Distance learning from projective measurements as an information-geometric probe of many-body physics

Abstract

The ability of modern quantum simulators--both digital and analogue--to generate large ensembles of single-shot projective "snapshots" has opened a data-rich avenue for the study of quantum many-body systems. Unsupervised machine learning analysis of such snapshots has gained traction, with numerous works reconstructing phase diagrams by learning and clustering low-dimensional representations of quantum states. Here, we forgo such representation learning in favour of distance learning: we infer the pairwise distances between quantum states--already sufficient for clustering--directly from snapshots. Specifically, we use a single neural discriminator to estimate Csiszar f-divergences--statistical distances between distributions--in an unsupervised manner. The resulting clusters reveal regimes with different dominant correlations, often coinciding with, but not limited to, conventionally defined phases of matter. Beyond phase-diagram exploration, we connect the infinitesimal limit of the inferred divergences to the Fisher information metric and analyse its finite-size scaling. This yields critical exponents of the discovered transitions and enables snapshot-based analysis of universality classes. We apply distance learning to a diverse set of systems characterised by conventional local order parameters (1D transverse-field and 2D classical Ising models), non-local topological order (extended toric code), and higher-order correlations (fermionic t-J model on a triangular lattice). In all cases, we correctly recover boundaries between distinct correlation regimes and, where applicable, quantitatively match established critical behaviour. Finally, we show that distances to suitably chosen reference snapshot distributions help identify the dominant correlations within the discovered clusters, positioning distance learning as a versatile information-geometric probe of quantum many-body physics.

Figures (13)

• Figure 1: The pipeline for discriminative distance learning. a. We sample the phase diagram by performing projective measurements on the ground state of a studied system (potentially in several bases). b. We prepare the classification dataset, where input vectors are the measurement results (potentially appended with the basis label), and the class labels indicate which phase diagram point the snapshot comes from. c. We train a classifier neural network to predict which phase diagram point a snapshot comes from. d. We use the posterior probability ratios produced by the trained classifier to estimate pairwise $f$-divergences between phase diagram points; to that end, we use an importance sampling scheme as described in Section \ref{['sec:discriminative_f_divergence_estimation']}.
• Figure 2: Schematic phase diagrams and representative snapshots for the studied systems. a. Transverse-field Ising model hosts two phases---ferromagnetic (FM) and paramagnetic (PM)---separated by a quantum critical point at $h_z/J = 1$. b. Similarly, the classical ferromagnetic Ising model hosts a ferro- and paramagnetic phase separated by the critical point at $T \approx 2.26$. c. The extended toric code hosts a topologically ordered deconfined phase at low external fields, which transitions into a confined (Higgs) phase as $h_x$ ($h_z$) grows. The last two phases can be adiabatically connected by a path around a first-order critical line (highlighted with yellow) linsel_percolation. The phase diagrams for triangular (2D) and cubic (3D) lattices are qualitatively similar and are adapted from Ref. linsel_independent_confinement. d. (i) the fermionic $t$--$J$ model on a triangular cylinder at small doublon doping ($\delta \approx 0.11$) hosts a multitude of regions primarily characterised by the position of the peak in the spin structure factor $S_{zz}(\mathbf{q})$. One can nevertheless distinguish a traditional ferromagnetic phase at low superexchange, a kinetically induced Nagaoka phase at $J/t$ from $0.1$ to $0.15$, and a commensurate spin-density wave at $J > 0.22$morera_nagaoka; (ii) the fully spin-polarised state of the fermionic $t$--$J$ model on a triangular ladder doped with holes and magnons hosts various multiparticle bound states morera_t_j_ladders.
• Figure 3: Discriminative distance learning of the transverse-field Ising model phase diagram. a--c. Matrices of pairwise Hellinger divergences between ground states of the TFIM for different system sizes $L$. d--f. Phase clustering results obtained by applying the HDBSCAN algorithm to the divergence matrices in panels a--c. Shaded areas of different colours indicate different clusters; black shaded areas indicate phase diagram points not attributed to any cluster; red points show the HDBSCAN clustering confidence for every phase diagram point. g. Finite-difference approximation to the Hellinger divergence susceptibility $\chi(h_z)$ defined in Eq. \ref{['equ:f_divergence_susceptibility']} for different system sizes $L$. Error bars are obtained via repeating distance learning for 19 random initialisations of the discriminator. h. Finite-size scaling collapse of the Hellinger divergence susceptibility for different system sizes $L$.
• Figure 4: Learnt $f$-divergence susceptibilities for the classical 2D Ising model. a. Finite-difference approximation to the Hellinger divergence susceptibility $\chi(T)$ defined in Eq. \ref{['equ:f_divergence_susceptibility']} for different lattice sizes $L$. Solid lines are produced using Ferrenberg-Swendsen histogram reweighting (see Appendix \ref{['sec:theoretical_heat_capacity_2d_ising']}). Error bars are obtained via repeating distance learning for 8 random initialisations of the discriminator. b. Finite-size scaling collapse of the curves from panel a using the known critical exponents of the 2D Ising universality class. The near-zero value of $a$ indicates that the logarithmic divergence is weakly visible for the system sizes considered and may be further obscured by the finite temperature step $\Delta T$.
• Figure 5: Discriminative distance learning of the extended toric code phase diagram. All phase diagrams are reconstructed from snapshots produced in both $x$ and $z$ bases, unless specified otherwise (figures b, c). a. Phase diagram at small external fields. b/c. Phase diagram at small external fields reconstructed from snapshots produced in $x$/$z$ basis only. d. Phase diagram at large external fields featuring a conjectured supercritical region not attributed to any cluster. e. Phase diagram reconstructed for the toric code on a 2D triangular lattice. f. Phase diagram reconstructed for the toric code on a 3D cubic lattice; each plaquette comprises four spins on a unit-cell face. g. The sum of $\sigma_x$ and $\sigma_z$ susceptibilities at high external fields; the borders of the found clusters from panel d are indicated with white dotted lines.