\langle n(T)\rangle$ or energy
steady-state engineering	distance to target fixed point plus Liouvillian gap checks
sensing protocol	Fisher information, contrast, or signal-to-noise objective

State-transfer objectives can be useful for calibration, but they do not certify a gate. A pulse that maps one initial state correctly may act badly on another input state. Gate or channel objectives must test a set of inputs spanning operator space, a process representation, or an equivalent fidelity formula. A computational basis alone tests populations but can miss errors in superpositions.

Constraints Are Physics

Constraints are not cosmetic. They are part of the model.

Constraint	Why it matters
amplitude bound	prevents unphysical instantaneous rotations or heating
finite bandwidth	models electronics, optics, filters, and pulse shaping
smoothness	reduces spectral leakage and calibration sensitivity
duration	trades speed against decoherence and adiabaticity
leakage penalty	discourages population outside the computational subspace
detuning uncertainty	tests robustness to calibration drift
noise model	prevents optimizing a closed-system fantasy
hardware transfer function	maps programmed waveform to delivered waveform

Without constraints, an optimizer can exploit features not present in the apparatus. A pulse that requires arbitrarily sharp edges or enormous amplitudes is a mathematical artifact unless the hardware can deliver it and the model remains valid at those scales.

Open-System Objectives

For a Markovian density-matrix model, the final state is

$$\rho(T) = \Phi_u(T,0)\rho(0),$$

where $\Phi_u$ is a completely positive trace-preserving map when the model is valid. If the target is a unitary gate $U_\star$, the open-system result is generally not unitary; it is a channel. Comparing only one input state can hide amplitude damping, dephasing, leakage, and nonunital drift.

For pure input states, a simple gate-style score averages their output overlaps:

$$F_{\mathcal S} = \frac{1}{|\mathcal S|} \sum_{\rho_j\in\mathcal S} \operatorname{Tr} \left[ U_\star\rho_jU_\star^\dagger \, \Phi_u(\rho_j) \right].$$

For a qubit, the six Pauli eigenstates $|\pm x\rangle,|\pm y\rangle,|\pm z\rangle$ form a state 2-design: their equally weighted score equals Haar-average gate fidelity for a linear trace-preserving qubit channel. An arbitrary small test set need not have this property. For mixed $\rho_j$, the displayed overlap is not generally a state fidelity. State whether population leaving the intended subspace counts as failure or is postselected; postselection changes the meaning of the score.

Gradients and Adjoints

For piecewise-constant controls, let $\rho_{n+1}=\Phi_n\rho_n$ with $\Phi_n=e^{\Delta t\mathcal L_n}$. Consider a terminal cost $J_T=1-\operatorname{Tr}(\rho_\star\rho_N)$ for a pure target, plus a control penalty $P$. Define the Hilbert–Schmidt inner product $\langle A,B\rangle=\operatorname{Tr}(A^\dagger B)$ and propagate

$$\Lambda_N=-\rho_\star,\qquad \Lambda_n=\Phi_n^\dagger\Lambda_{n+1}.$$

The chain rule then gives the exact discrete gradient

$$\frac{\partial J}{\partial u_{j,n}} =\operatorname{Re}\left\langle\Lambda_{n+1}, \frac{\partial\Phi_n}{\partial u_{j,n}}\rho_n\right\rangle +\frac{\partial P}{\partial u_{j,n}}.$$

The dagger on $\Phi_n$ denotes the adjoint superoperator, not a time-reversed physical evolution. One forward trajectory and one backward sequence of costates supply gradients for all slices. The derivative of a time-slice exponential is

$$\frac{\partial\Phi_n}{\partial u} =\int_0^{\Delta t}e^{(\Delta t-s)\mathcal L_n} \frac{\partial\mathcal L_n}{\partial u} e^{s\mathcal L_n}\,ds.$$

Replacing it by $\Delta t(\partial_u\mathcal L_n)\Phi_n$ is generally only a small-step approximation; it is exact when the relevant superoperators commute. Compare an implemented gradient with a central finite difference over a range of step sizes before relying on it. Automatic differentiation does not correct an inaccurate propagator or an incomplete physical model.

GRAPE and Krotov Preview

GRAPE, gradient ascent pulse engineering, is widely used when controls are piecewise constant and the objective gradient can be evaluated slice by slice. It is especially natural for finite-dimensional Hamiltonian control and has open-system extensions using Liouvillian propagators.

Krotov methods frame the update as an optimal-control variational problem with forward and backward propagation. They are often used when one wants monotonic improvement guarantees under the assumptions of the method.

Both families are algorithms for a chosen mathematical problem. Neither guarantees that the optimized pulse is physically robust, experimentally deliverable, or globally optimal.

Robust Optimization

A pulse optimized for a single parameter set may fail when the detuning, coupling, noise rate, or calibration shifts. Robust control replaces a single score by an ensemble score, for example

$$J_{\mathrm{rob}}[u] = \sum_m w_m \left[ 1-F_m[u] \right] + \lambda P[u],$$

where $m$ labels sampled parameter values and $w_m$ are weights. The pulse is then rewarded for working across the ensemble.

Robustness should be tested on parameter values not used during optimization. Otherwise the pulse may simply overfit the training grid.

Check the Pulse Beyond Its Optimization Grid

Recompute the final pulse with a finer propagation grid and tighter tolerances, using the waveform actually delivered after filtering. Compare it with a simple analytic pulse at the same duration and resource bounds. Report physical fidelity separately from penalties, and include leakage and constraint violations.

Then vary detuning, amplitude calibration, dissipation rates, and transfer-function parameters at values not used to train the pulse. Try several initial guesses: convergence from one guess establishes a local result, not a global optimum. Solving Lindblad Equations explains the relevant numerical checks.

Exercises

State transfer is not a gate

A pulse maps $\lvert0\rangle$ to $\lvert1\rangle$ with high fidelity. Why does this not prove it implements an $X_\pi$ gate?

Solution

A gate is a map on all input states, not one state. The same pulse might map $\lvert0\rangle$ correctly while adding the wrong phase to $\lvert1\rangle$, dephasing superpositions, leaking population, or acting nonlinearly after postselection. To certify a gate, one must test a process representation, an average gate fidelity, or enough input states to determine the channel within the intended model.

A basis-state test can miss a phase error

The desired qubit gate is the identity, but the device applies $Z$. Find its score on $|0\rangle,|1\rangle$ and on the six Pauli eigenstates.

Solution

Both computational basis states return to the same ray, so their mean overlap is 1. The four $x$ and $y$ eigenstates are mapped to their orthogonal partners, giving overlap 0, while the two $z$ eigenstates give 1. Thus the six-state average is $2/6=1/3$. A population-only test would label this severe phase error as a perfect gate.

Penalized objective

Suppose two pulses have physical fidelities $F_A=0.995$ and $F_B=0.990$. Pulse B is much smoother, so an objective $J=1-F+\lambda P$ assigns $J_B\lt J_A$. Which pulse is “better”?

Solution

Neither is universally better without the design goal. Pulse A has higher physical fidelity in the model. Pulse B has lower penalized objective because smoothness was rewarded. If smoothness represents a real hardware constraint or robustness requirement, B may be preferred. The result should report both $F$ and $J$ so readers can see the tradeoff.

Robust objective

A pulse is optimized at detuning $\Delta=0$ and works poorly at $\Delta=\pm\delta$. Write a simple robust objective using three detunings.

Solution

One simple choice is an equally weighted infidelity average:

$$J_{\mathrm{rob}} = \frac13 \left[ 1-F_{-\delta} + 1-F_0 + 1-F_{+\delta} \right] + \lambda P[u].$$

Equivalently,

$$J_{\mathrm{rob}} = 1 - \frac{ F_{-\delta}+F_0+F_{+\delta} }{3} + \lambda P[u].$$

The optimized pulse should then be tested at additional detunings not included in the three-point training set.

References

• C. Brif, R. Chakrabarti, and H. Rabitz, “Control of quantum phenomena: past, present and future,” New Journal of Physics 12, 075008 (2010).
• D. D’Alessandro, Introduction to Quantum Control and Dynamics, Chapman and Hall/CRC (2007).
• S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Kockenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, “Training Schrödinger’s cat: quantum optimal control,” European Physical Journal D 69, 279 (2015).
• N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, “Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms,” Journal of Magnetic Resonance 172, 296–305 (2005).
• V. F. Krotov, Global Methods in Optimal Control Theory, Marcel Dekker (1996).
• T. Schulte-Herbrüggen, A. Spörl, N. Khaneja, and S. J. Glaser, “Optimal control for generating quantum gates in open dissipative systems,” Journal of Physics B 44, 154013 (2011).