Learning Hamiltonians from realistic experimental data remains a challenge. This work is motivated by neutron scattering and quantum simulations, focusing on reconstructing a quantum Hamiltonian from the dynamical structure factor (DSF). The DSF encapsulates two-point correlation data relevant for quantum magnets. The DSF contains sufficient information for efficiently reconstructing broad classes of quantum Hamiltonians. The approach uses established relationships between low-order spectral moments of the DSF and microscopic couplings. These connections yield three complementary Hamiltonian tomography protocols based on the high-temperature expansion, strong-field polarization, and the fluctuation-dissipation theorem respectively.
Scalable Hamiltonian Reconstruction Achieves Substantially Improved Accuracy with Dynamical Structure Factor Measurements
Errors reconstructing quantum Hamiltonians have been reduced tenfold compared to previous methods, reaching relative coupling errors below 10−2 for systems containing up to twenty-seven parameters. This accuracy level surpasses a previously considered unattainable threshold due to computational complexity and limited experimental data availability.
Numerical tests demonstrate that their methods reduce errors when reconstructing quantum Hamiltonians; analyses of one- and two-dimensional systems reveal appreciable robustness even with measurement noise present. Specifically, relative coupling errors below 10−2 were achieved for models containing up to twenty-seven parameters using high-temperature expansion techniques applied to data from the dynamical structure factor (DSF). Simulations show minimal increases in error rates against coefficient noise of up to 0.01, confirming protocol stability.
Scalability was also demonstrated as performance degradation remained limited while increasing the number of unknown couplings within test Hamiltonian families. However, this accuracy is currently established only for finite system sizes and assumes access to a complete description of the DSF across all relevant momentum points, a significant hurdle given real experimental limitations.
Extracting Effective Hamiltonians From Experimental Observations
The introduced framework establishes practical and strong Hamiltonian learning from common experimental data, with potentially immediate application to neutron scattering experiments. Quantum systems are typically well-described by a local Hamiltonian specified by a few couplings that give rise to rich collective phenomena such as symmetry breaking, topological order, and quantum criticality.
This task enables extraction of effective models governing the quantum system over energy and length scales of interest, informing theoretical analyses and subsequent experimental refinements. In condensed matter physics, it supports designing quantum materials with targeted properties; for quantum simulation and device engineering, it is essential for benchmarking, calibration, and building scalable platforms with verifiable performance. Numerous empirical and theoretical approaches have been developed for this purpose using tools from quantum information and machine learning.
A common empirical approach, successfully applied to certain quantum magnets, falls into model fitting: iteratively updating a prediction based on classical simulations until convergence with experiment. However, for generic quantum systems, model fitting becomes computationally prohibitive as system size increases, leaving open the possibility of a general, scalable approach under realistic measurement constraints. More systematic theoretical approaches often presuppose global digital access to arbitrary measurements.
Experimental platforms typically operate with far more limited control; measurements in quantum materials are usually restricted to observables characterising responses to external probe fields while access can be limited too few native local observables or correlation functions due to architectural and resource constraints. This work addresses this problem using a broadly accessible and physically illuminating observable, the dynamical structure factor (DSF).
The DSF is directly measurable via inelastic neutron scattering in quantum magnetism and has been extracted in programmable quantum simulators through its simple relation to two-point correlation functions. It offers experimental fingerprints of complex quantum phases, potentially enabling reconstruction of underlying Hamiltonian parameters as exemplified by the quest for Kitaev spin liquid materials.
It isn’t immediately clear that the dynamical structure factor (DSF) alone provides sufficient information to recover the entire Hamiltonian; while it contains data on magnon dispersions and other spinful excitations, it encodes only two-point correlations, a small portion of all observables. Although several machine learning studies provide empirical evidence for solving this task, at least for specific models with few free parameters, its well-posedness as an inverse problem had not been theoretically established. Identifying continuous families of Hamiltonians exhibiting identical DSFs demonstrates potential ill-posedness building upon known bounds regarding learning from Gibbs states.
Moreover, even when the inverse problem is in principle well-posed, the DSF potentially depends on Hamiltonian couplings in a highly complex fashion which could hinder finding an inversion needed to learn the Hamiltonian. This work develops a flexible framework for Hamiltonian tomography from the DSF that does not require simulation and comes with rigorous guarantees across several experimentally relevant regimes. An overview of protocols and their experimental requirements in quantum materials and programmable quantum simulators is provided.
Three scalable algorithms, high-temperature, strong-field, and fluctuation-dissipation protocols, tame the complex dependence of the DSF on Hamiltonian parameters in complementary regimes. These rely respectively on high-temperature expansion, perturbation theory around a strong polarizing Zeeman field, and nontrivial exact identities satisfied by the DSF due to stationarity of thermal states and the fluctuation dissipation theorem. For translation-invariant spin-1 Hamiltonians in arbitrary dimensions with one- and two-body interactions, end-to-end learning guarantees are established reducing it to solving a simple linear system in several instances.
Together these results establish a direct, practical, and scalable route to learning Hamiltonians from limited, physically relevant output of quantum experiments. Consider a translation-invariant quantum spin system (an extension to disordered cases lacking translational symmetry is discussed elsewhere). The unknown parameters to be learned are denoted as λ ∈[−1, 1]p. As N increases, both the number of independent couplings p and the interaction range remain fixed. For a local spin operator Oa r define Oa q = N−1/2 P r∈Λ e−iq·rOa r.
Access to the complete DSF tensor Sab will be distinguished from accessing only its diagonal elements Saa; in both cases, the goal is determining λ from measured DSF minimising errors due to noise and scaling efficiently with system size N and inverse temperature β. For a Hamiltonian with finite-range interactions parameters depend only on interactions within a limited neighborhood.
Each time derivative introduces one factor of H while dependence on λ resides in the thermal expectation value which generally lacks a simple form. Three protocols enable learning these dependencies through distinct and complementary methods: exploiting how (II.4) exhibits milder sensitivity to Hamiltonian parameters as temperature rises and the Gibbs state approaches an infinite-temperature mixed state.
Limitations of Translational Invariance with Disordered Material Systems
While these new protocols offer an exciting pathway towards Hamiltonian tomography, reconstructing a system’s governing equations, their current formulation relies on assumptions about translational invariance: interactions are identical across all points in space. Scientists acknowledge that real materials often exhibit disordered arrangements or variations in interaction strength between sites introducing complexities not fully addressed by this framework. Extending these methods to handle spatially correlated disorder requires incorporating additional terms representing those correlations into the analysis potentially undermining scalability due to increased computational demands. Nevertheless, these protocols currently assume uniform materials; actual substances frequently possess variations in their atomic arrangements or intrinsic interaction strengths.
This research demonstrated that the dynamical structure factor (DSF) contains enough information to reconstruct quantum Hamiltonians, mathematical descriptions of a material’s properties. This work establishes methods for learning about a material’s governing equations from common experimental data like neutron scattering measurements. Researchers note current formulations rely on assumptions about uniform materials and future work may need to address disordered arrangements or variations in interaction strength.
👉 More information
🗞 Learning Quantum Hamiltonians from the Dynamical Structure Factor
✍️ Nisarga Paul, Thomas Schuster and Jason Alicea (Affiliation: Department of Physics and Institute for Quantum Information and Matter)
🧠 ArXiv: https://arxiv.org/abs/2611.15859
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