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The Journal of Chemical Physics

Introduction to modeling radical pair quantum spin dynamics with tensor networks

Kentaro Hino, Damyan Frantzov, Yuki Kurashige, Lewis M. Antill

Published Feb 23, 2026• 2 citations
Publisher

Abstract

Radical pairs are short-lived, spin-correlated intermediates that underpin processes in chemistry, biology, and emerging quantum technologies. Their behavior is governed by coupled electron-nuclear spin dynamics and is sensitive to weak magnetic fields, but full quantum treatments have been obstructed by the extreme computational cost of modeling many interacting spins. Here, this barrier is removed, demonstrating that open-system radical-pair dynamics can be resolved at nuclear-spin scales previously intractable, explicitly reaching regimes with tens of coupled nuclei and validated up to 60 spins. In biologically relevant flavin-tryptophan radical pairs, electron-transfer pathways and magnetic-field anisotropy are found to reshape spin evolution and, in turn, spin-selective reaction yields. The resulting directional responses expose a strong mechanistic link between the nuclear environment, magnetic geometry, and chemical outcome-a relationship central to hypotheses of avian magnetoreception and other magnetic-field effects in biology. This establishes a widely applicable simulation framework that removes a long-standing barrier in spin chemistry, quantum biology, and spin-based device science.

DOI
10.1063/5.0314129

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• A study published on arXiv introduces tensor network methodologies to resolve the chronic memory expansion problem in radical-pair spin dynamics simulations. • Utilizing Matrix Product States (MPS) and Matrix Product Density Operators (MPDO), the researchers accurately calculated open quantum dynamics for systems containing up to 60 spins. • Applied to the flavin-tryptophan (FAD-Trp) system, which is central to avian magnetoreception, the framework precisely captured anisotropic spin dynamics and electron-transfer pathways. [Quantum Biology Society] Simulating the quantum dynamics of radical pairs (pairs of spin qubits) has long faced a significant technological barrier in the fields of quantum biology and spin chemistry. When attempting to include all surrounding nuclear spins in calculations, the Hilbert space expands exponentially, quickly exceeding the memory limits of current computing systems. However, a groundbreaking study has recently been published that fundamentally overcomes this "curse of dimensionality" by introducing tensor network techniques previously used for many-body quantum systems. A paper titled "Introduction to modelling radical pair quantum spin dynamics with tensor networks," released on the preprint repository arXiv, presents a revolutionary solution to these massive computational limits. According to the original publication, the development of this framework was led by a joint research team including Kentaro Hino and Yuki Kurashige from Kyoto University (Japan), Damyan S. Frantzov from the University of Oxford (UK), and Lewis M. Antill from Sungkyunkwan University (Republic of Korea). ■ Memory Reduction via Tensor Networks To efficiently compress the explosively growing Hilbert and Liouville spaces, the research team introduced representations based on Matrix Product States (MPS) and Matrix Product Density Operators (MPDO). This enabled the simulation of full open quantum dynamics while explicitly accounting for dozens of nuclear spins—a feat previously considered impossible with traditional computing methods. The team successfully benchmarked the methodology on systems with up to 60 spins, proving the overwhelming efficiency of the tensor network approach. ■ Demonstration in the Flavin-Tryptophan System To verify the algorithm's validity in a real biological context, the researchers applied it to the flavin-tryptophan (FAD-Trp) radical-pair model, a key biological component linked to the avian magnetoreception mechanism. The analysis successfully identified minute dependencies on the direction of an external magnetic field and captured anisotropic spin dynamics and electron-transfer pathways with high precision. This effectively reproduced the complex quantum dynamics of an actual biological environment, where numerous nuclear spin hyperfine interactions are intricately entangled, without distortion. ■ New Horizons for Quantum Biology Computing This study demonstrates that tensor networks are not just theoretical tools for physics but serve as a highly powerful and practical computing framework that enables ultra-large-scale quantum information processing in spin chemistry and quantum biology. As this technology extends into fields like drug discovery and quantum sensing, it is expected to drastically accelerate the analysis of precise spin dynamics within large protein complexes—tasks that were once impossible due to computational constraints.

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Radical pairs are short-lived, spin-correlated intermediates that underpin processes in chemistry, biology, and emerging quantum technologies. Their behavior is governed by coupled electron-nuclear spin dynamics and is sensitive to weak magnetic fields, but full quantum treatments have been obstructed by the extreme computational cost of modeling many interacting spins. Here, this barrier is removed, demonstrating that open-system radical-pair dynamics can be resolved at nuclear-spin scales previously intractable, explicitly reaching regimes with tens of coupled nuclei and validated up to 60 spins. In biologically relevant flavin-tryptophan radical pairs, electron-transfer pathways and magnetic-field anisotropy are found to reshape spin evolution and, in turn, spin-selective reaction yields. The resulting directional responses expose a strong mechanistic link between the nuclear environment, magnetic geometry, and chemical outcome-a relationship central to hypotheses of avian magnetoreception and other magnetic-field effects in biology. This establishes a widely applicable simulation framework that removes a long-standing barrier in spin chemistry, quantum biology, and spin-based device science.
The development of the radical pair mechanism has allowed for theoretical explanation of the fact that magnetic fields are observed to have an effect on chemical reactions. The mechanism describes how an external magnetic field can alter chemical yields by interacting with the spin state of a pair of radicals. In the field of quantum biology, there has been some interest in the application of the mechanism to biological systems. This paper takes an open quantum systems approach to a model of the radical pair mechanism in order to derive a master equation in the Born-Markov approximation for the case of two electrons, each interacting with an environment of nuclear spins as well as the external magnetic field, then placed in a dissipative bosonic bath. This model is used to investigate two different cases relating to radical pair dynamics. The first uses a collective coupling approach to simplify calculations for larger numbers of nuclei interacting with the radical pair. The second looks at the effects of different hyperfine configurations of the radical pair model, for instance the case in which one of the electrons interact with two nuclei with different hyperfine coupling constants. The results of these investigations are analysed to see if they offer any insights into the biological application of the radical pair mechanism in avian magnetoreception.
The Radical Pair Mechanism, a process by which magnetic fields modulate the kinetics of chemical reactions by influencing spin dynamics, is thought to underlie a variety of biological mechanisms and is inherently both quantum and noisy. Model simulations with limited electron-nuclear interactions have been conducted on classical devices resulting in experimentally verified final state results, however these simulations become too complex to simulate as the systems scale in size. This motivates us to propound the simulation of the relevant spin dynamics in a noisy quantum device to probe the currently unexplored open quantum systems aspects of the radical pair mechanism and to provide a novel theoretical tool for us to be able to tackle bottlenecks in simulating larger and more physically relevant biological spin systems. We employ a hybrid quantum machine learning approach where we evaluate a parametrized quantum circuit representing the evolution of the system on the quantum device and then update the parameters according to our objective cost function on a classical computer using a gradient optimizer and a noisy gradient free (SPSA) optimizer. We show that a 3 qubit circuit with a series of parametrized rotation gates representing the Hyperfine and Zeeman interactions can be optimized, resulting in a circuit that evolves a singlet state under different magnetic field angles. Moreover, we show that the noisy gradient free optimizer is better suited for optimizing on a noisy quantum device. However, the precision in the quantum circuit outputs is only sub-par at present, leaving room for further improvements.
The dynamics of many chemical systems can be influenced significantly by interactions with their environment (bath). An exact treatment of this is often infeasible due to the exponential scaling of Hilbert space with the number of degrees of freedom. In this thesis, dynamical methods, which overcome this difficulty, are considered for spin systems coupled to spin baths as well as systems coupled to harmonic oscillator baths. The thesis begins by considering recombination reactions of radical pairs, in which two electron spins interact with baths of nuclear spins. Motivated by inaccuracies observed in semiclassical treatments of the dynamics of a series of cryptochrome-based radical pairs, a new approach capable of treating the quantum dynamics of spin systems coupled to large baths of spins is presented. This method is validated for a series of central spin models, and is found to correctly describe the quantum dynamics of spin systems containing up to 1000 spins. Following this, the validity of the conventional treatment of radical pair recombination reactions is explored by considering the coupled electronic, nuclear and spin dynamics of a model radical ion pair. The dynamics of this model is treated using the hierarchical equations of motion (HEOM) formalism, and the results obtained are used to validate a recently proposed series of master equations. The HEOM formalism exactly captures the quantum dynamics of systems that are coupled to harmonic baths. However, due to the scaling of its computational cost with the strength of the coupling, it often becomes impractical. To overcome this difficulty, an efficient tree tensor network-based approach is presented and valided by treating a series of simple models. Following this, it is applied to the calculation of electron transfer rates for a series of models spanning a wide range of physical regimes in order to validate a recently proposed interpolation formula for the calculation of electron transfer rates.
It is a remarkable fact that the interactions between electron and nuclear spins, and their interactions with a magnetic field, can have a profound effect on chemical reactivity, despite the fact that the characteristic interaction energies are orders of magnitude smaller than thermal energy. These observations are explained by the existence of long-lived quantum coherences between spins in certain molecular species (in particular radical pairs), and the study of these spin effects constitutes the field of spin chemistry, a field which spans a wide range of systems, from biochemical reactions to device physics. Because spin chemistry involves subtle quantum effects, many intuitive models used by Physical Chemists, in particular classical kinetic equations, cannot be universally applied. On the other hand, complete first principles modelling of reactions involving spin effects comes at an impossibly high computational cost in complex systems, and often offers little physical insight. In order to address the shortcomings of these approaches, quantum master equations (QMEs) are employed to describe spin chemical phenomena. These are equations that describe a reduced set of degrees of freedom quantum mechanically, accounting for the effects of remaining degrees freedom implicitly. In this thesis I explore the use of QME techniques to resolve a range of problems in spin chemistry. These techniques can be used to obtain the correct radical pair spin QME, resolving questions about the role of quantum measurement in these systems. As well as aiding the development of the theory of spin chemistry, QMEs provide a practical tool for modelling radical pair reactions including spin relaxation effects, as I also demonstrate. As a final study of QME techniques in spin chemistry, I use these methods to show when the classical kinetic description of radical pair reactions is and is not valid.
Radical pairs and the dynamics they undergo are prevalent in many chemical and biological systems. Specifically, it has been proposed that the radical-pair mechanism results from a relatively strong hyperfine interaction with its intrinsic nuclear spin environment. While the existence of this mechanism is undisputed, the nanoscale details remain to be experimentally shown. Here, we analyze the role of a quantum sensor in detecting the spin dynamics (non-Markovian) of individual radical pairs in the presence of a weak magnetic field. We show how quantum control methods can be used to set apart the dynamics of the radical-pair mechanism at various stages of the evolution. We expect these findings to have implications to the understanding of the physical mechanism in magnetoreception and other biochemical processes with a microscopic detail.
• A study published on arXiv introduces tensor network methodologies to resolve the chronic memory expansion problem in radical-pair spin dynamics simulations. • Utilizing Matrix Product States (MPS) and Matrix Product Density Operators (MPDO), the researchers accurately calculated open quantum dynamics for systems containing up to 60 spins. • Applied to the flavin-tryptophan (FAD-Trp) system, which is central to avian magnetoreception, the framework precisely captured anisotropic spin dynamics and electron-transfer pathways. [Quantum Biology Society] Simulating the quantum dynamics of radical pairs (pairs of spin qubits) has long faced a significant technological barrier in the fields of quantum biology and spin chemistry. When attempting to include all surrounding nuclear spins in calculations, the Hilbert space expands exponentially, quickly exceeding the memory limits of current computing systems. However, a groundbreaking study has recently been published that fundamentally overcomes this "curse of dimensionality" by introducing tensor network techniques previously used for many-body quantum systems. A paper titled "Introduction to modelling radical pair quantum spin dynamics with tensor networks," released on the preprint repository arXiv, presents a revolutionary solution to these massive computational limits. According to the original publication, the development of this framework was led by a joint research team including Kentaro Hino and Yuki Kurashige from Kyoto University (Japan), Damyan S. Frantzov from the University of Oxford (UK), and Lewis M. Antill from Sungkyunkwan University (Republic of Korea). ■ Memory Reduction via Tensor Networks To efficiently compress the explosively growing Hilbert and Liouville spaces, the research team introduced representations based on Matrix Product States (MPS) and Matrix Product Density Operators (MPDO). This enabled the simulation of full open quantum dynamics while explicitly accounting for dozens of nuclear spins—a feat previously considered impossible with traditional computing methods. The team successfully benchmarked the methodology on systems with up to 60 spins, proving the overwhelming efficiency of the tensor network approach. ■ Demonstration in the Flavin-Tryptophan System To verify the algorithm's validity in a real biological context, the researchers applied it to the flavin-tryptophan (FAD-Trp) radical-pair model, a key biological component linked to the avian magnetoreception mechanism. The analysis successfully identified minute dependencies on the direction of an external magnetic field and captured anisotropic spin dynamics and electron-transfer pathways with high precision. This effectively reproduced the complex quantum dynamics of an actual biological environment, where numerous nuclear spin hyperfine interactions are intricately entangled, without distortion. ■ New Horizons for Quantum Biology Computing This study demonstrates that tensor networks are not just theoretical tools for physics but serve as a highly powerful and practical computing framework that enables ultra-large-scale quantum information processing in spin chemistry and quantum biology. As this technology extends into fields like drug discovery and quantum sensing, it is expected to drastically accelerate the analysis of precise spin dynamics within large protein complexes—tasks that were once impossible due to computational constraints.