Parametric Quantum State Tomography with HyperRBMs

TL;DR

The paper tackles the scalability barrier of quantum state tomography by introducing HyperRBM, a parametric QST framework where a hypernetwork conditions an RBM on Hamiltonian parameters to represent an entire ground-state family in a single model. Applied to the transverse-field Ising model in 1D and 2D geometries, the approach achieves high-fidelity reconstructions from local measurements across phases and the critical region, and enables direct extraction of fidelity susceptibility to identify quantum phase transitions without prior knowledge of the critical point. It demonstrates accurate local observable estimation, global state fidelities, and non-local entanglement measures (Rényi entropy) within a coherent parametric family, while maintaining tractable training and sampling on CPU resources. The results suggest a scalable route to tomographic reconstruction across full phase diagrams and invite extension to larger systems and non-stoquastic or complex-valued states, with interpretability benefits from the explicit parametric dependence on g.

Abstract

Quantum state tomography (QST) is essential for validating quantum devices but suffers from exponential scaling in system size. Neural-network quantum states, such as Restricted Boltzmann Machines (RBMs), can efficiently parameterize individual many-body quantum states and have been successfully used for QST. However, existing approaches are point-wise and require retraining at every parameter value in a phase diagram. We introduce a parametric QST framework based on a hypernetwork that conditions an RBM on Hamiltonian control parameters, enabling a single model to represent an entire family of quantum ground states. Applied to the transverse-field Ising model, our HyperRBM achieves high-fidelity reconstructions from local Pauli measurements on 1D and 2D lattices across both phases and through the critical region. Crucially, the model accurately reproduces the fidelity susceptibility and identifies the quantum phase transition without prior knowledge of the critical point. These results demonstrate that hypernetwork-modulated neural quantum states provide an efficient and scalable route to tomographic reconstruction across full phase diagrams.

Figures (6)

• Figure 1: HyperRBM reconstruction of Rényi entropy and susceptibility in the TFIM. (a) Second Rényi entropy $S_2(g)$ for the ground state of the $4\times4$ transverse-field Ising model for various subsystem sizes and transverse field strengths $g$. All results are obtained from a single HyperRBM trained on local Pauli measurements. Solid lines indicate exact values from ED. (b) HyperRBM architecture. A hypernetwork acts as conditioner and takes the control parameter $g$ and outputs FiLM coefficients $\mathbf{\gamma}$ and $\mathbf{\beta}$, which affinely modulate the RBM biases in visible and hidden layers. (c) Fidelity susceptibility $\chi_F(g)$ of the $4\times4$ TFIM. Orange markers show HyperRBM estimates from free-energy gradient variance (Sec. \ref{['sec:fidelity_susceptibility']}), gray dashed curves indicate ED references. Vertical gray dotted lines mark support field strengths, and the red dotted line indicates the finite-size critical field $g_c$. The reconstructed $\chi_F(g)$ reproduces the ED peak in the crossover region.
• Figure 2: Magnetization reconstruction and interpolation in $g$. TFIM on the $4\times4$ lattice: longitudinal magnetization $\langle\sigma^z(g)\rangle$ (orange) and transverse magnetization $\langle\sigma^x(g)\rangle$ (blue) as a function of the transverse field $g$. Gray dashed lines show the exact-diagonalization (ED) reference. Markers show estimates from the learned conditional RBM state: filled circles are support fields used for training, open diamonds are novel fields used only for evaluation. Vertical dotted lines indicate the support locations. Panels (a)-(c) correspond to increasingly dense support sets, illustrating that adding support points improves the accuracy of interpolation on novel $g$ values.
• Figure 3: Exact overlap and sample efficiency. Exact overlap $|\langle \Psi_{\mathrm{ED}}(g)\mid \Psi_\theta(g)\rangle|$ between the ED ground state and the reconstructed RBM state as a function of the transverse field $g$ (here for the $4\times4$ TFIM). Colors denote the number of training samples per support point, $N_s\in\{2\,000,5\,000,20\,000\}$. Filled circles indicate support field strengths used for training, while open diamonds indicate novel fields strengths used only for evaluation. Gray vertical dotted lines mark the support locations, and the red dotted line marks the finite-size critical field $g_c$. The overlap remains close to unity throughout the sweep, with the largest deviations concentrated near the crossover region and a systematic improvement as $N_s$ increases.
• Figure 4: Fidelity susceptibility and exact overlap. Fidelity susceptibility $\chi_F(g)$ of the $3\times3$ TFIM as a function of the transverse field $g$. Orange diamonds show the RBM estimate obtained from variance of the free energy gradients (see Sec. \ref{['sec:fidelity_susceptibility']}), while the gray dashed curve is the ED reference. The blue curve (right axis) reports the ED-RBM overlap on the training support points, demonstrating near-unity state fidelity across all field strengths. Vertical gray dotted lines indicate the support field strengths, and the red dotted line marks the finite-size critical field $g_c$. The reconstructed $\chi_F(g)$ reproduces the ED peak in the crossover region.
• Figure 5: Second Rényi entropy across the TFIM transition. TFIM on the $1\times16$ chain: second Rényi entropy $S_2(\ell,g)$ of a contiguous subsystem of size $\ell$ (Sec. \ref{['sec:renyi_entropy']}). (a) RBM estimate evaluated on a dense grid ($\Delta g=0.05$) on control parameter $g$, revealing a steep gradient around $g\approx1$ that is captured smoothly as a function of $g$. Colored overlay curves indicate the training support fields strengths. (b) Overlaid $S_2(\ell)$ slices at different support fields $g$: solid colored curves (markers) are RBM estimates and black dashed curves show the exact-diagonalization (ED) reference. The RBM reproduces both the transition-associated drop and the expected entropy scale, saturating near $S_2\approx\ln 2$ for this finite system.