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arXiv:2508.01806 · Quantum

Quantum Optimal Control for Coherent Spin Dynamics of Radical Pairs via Pontryagin Maximum Principle

Ugur G. Abdulla, José H. Rodrigues, Jean-Jacques E. Slotine

Published May 6, 2026• 0 citations
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Abstract

This paper aims to devise the shape of the external electromagnetic field that drives the spin dynamics of radical pairs to a quantum coherent state through maximization of the triplet-born singlet yield in biochemical reactions. The model is a Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms. We introduce a one-parameter family of optimal control problems by coupling the Schrödinger system to a control field through filtering equations for the electromagnetic field. Fréchet differentiability and the Pontryagin Maximum Principle in Hilbert space are proved, and the bang-bang structure of the optimal control is established. A new iterative Pontryagin Maximum Principle (IPMP) method for the identification of the bang-bang optimal control is developed. Numerical simulations based on IPMP and the gradient projection method (GPM) in Hilbert spaces are pursued, and the convergence, stability, and the regularization effect are demonstrated. Comparative analysis of filtering with regular optimal electromagnetic field versus non-filtering with bang-bang optimal field ( Abdulla et al, Quantum Sci. Technol., 9 , 4, 2024 ) demonstrates that the change of the maxima of the singlet yield is less than 1%. The results open a venue for a potential experimental work on magnetoreception as a manifestation of quantum biological phenomena.

DOI
10.22331/q-2026-05-06-2096

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This paper aims to devise the shape of the external electromagnetic field that drives the spin dynamics of radical pairs to a quantum coherent state through maximization of the triplet-born singlet yield in biochemical reactions. The model is a Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms. We introduce a one-parameter family of optimal control problems by coupling the Schrödinger system to a control field through filtering equations for the electromagnetic field. Fréchet differentiability and the Pontryagin Maximum Principle in Hilbert space are proved, and the bang-bang structure of the optimal control is established. A new iterative Pontryagin Maximum Principle (IPMP) method for the identification of the bang-bang optimal control is developed. Numerical simulations based on IPMP and the gradient projection method (GPM) in Hilbert spaces are pursued, and the convergence, stability, and the regularization effect are demonstrated. Comparative analysis of filtering with regular optimal electromagnetic field versus non-filtering with bang-bang optimal field ( Abdulla et al, Quantum Sci. Technol., 9 , 4, 2024 ) demonstrates that the change of the maxima of the singlet yield is less than 1%. The results open a venue for a potential experimental work on magnetoreception as a manifestation of quantum biological phenomena.
Abstract Optimal control of the external electromagnetic field input for the maximization of the quantum triplet-singlet yield of the radical pairs in biochemical reactions modeled by Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms are analyzed. Fréchet differentiability and Pontryagin Maximum Principle in Hilbert space is proved and the bang-bang structure of the optimal control is established. A closed optimality system of nonlinear differential equations for the identification of the bang-bang optimal control is revealed. Numerical methods for the identification of the bang-bang optimal control based on the Pontryagin maximum principle are developed. Numerical simulations are pursued, and the convergence and stability of the numerical methods are demonstrated. The results contribute towards understanding the structure-function relationship of the putative magnetoreceptor to manipulate and enhance quantum coherences at room temperature and leveraging biofidelic function to inspire novel quantum devices.
Identification of the external electromagnetic fields and internal hyperfine parameters which optimize the quantum singlet-triplet yield of simplified radical pairs modeled by Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms are analyzed. A method that combines sensitivity analysis with Tikhonov regularization is implemented. Numerical results demonstrate that the quantum singlet-triplet yield of the radical pair system can be significantly reduced if optimization is pursued simultaneously for both external magnetic fields and internal hyperfine parameters. The results may contribute towards understanding the structure-function relationship of a putative magnetoreceptor to manipulate and enhance quantum coherences at room temperature and leveraging biofidelic function to inspire novel quantum devices.
The quantum control of spin-correlated radical pairs (SCRPs) holds promise for the targeted manipulation of magnetic field effects (MFEs), with potential applications ranging from the design of noise-resilient quantum information processors to genetically encodable quantum sensors. However, achieving precise handles over the intricate interplay between coherent electron spin dynamics and incoherent relaxation processes in photo-excited radical-pair reactions requires tractable approaches for numerically obtaining controls for large, complex open quantum systems. To that end, we realise optimal control of SCRPs in the low-magnetic-field regime, through first generalising the gradient-ascent pulse engineering (GRAPE) paradigm to allow for optimising time-integral reaction yields for coherent control of SCRP spin dynamics. We then explore how interradical interactions which are unavoidable in a biophysical setting, such as electron-electron dipolar and exchange couplings, alongside driving resulting from intrinsic structural motion, can push the precision of the proposed radical pair quantum compass in cryptochrome closer to approaching the quantum Cram\'{e}r-Rao bound. Remarkably, this approach to optimality is amplified under environmental noise and persists with increasing complexity of the spin system, suggesting that perturbations inherent to such natural systems have enabled them to operate closer to the quantum limit to more effectively extract information from the weak geomagnetic field. Bolstered by these findings, we furthermore demonstrate how a control engineering approach based on the Pontryagin Maximum Principle (PMP) can offer a more computationally feasible route towards realising not only coherent, but also incoherent control, of noisy SCRP open-system spin dynamics. This raises prospects for experimental realisation of spin control of actual radical-pair systems in ambient magnetic fields, by suppressing or boosting radical reaction yields using purpose-specific radio-frequency wave forms, paving the way for inspiring more noise-robust quantum magnetometry via chemical reaction-yield-dependence, and potentially applications of quantum control to physiologically relevant radical-pair reaction intermediates.
The quantum control of spin-correlated radical pairs may enable the targeted manipulation of magnetic field effects, with potential long-term applications across molecular quantum technologies, from prospective noise-resilient quantum information processors to genetically encodable quantum sensors. However, achieving precise handles over the intricate interplay between coherent electron spin dynamics and incoherent relaxation processes in photoexcited radical-pair reactions requires tractable approaches for numerically obtaining controls for large, complex open quantum systems. Employing techniques relying on full Liouville-space propagators to that end becomes computationally infeasible for large spin systems of realistic complexity. Here, we demonstrate how a control engineering approach based on the Pontryagin Maximum Principle (PMP) can offer a viable alternative by reporting on the successful application of PMP-optimal control to steer the coherent and incoherent spin dynamics of noisy radical pairs. This enables controls for prototypical radical-pair models that exhibit robustness in the face of relevant noise sources and paves the way to incoherent control of radical-pair spin dynamics.
We realize arbitrary-wave-form-based control of spin-selective recombination reactions of radical pairs in the low-magnetic-field regime. To this end, we extend the gradient-ascent pulse engineering (GRAPE) paradigm to allow for optimizing reaction yields. This overcomes drawbacks of previously suggested time-local optimization approaches for the reaction control of radical pairs, which were limited to high biasing fields. We demonstrate how efficient time-global optimization of the recombination yields can be realized by gradient-based methods augmented by time blocking, sparse sampling of the yield, and evaluation of the central single time-step propagators and their Fréchet derivatives using iterated Trotter-Suzuki splittings. Results are shown for both a toy model, previously used to demonstrate coherent control of radical-pair reactions in the simpler high-field scenario and, furthermore, for a realistic exciplex-forming donor-acceptor system comprising 16 nuclear spins. This raises prospects for the spin control of actual radical-pair systems in ambient magnetic fields, by suppressing or boosting radical reaction yields using purpose-specific radio-frequency wave forms, paving the way for reaction-yield-dependent quantum magnetometry and potentially applications of quantum control to biochemical radical-pair reactions. We demonstrate the latter aspect for two radical pairs implicated in quantum biology. Published by the American Physical Society 2024
We address a control-constrained optimal control problem pertaining to the transformation of quantum states. Our objective is to steer a quantum system from an initial state to a desired target state while adhering to the principles of the Liouville-von Neumann equation. To achieve this, we introduce a cost functional that balances the dual goals of fidelity maximization and energy consumption minimization. We derive optimality conditions in the form of the Pontryagin Maximum Principle (PMP) for the matrix-valued dynamics associated with this problem. Subsequently, we present a time discretized computational scheme designed to solve the optimal control problem. This computational scheme is rooted in an indirect method grounded in the PMP, showcasing its versatility and efficacy. To illustrate the practicality and applicability of our methodology, we employ it to address the case of a spin ½ particle subjected to interaction with a magnetic field. Our findings shed light on the potential of this approach to tackle complex quantum control scenarios and contribute to the broader field of quantum state transformations.