Quantum data can be obtained from a diverse range of sources, including direct measurements from noisy quantum processors, cold-atom simulators, and classical approximations such as variational neural-network states. However, our ability to characterize these systems is fundamentally limited, as the available measurement data are often sparse compared to the exponentially large Hilbert space of the system. To address this, we propose using the expected minimal Hamming distance calculated for a set of unique bitstrings as a robust metric revealing a distinct power-law behaviour. Crucially, we have established a direct connection between the empirical power-law parameters and the intrinsic dimension of the quantum measurement outcomes. Through various examples of real experiments and simulations, we show that the power-law parameters reliably characterize the underlying geometric structure of measurement outcomes and identify quantum phase transitions from limited quantum information, without the need for accumulating extensive statistics or explicitly calculating physical observables. This enables the analysis of completely different quantum experiments within a single framework. Furthermore, we demonstrate the versatility of our approach on classical high-dimensional data, showing that the power-law parameters can successfully discriminate different classes of handwritten digits from the MNIST dataset.
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Updated version with an expanded theoretical framework linking scaling to intrinsic dimension, updated state discrimination sections, and new benchmarks on the classical MNIST dataset
Quantum data can be obtained from a diverse range of sources, including direct measurements from noisy quantum processors, cold-atom simulators, and classical approximations such as variational neural-network states. However, our ability to characterize these systems is fundamentally limited, as the available measurement data are often sparse compared to the exponentially large Hilbert space of the system. To address this, we propose using the expected minimal Hamming distance calculated for a set of unique bitstrings as a robust metric revealing a distinct power-law behaviour. Crucially, we have established a direct connection between the empirical power-law parameters and the intrinsic dimension of the quantum measurement outcomes. Through various examples of real experiments and simulations, we show that the power-law parameters reliably characterize the underlying geometric structure of measurement outcomes and identify quantum phase transitions from limited quantum information, without the need for accumulating extensive statistics or explicitly calculating physical observables. This enables the analysis of completely different quantum experiments within a single framework. Furthermore, we demonstrate the versatility of our approach on classical high-dimensional data, showing that the power-law parameters can successfully discriminate different classes of handwritten digits from the MNIST dataset.
Analyzing the properties of complex quantum systems is crucial for further development of quantum devices, yet this task is typically challenging and demanding with respect to the required amount of measurements. Special attention to this problem appears within the context of characterizing outcomes of noisy intermediate-scale quantum devices, which produce quantum states with specific properties so that it is expected to be hard to simulate such states using classical resources. In this work, we address the problem of characterization of a boson sampling device, which uses the interference of input photons to produce samples of nontrivial probability distributions that at certain condition are hard to obtain classically. For realistic experimental conditions the problem is to probe multiphoton interference with a limited number of the measurement outcomes without collisions and repetitions. By constructing networks on the measurement outcomes, we demonstrate the possibility to discriminate between regimes of indistinguishable and distinguishable bosons by quantifying the structures of the corresponding networks. Based on this, we propose a machine-learning-based protocol to benchmark a boson sampler with unknown scattering matrix. Notably, the protocol works in the most challenging regimes of having a very limited number of bitstrings without collisions and repetitions. As we expect, our framework can be directly applied for characterizing boson sampling devices that are currently available in experiments.
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.
Quantum neural networks generalize classical artificial neural networks into the quantum domain. They are formulated as parameterized quantum circuits which are optimized by measuring and minimizing a suitably chosen loss function. The core challenge in understanding, implementing and ultimately using quantum neural networks is that they represent many-body systems with an exponentially large Hilbert space, in combination with a large parameter search space. Moreover, noise -- which is inherent to any quantum measurement -- sets practical limits for the estimation of training loss. Here, we study physics-informed large-scale quantum neural networks that are trained through a finite number of noisy loss function measurements. We show that this architecture permits the construction of nontrivial decision boundaries that enable the classification of quantum states through measuring an order parameter. Our approach can directly process quantum data that is output from quantum simulators and computers and is well suited for implementation on current hardware. Moreover, owed to a close link between the neural network dynamics and the evolution of Markovian open many-body quantum systems, one may expect a certain robustness to noise, which is ubiquitous in the current NISQ era.
Scaling laws, the power-law relations between loss, architecture size, and compute observed in modern neural networks, offer a quantitative way to characterize the complexity of a learning problem, with the exponent governing the decay of the loss reflecting how rapidly additional resources translate into improved accuracy, and thus how hard the target is to learn. Whether an analogous framework can characterize the complexity of physical problems remains open. We address this question for Neural-Network Quantum States, a leading variational approach for strongly correlated quantum many-body systems. Using transformer wave functions to approximate ground states of the J1-J2 Heisenberg model on triangular and square lattices with up to 20×20 sites, we find that the V-score, a measure of accuracy of a variational state, decays as a power law in training compute. Under an appropriate rescaling of compute, results for different system sizes collapse onto a single curve, analogous to scaling collapse in critical phenomena. The resulting power law is, to a good approximation, independent of the number of sites, showing that the transformer Ansatz is size-consistent for the systems considered. The exponent decreases systematically with frustration, identifying it as a quantitative measure of representational difficulty of the ground state and establishing scaling laws as a general framework for benchmarking variational ansätze.
Monitored quantum circuits host a rich variety of exotic non-equilibrium phases. Among the most representative examples are measurement-induced phase transitions between distinct area-law entangled states. However, because these transitions are characterized by specific entanglement quantities such as mutual information or topological entanglement entropy that are nonlinear functionals of the density matrix, their experimental observation requires multiple identical quantum trajectories via post-selection, which becomes exponentially unfeasible for large systems. Here, we leverage modern machine learning tools to address this challenge. We devise a neural network architecture combining a convolutional neural network with an attention mechanism, and use raw measurement outcomes directly as input to classify trivial, long-range entangled, and symmetry-protected topological phases. We show that the system's relaxation to a steady-state phase manifests as a sharp convergence in the classifier's accuracy, entirely bypassing the need for quantum state reconstruction. We systematically study the performance of our network as a function of sample size, input data, spatial and temporal constraints, and system size scalability. Our results demonstrate that this approach is robust and post-selection free, offering a practical pathway for experimentally probing measurement-induced phases.
The fate of ergodicity and localization in disordered, interacting quantum systems in two spatial dimensions remains an outstanding open problem in non-equilibrium physics. Theoretical and numerical approaches are severely constrained by exponential Hilbert space growth. Here, we leverage the high data-acquisition rates of superconducting quantum processors to extensively sample Hilbert space configurations, effectively imaging the state as a wavefunction network (WFN). Tracking the bitstring return probability across varied disorder strengths, we find heavy-tailed distributions and power-law typical decay characteristic of glassy relaxation. Furthermore, the distribution of bitstring probabilities transitions from a Porter-Thomas form at low disorder to an eight-decade power law at stronger disorder. In the reconstructed configuration space, WFN connectivity is compatible with a random geometric network at low disorder and becomes scale-free at stronger disorder. Finally, a multifractal analysis based on the binary intrinsic dimension (BID) reveals a non-ergodic regime where the wavefunction support saturates to a finite fraction of the total Hilbert space. Together, these results rule out a direct ergodic-to-localized transition, pointing instead to an extended multifractal regime. By visualizing wavefunction dynamics directly in configuration space, this work complements traditional real-space studies, showing the potential of quantum processors to resolve long-standing questions in non-equilibrium physics.