Gradient-free pulse optimization for adiabatic control in open few-body quantum systems

We present a robust pulse optimization method for adiabatic population transfer and adiabatic quantum computation. The approach relies on identifying control pulses that keep the evolving quantum system close to its instantaneous ground state. By combining advanced gradient-free optimization tools with specialized cost functions for adiabatic control, it achieves both efficiency and robustness. To demonstrate its generality, we apply the method to three examples involving both atomic and superconducting qubits. We test different optimization cost functions and discretization bases, showing that the approach outperforms ensemble optimization. Finally, to verify its performance on real quantum hardware, we implement digitized adiabatic qubit control using the optimized pulses on the IBM Quantum cloud.

💡 Research Summary

The paper introduces a novel, gradient‑free pulse‑optimization framework tailored for accelerating adiabatic protocols in open quantum systems. Traditional adiabatic population transfer (APT) and adiabatic quantum computation (AQC) rely on slow evolution relative to the instantaneous spectral gap, which limits speed and makes them vulnerable to decoherence. The authors address this limitation by integrating quantum optimal control (QOC) with gradient‑free optimization techniques and by defining cost functions that directly penalize diabatic errors.

The dynamics of the system are modeled with a Lindblad master equation, assuming a Markovian environment. Control fields (e.g., Rabi frequency Ω(t) and detuning Δ(t)) are collected in a vector v(t). The total cost is C = C_i + η C_0, where C_i quantifies infidelity (diabatic error) and C_0 penalizes total pulse power. Three distinct infidelity measures are introduced:

• C₁ – the final‑time diabatic error, essentially 1 minus the fidelity with the instantaneous ground state.
• C₂ – the time‑averaged diabatic error, integrating the same quantity over the whole protocol duration.
• C₃ – an ensemble‑averaged C₁, evaluated over a set of scaled or shifted control parameters to capture robustness against amplitude and frequency fluctuations.

Minimizing C₁ alone reproduces conventional QOC (GRAPE‑type) results, while minimizing a weighted sum C₁ + λ C₂ yields “adiabatic QOC” pulses that stay close to the ground state throughout the evolution, thereby improving robustness. C₃ is used for ensemble optimization, a technique previously shown to increase tolerance to control‑parameter noise.

Pulse shapes are expressed as a reference pulse v₀(t) plus a correction expanded in a finite basis: Gaussian functions, sinusoidal functions, or Chebyshev polynomials. The authors compare these bases, finding that Chebyshev polynomials consistently give the lowest final infidelity, Gaussian bases converge fastest, and sinusoidal bases provide a middle ground. The expansion coefficients are optimized using the Covariance Matrix Adaptation Evolution Strategy (CMA‑ES), a gradient‑free evolutionary algorithm well‑suited for multi‑objective problems. Implementation relies on the open‑source Python library Nevergrad for sampling and on QuTiP for propagating the Lindblad dynamics. At each iteration, a batch of coefficient samples is generated, the corresponding pulses are built, the system is evolved, and the cost (including C₂ if applicable) is evaluated. The best‑performing samples are used to update the mean and covariance of the sampling distribution, and the process repeats until convergence criteria are met. Convergence analysis in the supplementary material shows rapid reduction of cost and avoidance of local minima.

The methodology is demonstrated on three physically distinct examples:

1. Rapid Adiabatic Passage (RAP) in a two‑level atom – Here Ω(t) and Δ(t) are the control fields. Three pulse families are compared: an unoptimized polynomial pulse, a pulse optimized with traditional QOC (minimizing C₁), and a pulse optimized with adiabatic QOC (minimizing C₁ + C₂). Simulations reveal that the adiabatic‑QOC pulse maintains high fidelity over a broad range of amplitude scaling (ε) and detuning offsets (δ), whereas the other two exhibit narrow “high‑fidelity islands” and sharp fidelity loss for non‑zero δ. The authors further digitize the pulses and run them on IBM’s Brisbane quantum processor, generating 625 parameter sets (25 detuning values × 25 pulse‑area values). Experimental results mirror the simulation trends, confirming the robustness of the adiabatic‑QOC design on real hardware despite limited shot counts.

2. Multilevel STIRAP‑like transfer between superconducting qubits coupled via a multimode waveguide – The system consists of two qubits (A and B) each coupled to a set of guided modes with time‑dependent couplings g_ac(t) and g_bc(t). The protocol follows a counter‑intuitive sequence (first g_bc, then g_ac) to move an excitation from A to B while remaining in the instantaneous ground state. The authors compare four pulse sets: (i) analytically derived super‑adiabatic transitionless driving (SA‑TD), (ii) pulses minimizing C₁, (iii) pulses minimizing C₁ + ½ C₂, and (iv) ensemble‑optimized pulses (C₃). Using Chebyshev expansions for the latter three, they find that the C₁+½C₂ pulses are markedly more tolerant to overall amplitude scaling (ε) than SA‑TD or C₁‑only pulses, achieving infidelities below 10⁻³ across a wide ε range. However, when dephasing time T_φ is varied, all pulse families show comparable sensitivity, indicating that the adiabatic‑QOC approach primarily improves robustness to control‑amplitude errors rather than intrinsic decoherence.

3. Maximum Independent Set (MIS) problem solved via AQC – The authors encode the MIS Hamiltonian and apply the same gradient‑free adiabatic optimization to generate control schedules that respect a total pulse area constraint of 4π (a practical limit for many experimental platforms). The adiabatic‑QOC schedules achieve the same success probability as conventional annealing but with a ≈30 % reduction in total runtime, demonstrating that the method can accelerate genuine quantum‑algorithmic tasks while keeping power consumption realistic.

Across all examples, the gradient‑free approach outperforms ensemble optimization in terms of computational efficiency: the number of function evaluations required to reach a target infidelity is reduced by roughly an order of magnitude, thanks to the CMA‑ES’s adaptive sampling. Moreover, because no gradient of the quantum dynamics is needed, the method scales to larger Hilbert spaces where analytic gradients become prohibitive.

The paper also discusses practical implementation aspects. The optimized continuous‑time pulses are digitized (Trotterized) for execution on gate‑based hardware, showing that the method integrates seamlessly with existing quantum‑cloud platforms. The authors note that while the Markovian Lindblad model suffices for the systems studied, extensions to non‑Markovian environments (e.g., colored noise in long waveguides) are a natural next step. They also suggest exploring richer basis sets such as wavelets or neural‑network‑generated functions to further enhance expressivity, and leveraging parallel GPU computation to mitigate the increased sampling cost for very high‑dimensional control spaces.

In summary, the work delivers a versatile, gradient‑free pulse‑optimization framework that directly targets diabatic errors, combines multi‑objective cost functions, and works with a variety of basis expansions. It demonstrates superior robustness and computational efficiency over traditional QOC and ensemble methods, validates the approach on real quantum hardware, and opens pathways for fast, high‑fidelity adiabatic control in both analog and digital quantum‑computing contexts.