Can a probability distribution measured in one basis determine complex quantum geometry? The answer is affirmative for a lattice Laughlin quasihole. The exact occupation law at one generic reference position, together with the known analytic quasihole factor, fixes the complete complex Gram kernel. A finite snapshot ensemble estimates this kernel without preparing another member of the family, and thereby determines finite-distance overlaps, Bargmann phases, the quantum metric, and the Berry curvature. The same analytic structure confines the family to an exact projective subspace whose dimension grows at most linearly with particle number. Exact enumeration of a Nielsen-Cirac-Sierra state demonstrates finite-shot reconstruction of both the metric and a geometric phase. Moreover, a Rényi-2 divergence sets the statistical range of the reconstruction while, at nondegenerate points, occupation readout also attains the local single-copy multiparameter information bound.
Can a probability distribution measured in one basis determine complex quantum geometry? The answer is affirmative for a lattice Laughlin quasihole. The exact occupation law at one generic reference position, together with the known analytic quasihole factor, fixes the complete complex Gram kernel. A finite snapshot ensemble estimates this kernel without preparing another member of the family, and thereby determines finite-distance overlaps, Bargmann phases, the quantum metric, and the Berry curvature. The same analytic structure confines the family to an exact projective subspace whose dimension grows at most linearly with particle number. Exact enumeration of a Nielsen-Cirac-Sierra state demonstrates finite-shot reconstruction of both the metric and a geometric phase. Moreover, a Rényi-2 divergence sets the statistical range of the reconstruction while, at nondegenerate points, occupation readout also attains the local single-copy multiparameter information bound.
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.
Analog quantum simulators natively return large collections of single-shot projective measurements, the wave-function snapshots. These are high-dimensional configurations drawn from the Born distribution of the state they prepare. These collections carry far more information than is retained when they are simply compressed into low-order observables fixed in advance. Moreover, this procedure requires knowing beforehand what one is looking for. This thesis chooses to take a different route, asking what structure a dataset of snapshots possesses as a set of points in configuration space, and what kind of physics that structure reflects. The tools used here are drawn from un supervised learning and network theory, namely principal component analysis (PCA), intrinsic-dimension estimators (TWO-NN and BID), and the wave-function networks obtained by promoting snapshots to nodes and activating links according to the met ric structure of configuration space. They are chosen under two constraints, namely interpretability at every step, and the smallest possible amount of physical input: no order parameter, no symmetry, and no knowledge of the Hamiltonian. At equilibrium, wave-function networks are used to build a classification of one dimensional quantum phases based on snapshots alone. The scheme first selects, in the spirit of Occam’s razor, the measurement basis carrying the minimal information content, and then analyses the networks constructed in that basis. Paramagnetic and symmetry-broken phases display low complexity and “probability” networks, symmetry protected topological phases high complexity and Erdős–Rényi networks, and critical points high complexity and scale-free networks. Interestingly, gapped and gapless mat ter are treated on the same footing, and real-space decimation enters the scheme to resolve massive phases rather than flowing towards criticality. Out of equilibrium, the dynamical exponent governing transport of a conserved quan tity is shown to be imprinted on the singular-value spectrum of the snapshot matrix. In particular, it is shown that the PCA entropy, and in fact the leading eigenvalue alone, grows with the same exponent as the underlying spin or energy transport, at modest system sizes, evolution times and sample numbers. Ergodicity breaking is then ad dressed through the binary intrinsic dimension (BID) of the snapshots, in the quantum sun model: the BID grows and saturates at the Thouless time, reaching its maximum in the ergodic phase, remaining system-size independent in the non-ergodic one, and growing sub-maximally at criticality, where the data occupy an extensive but lower dimensional manifold. The associated networks reproduce the equilibrium dichotomy, being random in the ergodic phase and scale-free at the transition, whose location can be identified by the BID within a few percent of spectral estimates. Preliminary experimental observations, not reported here, show analogous features. Taken together, these results indicate that datasets orders of magnitude smaller than the Hilbert-space dimension suffice to identify collective properties of quantum matter through the study of the structure of the data points in configuration space. Moreover, the same structural classes recur across ground states and quenched dynamics, with no order parameter supplied and under realistic experimental imperfections. The final chapter elaborates on another aspect of quantum simulator capabilities, that is, that to realize gauge theory dynamics. There, one-dimensional U(1) lattice gauge theories with a topological θ term are mapped exactly onto kinetically con strained fermions in an emergent Stark potential. The mapping accounts for the slow relaxation reported in recent Rydberg experiments: prethermalization near an inte grable point at θ = 0, and Stark localization with Hilbert-space fragmentation at θ ̸ =0.
A projective measurement of a many-body state produces a microscopic snapshot, usually viewed as random classical data. We show that partial occupation snapshots of the critical XX chain contain a universal angle with a precise conformal meaning. Dividing the ring into two measured and two unmeasured arcs, we assign geometry-dependent conformal side weights to the observed occupations and obtain a compact variable δL. At every finite size, δL is fixed by the measured sites alone; the particular complete-configuration lift XL used in the proof additionally depends on unobserved particles. This angle is an exact microscopic compact coordinate whose scaling-limit law is that of the relative Dirichlet phase of the associated conformal quadrilateral---the boundary coordinate conjugate to charge in continuum post-measurement descriptions. Exact free-fermion determinants yield all of its Fourier moments. We prove that the lift becomes Gaussian with variance 2h(ζ), where h(ζ) is the rectangle modulus, and hence $\langle e^{\ii qδ_L}\rangle\to e^{-h(ζ)q^2}$. Thus raw quantum snapshots realize the heat kernel on a circle and provide an outcome-level microscopic foundation for the compact zero-mode sector of Born averages over fluctuating conformal boundary conditions.
The value of a multiqubit geometric phase at a single operating point does not reveal whether it was generated directly or through a connected sequence of lower-body interactions. For analytic, gapped, nondegenerate Abelian holonomies, we jointly resolve logical support and independently calibrated coupling support. Every nonzero connected response must involve an active set that connects and covers the target and must contain one factor of each active coupling, yielding the sharp onset bound νS≥τE(S). The same selection law applies locally to projected Berry curvature. Conversely, an unrestricted finite-dimensional construction realizes all responses allowed by the connected-cover condition simultaneously, making the condition necessary and sufficient within this theorem class. Operationally, simultaneous confidence bands certify detected mixed responses while controlling false positives; a factor-of-two separation condition additionally recovers exact library-relative response minima. In a complete dictionary, nontriviality of every minimal-cover response is generically necessary and sufficient for these minima to coincide with the minimal calibrated mechanisms. Three-qubit Wilson calculations distinguish direct and pair-mediated routes with the same endpoint phase, and a synthetic Ramsey audit locates the finite-resolution boundary.
A composite system of N local q-level factors has dimension D=q^N. Although at most D vectors can be exactly orthogonal, a fixed squared-overlap tolerance \eps permits M_\eps(D)\gtrsim\exp(c_\eps D) mutually quasi-orthogonal directions. Consequently M_\eps(q^N)\gtrsim\exp(c_\eps q^N): the dimension grows exponentially in N, but its quasi-orthogonal capacity grows doubly exponentially. Lévy's lemma explains the accompanying concentration of regular observables, and Johnson--Lindenstrauss scaling gives the complementary finite-set account of exponential capacity. For Haar-random pure states the sharper exact law is \mathbb P(|\langleφ|ψ\rangle|^2\geq\eps)=(1-\eps)^{D-1}, while typical Fubini--Study angles lie within O(D^{-1/2}) of π/2: capacity explodes as angular structure homogenizes. We turn these facts into simultaneous overlap and trace-distance bounds for finite decoherence branch families, including mixed environments and collective weak coherences. The results are conditional on typical relative environmental dynamics. They neither select a pointer basis nor identify coherence suppression with readable or redundant records.
We show that the Hilbert space spanned by a continuously parametrized wavefunction family---i.e., a quantum state manifold---is dominated by a subspace, onto which all member states have close to unity projection weight. Its characteristic dimensionality DP is much smaller than the full Hilbert space dimension, and is equivalent to a statistical complexity measure eS2, where S2 is the 2nd Renyi entropy of the manifold. In the thermodynamic limit, DP closely approximates the quantum geometric volume of the manifold under the Fubini-Study metric, revealing an intriguing connection between information and geometry. This connection persists in compact manifolds such as a twisted boundary phase, where the corresponding geometric circumference is lower bounded by a term proportional to its topological index, reminiscent of entanglement entropy.