Also known as Nelder-Mead · CMA-ES · genetic algorithm · particle swarm · simplex · derivative-free
Gradient-free optimizers need only the value of the figure of merit, never its derivative. They are the
natural choice when gradients are unavailable or unreliable. Examples: the fidelity is measured on an experiment
and is noisy, the simulation is a black box, or the objective is not differentiable. They are also useful when the
landscape is rugged enough that local gradient methods get stuck. The price is poor scaling with the number of
parameters, so they are usually combined with a compact pulse description such as CRAB.
The main families
Direct search — Nelder–Mead. A simplex of \(n+1\) points in parameter space is reflected, expanded and
contracted towards better values [1]. Simple, with few settings, and effective for up to a few tens
of parameters. It is the classic partner of CRAB[2].
Evolution strategies — CMA-ES. Candidates are sampled from a multivariate Gaussian whose mean and covariance
matrix adapt to the landscape [3]. CMA-ES learns correlations between parameters and copes well with
noise and rugged landscapes.
Genetic and evolutionary algorithms. Populations of pulses are recombined and mutated. They were used in the
first proposal of learning control, in which an algorithm drives a laser experiment directly
[4]. For fast gates where greedy algorithms are trapped — a qutrit phase gate and a CNOT — an
evolutionary algorithm has been shown to succeed [5].
Surrogate-based optimization. Bayesian optimization and other machine-learning surrogates build a model of the
landscape from few, expensive evaluations, for example to optimize ultracold-atom experiments online
[6]; see Reinforcement & machine learning for control.
Where they shine
Closed-loop experiments. Optimizing directly on the device sidesteps model errors. Examples include tuning
superconducting-qubit gates with randomized-benchmarking fidelities as the objective [7], and
refining model-based optimal pulses in the laboratory (ad-HOC) [8]. See Closed-loop & in-situ optimization.
Black-box simulations whose gradients are unavailable, for example tensor-network or mean-field dynamics in
Many-body optimal control.
Global exploration. Population methods can escape local optima that trap greedy methods. Even human
intuition, collected through a citizen-science game, has been used to seed optimizers in such landscapes
[9].
Limitations
The number of evaluations needed grows quickly with the number of parameters, so the pulse description must stay
compact (a few to a few dozen parameters).
With cheap exact gradients, gradient-based methods (GRAPE, GOAT with
L-BFGS) almost always converge in far fewer evaluations.
“Global” methods such as genetic algorithms or CMA-ES explore more widely, but give no guarantee of finding the global
optimum. For a certified global optimum, see QCPOP.
Key papers
Richard S. Judson and Herschel RabitzTeaching lasers to control moleculesPhys. Rev. Lett.68, 1500-1503 (1992)doi
Proposes learning control. A closed loop between experiment and an evolutionary algorithm finds laser pulses without a model.
J. A. Nelder and R. MeadA Simplex Method for Function MinimizationThe Computer Journal7, 308-313 (1965)doi
The Nelder–Mead simplex method, the direct-search workhorse of CRAB and of many closed-loop experiments.
Nikolaus Hansen and Andreas OstermeierCompletely Derandomized Self-Adaptation in Evolution StrategiesEvolutionary Computation9, 159-195 (2001)doi
CMA-ES, a robust evolution strategy that adapts its search distribution; well suited to noisy, rugged objectives.
Ehsan Zahedinejad, Sophie Schirmer and Barry C. SandersEvolutionary algorithms for hard quantum controlPhys. Rev. A90, 032310 (2014)doi
Shows greedy, local optimization getting trapped on fast gates (a qutrit phase gate, a CNOT) and an evolutionary algorithm succeeding.
Nikolaus Hansen and Andreas OstermeierCompletely Derandomized Self-Adaptation in Evolution StrategiesEvolutionary Computation9, 159-195 (2001)doi
Richard S. Judson and Herschel RabitzTeaching lasers to control moleculesPhys. Rev. Lett.68, 1500-1503 (1992)doi
Ehsan Zahedinejad, Sophie Schirmer and Barry C. SandersEvolutionary algorithms for hard quantum controlPhys. Rev. A90, 032310 (2014)doi
P. B. Wigley, P. J. Everitt, A. van den Hengel, J. W. Bastian, M. A. Sooriyabandara, et al.Fast machine-learning online optimization of ultra-cold-atom experimentsSci Rep6, 25890 (2016)doi
J. Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen, et al.Optimal Quantum Control Using Randomized BenchmarkingPhys. Rev. Lett.112, 240504 (2014)doi
D. J. Egger and F. K. WilhelmAdaptive Hybrid Optimal Quantum Control for Imprecisely Characterized SystemsPhys. Rev. Lett.112, 240503 (2014)doi
Jens Jakob W. H. Sørensen, Mads Kock Pedersen, Michael Munch, Pinja Haikka, Jesper Halkjær Jensen, et al.Exploring the quantum speed limit with computer gamesNature532, 210-213 (2016)doi
Recent papers
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Frequency crowding, the close spacing of qubit transitions, limits superconducting quantum processors through crosstalk, leakage, and spectator dependent errors. We introduce a Hamiltonian based framework that maps two- and three-body collision mechanisms onto a Max-k-Cut problem on qubit frequency allocation. In order to make larger lattices tractable, we introduce a cluster-stitching protocol that decomposes larger lattices into overlapping subgraphs and reconciles their local frequency assignments into globally consistent configurations. We tested our proposed framework across multiple qubit architectures to suppress dominant collision penalties by revealing a hardware residual crosstalk. We benchmark digitized counterdiabatic QAOA (DC-QAOA), standard QAOA, and simulated annealing. Counterdiabatic terms improve QAOA optimization near steep collision penalties, while simulated annealing achieves the lowest costs on the tested instances. These results indicate that our hardware-aware optimization connects microscopic interaction physics to processor-level frequency planning. These results enable a systematic solver for benchmarking of scalable software hardware co design.
We revisit an established Quantum Control objective, characterize its exact local metric geometry, and reposition it as a rigorous benchmark for Search and Optimization. The trap-free topology of the landscape, established two decades ago, guarantees unhindered optimal pathways, but global topology alone does not govern convergence speed. Although the so-called Quantum Sphere is defined by the fourth power of the control field, we show that its yield gap admits an exact representation as a nonnegative linear combination of $\sin^2$ terms. From this single identity, we derive a two-sided quadratic sandwich with ratio $4/π^2$ on a certified, worst-case quarter-wave region, proving the optimal peak nondegenerate modulo its affine gauge and well-conditioned. For white-box optimization, on a halved (eighth-wave) region, the objective satisfies a Polyak-Łojasiewicz inequality with an explicit constant, granting Gradient Descent a linear rate of $1 - 16/(π^4κ_r)$, where $κ_r$ is the condition number of the relevant Hessian block. For black-box optimization, we prove that the elitist single-parent Evolution Strategy converges linearly, almost surely and in expected hitting time, from a certified sublevel set, despite the landscape's continuous gauge symmetry and its lattice of periodic optima; the certified level is explicit whenever the landscape is trap-free, and near the optimum the strategy provably sees only the quadratic tangent of the peak. Systematic numerical simulations corroborate the theory. Finally, we show that additive input (actuator) noise reweights each term of the representation by a Debye-Waller factor without disrupting the landscape's structure, and outline open challenges: output (signal) noise, far-field plateau escape, angular-synchronization certificates, and another physically scalable benchmark (SHG with Glass).
Optimal control in cancer chemotherapy is challenged by tumor heterogeneity and mutations, which complicate treatment effectiveness. Traditional methods, such as Pontryagin's maximum principle (PMP), are often hindered by their reliance on an initial guess for the costate equation, affecting accuracy and convergence. To address these limitations, this work introduces a hybrid quantum particle swarm optimization (QPSO) method based on regularization. QPSO is employed for global exploration to approximate the optimal control trajectory, followed by a regularization-based refinement to ensure smoothness and consistency with optimality conditions. The Hamiltonian function is used for first- and second-order optimality checks, verifying solution quality. Numerical case studies explore various drug effectiveness functions, demonstrating the role of periodic and localized drug delivery in achieving robust tumor suppression and providing insights into the impact of different drug combinations on optimal chemotherapy strategies.
Precise control of multi-qubit architectures remains a critical bottleneck in superconducting quantum processors. In this work, we investigate the synthesis of high-fidelity quantum operations and state transfer protocols within an extended superconducting linear chain, scaling from three to seven sites. Using Floquet theory, we model the periodic drive as a train of delta-like pulses, mapping the quantum control problem onto quasi-energy resonance conditions. Combining the Baker-Campbell-Hausdorff expansion with Floquet spectral decomposition, we analytically identify optimal driving parameters, refined via the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). In the three-qubit architecture, this enables high-fidelity synthesis of the iSWAP gate. Extending to a seven-site chain, we implement periodic trains of finite-width Gaussian pulses to activate distinct double-excitation transport channels with ultra-short gate durations t_gate (~170 ns). This achieves a clear scale separation from energy-relaxation times (T1) typical of fixed-frequency transmon devices with tunable couplers, such as IBM Quantum hardware. Finally, we benchmark stability under realistic imperfections, revealing a heightened sensitivity to static parameter disorder at the sub-percent level (eta ~ 10^-3) driven by spectral crowding, and discuss how closed-loop topologies could mitigate this constraint. This framework bridges time-periodic control theory and practical quantum gate engineering.
Optimal control problems arise in a wide range of scientific disciplines, but the corresponding optimization algorithms often display a strong dependence on hyperparameters that significantly influence performance and convergence. For the optimal implementation of quantum algorithms, these challenges are further amplified by high-dimensional control landscapes and the need for high-fidelity operations. Here, we propose an algorithm for optimal control problems in quantum computing to efficiently generate high-fidelity control protocols for multi-qubit systems in a hyperparameter and gradient free manner. The method, referred to as Pulse Engineering via Projection of response functions at infinite nonlinear order (PEPRino), leverages the framework of response theory to navigate the control landscape to find high-fidelity implementations. This is achieved by determining the control landscape via response functions to infinite order, efficiently evaluated by resummation in terms of the first and second order response function. To demonstrate the approach, we apply it to quantum systems consisting of two and three qubits for the optimal implementation of the Quantum Fourier Transform (QFT). We benchmark the proposed algorithm against the Chopped Random Basis (CRAB) algorithm utilizing the Nelder-Mead method, focusing on the 2-qubit scenario. The results indicate faster convergence regarding iteration steps and computational time, highlighting the advantages of our approach.