Learning Quantum Hamiltonians from the Dynamical Structure Factor

Introduction

Quantum systems are typically well-described by a local Hamiltonian, specified by a small number of couplings, from which rich collective phenomena such as symmetry breaking, topological order, and quantum criticality can emerge. Constraining and characterizing the Hamiltonian from experimental data presents a recurring challenge across physics, from particle phenomenology to quantum materials. This task of Hamiltonian learning enables one to extract effective models governing the quantum system over energy and length scales of interest, informing theoretical analyses and subsequent experimental refinements. In condensed matter, Hamiltonian learning facilitates the design of quantum materials with targeted properties [1, 2, 3, 4, 5]; in quantum simulation and device engineering, it is essential for benchmarking, calibration, and building scalable platforms with verifiable performance [6, 7, 8, 9, 10, 11, 12, 13].

Hamiltonian learning has prompted many empirical and theoretical approaches, drawing on tools from quantum information and machine learning [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. A common empirical approach, applied successfully to certain quantum magnets [28, 29, 30], falls into the framework of model fitting, involving iteratively updating a model prediction and classical simulation until convergence to experiment. For generic quantum systems, however, model fitting is computationally prohibitive to scale to large system size, leaving open the existence of a general, scalable approach under realistic experimental measurement constraints. More systematic theoretical approaches often presuppose global digital access to arbitrary measurements. Experimental platforms nevertheless typically operate with far more limited control: Measurements in quantum materials are usually limited to observables characterizing responses to external probe fields, while in quantum simulators, access can be restricted to a few native local observables or correlation functions due to architectural and resource constraints.

Can one learn the Hamiltonian from limited, realistic experimental outputs? This fundamentally and practically important question has drawn significant recent attention [31, 32, 33, 34, 35]. In this work, we address the problem for a broadly accessible and physically illuminating observable in quantum experiments: the dynamical structure factor (DSF). This quantity [Eq. (II.2)] is directly measurable via inelastic neutron scattering in quantum magnetism and has been extracted in programmable quantum simulators using its simple relation to two-point correlation functions [36, 37, 38, 39, 40]. The DSF offers experimental fingerprints of complex quantum phases which can potentially enable the reconstruction of underlying Hamiltonian parameters, as perhaps best exemplified by the quest for Kitaev spin liquid materials [41, 42, 43, 44].

It is a priori far from clear that the DSF alone provides sufficient information to recover the entire Hamiltonian. Indeed, while the DSF contains information on magnon dispersions and other spinful excitations, it encodes only a slice of two-point correlation data—a small portion of all observables. Although several machine-learning studies provide empirical evidence of this task’s tractability, at least for specific models with a few free parameters [45, 46], the well-posedness of this inverse problem has not been theoretically established. Indeed, building on known hardness bounds for learning from Gibbs states [19], we rigorously demonstrate that it can be ill-posed: We identify continuous families of few-body Hamiltonians that exhibit identical DSFs. Moreover, even when the inverse problem is in principle well-posed, the DSF potentially depends on Hamiltonian couplings in a highly complex fashion that can in practice stymie finding the inversion needed to learn the Hamiltonian.

In this work, we develop a versatile framework for Hamiltonian tomography from the DSF [Figure 1a]. Our framework does not require simulation of the Hamiltonian and comes with rigorous guarantees for general classes of Hamiltonians across several broad experimentally relevant regimes, thus providing strong evidence that the aforementioned challenges do not impede practical learnability. Our central insight is that the microscopic Hamiltonian parameters can be extracted from explicit relations, or sum rules, among integrated frequency moments of the DSF [Figure 1b]. In particular, we provide three scalable Hamiltonian learning algorithms that we dub high-temperature, strong-field, and fluctuation-dissipation protocols—each taming the complex dependence of the DSF on Hamiltonian parameters in complementary regimes [Figure 1c]; see Table I for an executive summary]. These protocols respectively rely on a 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 the concrete case of translation-invariant spin-$\frac{1}{2}$ Hamiltonians in arbitrary dimensions with one- and two-body interactions, we establish end-to-end learning guarantees and in several instances reduce Hamiltonian learning to solving a simple linear system. Extensive numerical tests across one- and two-dimensional Hamiltonian families demonstrate appreciable robustness of our protocols to measurement noise and experimental imperfections. Together, these results establish a direct, practical, and scalable route to learning Hamiltonians from limited, physically relevant output of quantum experiments.

Protocol	High-temperature (HT)	Strong-field (SF)	Fluctuation–dissipation (FD)
Executive summary	Leverages a high-$T$ expansion of DSF spectral moments to construct a low-degree polynomial system for the Hamiltonian couplings [Sec. III].	Leverages strong-field polarization and a $1/B$ expansion of DSF spectral moments to construct a low-degree polynomial system for the couplings [Sec. V].	Uses exact DSF identities derived from FD theorem to construct a linear system for the couplings, without requiring high temperatures or strong fields [Sec. VI].
Experimental requirements	Quantum materials: DSF measurements (e.g., neutron scattering) at multiple high temperatures ($T \gg$ couplings). Quantum simulators: Tunable high-$T$ Gibbs state preparation and short-time two-point correlation data.	Quantum materials: DSF measurements at strong fields ($B \gg T$, couplings), with tunable field strength & orientation. Quantum simulators: Polarized state preparation with a tunable global field, short-time two-point correlation data.	Quantum materials: DSF measurements without requiring high temperatures or strong applied fields, one- and two-body Hamiltonian. Quantum simulators: Short-time two-point correlation data in a stationary state (e.g., Gibbs state), one- and two-body Hamiltonian.

Hamiltonians and spectral moments

Consider a translation-invariant quantum spin system (see §VII for an extension to the disordered case lacking translation symmetry). On a lattice $\Lambda$ of $N$ sites, the Hamiltonian reads

Here each $H^{j} = \sum_{r\in\Lambda} h^{j}_{r}$ is an extensive interaction term—e.g., involving exchange or Zeeman fields—arising from a sum of translated local operators $h^{j}_{r}$ with $O(1)$ magnitudes $\lVert h^{j}_{r}\rVert_{\infty}$. We take $\lambda\in[-1,1]^{p}$ to be the unknown parameters to be learned. The number of independent couplings $p$ and the interaction range are fixed as $N$ increases. For a local spin operator $O^{\alpha}_{r}$, define $Q^a_q = N^{-1/2} \sum_{r\in\Lambda} e^{-i\mathbf{q}\cdot\mathbf{r}} O^a_r$. Below, $a,b,c$ indices run over $x,y,z$ with repeated indices summed. The DSF at inverse temperature $\beta$ is then

where $\langle A\rangle_\beta:= \operatorname{Tr}(e^{-\beta H}A)/\operatorname{Tr}(e^{-\beta H})$ indicates a thermal quantum average. At later stages, we distinguish access to the full tensor $S^{ab}$ from diagonal access to $S^{aa}$. In either case, the task is to recover $\boldsymbol{\lambda}$ from the DSF at known temperatures in a manner robust to noise and scaling efficiently in $N$ and $\beta$.

Next, define DSF spectral moments

which are equivalent to short-time derivatives of the two-point spin correlation function: $m^{ab}_n(\mathbf{q}) = 2\pi i^n \partial_t^n\langle O^a_{\mathbf{q}}(t)O^b_{-\mathbf{q}}\rangle_\beta|_{t=0}$, with $m^{ab}_0(\mathbf{q})/2\pi$ the static spin structure factor. Additionally Fourier transforming back to real space yields the particularly useful quantities

Equivalently, $m^{ab}_{n,d} = 2\pi\langle[[\cdots[O^a_0,H],H],\cdots,H]O^b_d\rangle_\beta$, with the right-hand comprising $n$ commutators. For a finite-range Hamiltonian, $m^{ab}_{n,d}$ at fixed $n$ depends only on interactions within a finite neighborhood of 0. Each time derivative introduces one factor of $H$, and the remaining dependence on $\boldsymbol{\lambda}$ lies in the thermal expectation value, which in general does not take a simple form. Our three protocols will render the dependence of the $m^{ab}_{n,d}$ on $\boldsymbol{\lambda}$ learnable in three distinct but complementary ways, as illustrated in Figure 1c.

High-temperature tomography

Our high-temperature (HT) protocol leverages the fact that Eq. (II.4) depends more mildly on Hamiltonian parameters as temperature increases and the Gibbs state approaches a featureless infinite-temperature mixed state. This trend forms the basis of the high-temperature series expansion in quantum lattice models [47] and existing digital quantum algorithms for learning Hamiltonians from high-temperature Gibbs states [19], and here allows us to tame the complexity of Hamiltonian learning. In particular, a high-temperature expansion allows us to derive low-degree polynomial relations between Hamiltonian couplings and DSF spectral moments. At small $\beta$, we expand Eq. (II.4) as

with each nonzero coefficient $m^{ab}_{n,d,i}(\boldsymbol{\lambda})$ a homogeneous polynomial of degree $n+i$ in the couplings. For our choice of bounded, finite-range interactions, truncating Eq. (III.1) at order $R$ incurs a controlled $O((\beta/\beta_c)^{R+1})$ error as $\beta/\beta_c \to0$, where $\beta_c > 0$ is an $O(1)$ threshold determined by Hamiltonian details. The $\boldsymbol{\lambda}$-dependent coefficients on the right side of Eq. (III.1) can be evaluated efficiently using local operator algebra without full many-body simulation [App. B].

We collect a finite set of $n_f$ theoretically derived coefficients $m^{ab}_{n,d,i}(\lambda)$ and denote the resulting collection by $F_{\mathrm{th}}(\lambda)$, which comprises different spin channels $(ab)$, moment orders $(n)$, spatial harmonics $(d)$, and high-$T$ expansion orders $(i)$. We refer to $F_{\mathrm{th}}(\lambda)$ as the feature map. The experimental data corresponding to the feature map, which we denote by $F_{\mathrm{exp}}$, is extracted from the measured DSF, assumed to be available across multiple high temperatures and the full Brillouin zone. The HT protocol entails solving the polynomial system $F_{\mathrm{exp}}=F_{\mathrm{th}}(\lambda)$. Crucially, for fixed feature map, interaction range and parameter count, the size and degree of this polynomial system does not scale with $N$. Hence the computational cost of the HT protocol is independent of system size, in contrast to model fitting approaches requiring repeated simulations of the DSF [48].

The remaining question is whether the polynomial system $F_{\mathrm{exp}}=F_{\mathrm{th}}(\lambda)$ uniquely determines the Hamiltonian couplings. We identify three possible scenarios analyzed in detail in App. § 0.10. First, the system may be underdetermined—e.g., if the feature map includes fewer than $p$ components—and admit a continuously infinite set of solutions. Expanding the feature set can remedy this non-solvability. Second, the system may admit a discrete set of solutions—a case we call locally solvable. Local solvability can arise from spurious solutions or unavoidable degeneracy that we term measurement equivalence arising from distinct Hamiltonians which are indistinguishable under $F_{\mathrm{th}}(\lambda)$. A powerful property of polynomial feature maps is that local solvability can be certified for almost all couplings by checking that the Jacobian of $F_{\mathrm{th}}(\lambda)$ has full rank at a random $\lambda$ [34]. Third, the system may have a unique solution, in which case we say it is globally solvable. Instances of both global solvability and local solvability with measurement equivalence are discussed below.

To illustrate the HT protocol concretely, let us specialize to spin-$\frac{1}{2}$ Hamiltonians with arbitrary one- and two-body interactions in any dimension. Consider, for a finite set of bond vectors $\mathcal{R}$,

where $\sigma^{a}$ are Pauli matrices. Direct calculation of the feature map $F_{\mathrm{th}}$ reveals $b^{a}=\frac{1}{4\pi}\varepsilon_{abc}m_{0,0,1}^{bc}$ and $\lambda_{d}^{ab}=-\frac{1}{2\pi}m_{0,d,1}^{ab}$. Therefore, spatial harmonics of the static structure factor ($n=0$) alone, expanded to first order in $\beta$, contain the information of all one- and two-qubit couplings.

Frequently in neutron scattering experiments, access is limited to unpolarized neutron beams, whereby only the diagonal DSF is available [49]. In such cases, we may still learn diagonal exchange interactions using $\lambda_{d}^{aa}=-\frac{1}{2\pi}m_{0,d,1}^{aa}$. However, the off-diagonal exchanges $\lambda_{d}^{ab}$ and fields $b^{a}$ must be inferred from higher DSF moments. For instance, direct calculation shows