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Chiral graviton modes are hallmark collective excitations of fractional quantum Hall (FQH) liquids. However, their existence on the lattice, where continuum symmetries that protect them from decay are lost, is still an open and urgent question, especially considering the recent advances in the realization of fractional Chern insulators (FCI) in transition metal dichalcogenides and rhombohedral pentalayer graphene. Here we present a comprehensive theoretical and numerical study of graviton-modes in fermionic FCI, and thoroughly demonstrate their existence. We first derive a lattice stress tensor operator in the context of the fermionic Harper–Hofstadter (HH) model which captures the graviton in the flat band limit. Importantly, we discover that such lattice stress-tensor operators are deeply connected to correlation hole dynamics, readily generalizable to generic Chern bands. We then explicitly show the adiabatic connection between FQH and FCI chiral graviton modes by interpolating from a low flux HH model to a Checkerboard lattice model that hosts a topological flat band. In particular, using state-of-the-art matrix product state and exact diagonalization simulations, we provide strong evidence that chiral graviton modes are long-lived excitations in FCIs despite the lack of continuous symmetries and the scattering with a two-magnetoroton continuum. By means of a careful finite-size analysis, we show that the lattice generates an intrinsic decay rate for the graviton mode which is small compared to its energy. We discuss the relevance of our results for the exploration of graviton modes in FCI phases realized in solid state settings, experimental utility on diagnosis of fractional topological phases, as well as cold atom experiments.

Neural Quantum States (NQS) are powerful tools used to represent complex quantum many-body states in an increasingly wide range of applications. However, despite their popularity, at present only a rudimentary understanding of their limitations exists. In this work, we investigate the dependence of NQS on the choice of the computational basis, focusing on restricted Boltzmann machines. Considering a family of rotated Hamiltonians corresponding to the paradigmatic transverse-field Ising model, we discuss the properties of ground states responsible for the dependence of NQS performance, namely the presence of ground state degeneracies as well as the uniformity of amplitudes and phases, carefully examining their interplay. We identify that the basis-dependence of the performance is linked to the convergence properties of a cluster or cumulant expansion of multi-spin operators—providing a framework to directly connect physical, basis-dependent properties, to performance itself. Our results provide insights that may be used to gauge the applicability of NQS to new problems and to identify the optimal basis for numerical computations.

Understanding the interplay between nonstabilizerness and entanglement is crucial for uncovering the fundamental origins of quantum complexity. Recent studies have proposed entanglement spectral quantities, such as antiflatness of the entanglement spectrum and entanglement capacity, as effective complexity measures, establishing direct connections to stabilizer Rényi entropies. In this work, we systematically investigate quantum complexity across a diverse range of spin models, analyzing how entanglement structure and nonstabilizerness serve as distinctive signatures of quantum phases. By studying entanglement spectra and stabilizer entropy measures, we demonstrate that these quantities consistently differentiate between distinct phases of matter. Specifically, we provide a detailed analysis of spin chains including the XXZ model, the transverse-field XY model, its extension with Dzyaloshinskii-Moriya interactions, as well as the Cluster Ising and Cluster XY models. Our findings reveal that entanglement spectral properties and magic-based measures serve as intertwined, robust indicators of quantum phase transitions, highlighting their significance in characterizing quantum complexity in many-body systems.

Chiral graviton modes are elusive excitations arising from the hidden quantum geometry of fractional quantum Hall states. It remains unclear, however, whether this picture extends to lattice models, where continuum translations are broken and additional quasiparticle decay channels arise. We present a framework in which we explicitly derive a field theory incorporating lattice chiral graviton operators within the paradigmatic bosonic Harper-Hofstadter model. Extensive numerical evidence suggests that chiral graviton modes persist away from the continuum and are well captured by the proposed lattice operators. We identify geometric quenches as a viable experimental probe, paving the way for the exploration of chiral graviton modes in near-term quantum simulation experiments.

We study the behavior of magic as a bipartite correlation in the quantum Ising chain across its quantum phase transition and at finite temperature. To quantify the magic of partitions rigorously, we formulate a hybrid scheme that combines stochastic sampling of reduced density matrices via quantum Monte Carlo, with state-of-the-art estimators for the robustness of magic - a bona fide measure of magic for mixed states. This allows us to compute the mutual robustness of magic for partitions up to eight sites, embedded into a much larger system. We show how mutual robustness is directly related to critical behaviors: at the critical point, it displays a power-law decay as a function of the distance between partitions, whose exponent is related to the partition size. Once finite temperature is included, mutual magic retains its low temperature value up to an effective critical temperature, whose dependence on size is also algebraic. This suggests that magic, differently from entanglement, does not necessarily undergo a sudden death.

We study the dynamics of hard-core bosons on ladders, in the presence of strong kinetic constrains akin to those of the Bariev model. We use a combination of analytical methods and numerical simulations to establish the phase diagram of the model. The model displays a paired Tomonaga-Luttinger liquid phase featuring an emergent dipole symmetry, which encodes the local pairing constraint into a global, nonlocal quantity. We scrutinize the effect of such emergent low-energy symmetry during quench dynamics including single-particle defects. We observe that, despite being approximate, the dipole symmetry still leads to very slow relaxation dynamics, which we model via an effective field theory. The model we discuss is amenable to realization in both cold atoms in optical lattices and Rydberg atom arrays with dynamics taking place solely in the Rydberg manifold. To observe the unusual dynamics of excitations in such experimental platforms, we propose a two-step protocol, which starts with the quasi-adiabatic preparation of low-energy states, followed by the local creation of defects and their study under quench dynamics.

Locality is a transversal principle that governs quantum dynamics of many-body systems. However, for cavity-embedded systems, such a fundamental notion is hindered by the presence of nonlocal cavity modes, leaving space for new possible dynamical behavior. Here, we investigate the real-time dynamics of low-energy excitations in one-dimensional Rydberg atom arrays coupled to a global cavity mode. We derive an effective description in terms of a Tavis-Cummings-Ising model, whose phase diagram features ordered and disordered phases. The nonlocal nature of the cavity mode drastically affects the emergent meson and string dynamics. Mesons hybridize coherently with the cavity photons, leading to composite meson-polariton excitations. Strings, differently from local interacting theories, acquire a finite kinetic energy thanks to nonlocal cavity-mediated interactions between the underlying domain walls. We then conclude by presenting a new concrete experimental blueprint for a cavity QED Rydberg atom array simulator where the physics outlined in this Letter can be realized.

Understanding how entanglement can be reduced through simple operations is crucial for both classical and quantum algorithms. We investigate the entanglement properties of lattice models hosting conformal field theories cooled via local Clifford operations, a procedure we refer to as stabilizer disentangling. We uncover two distinct regimes: a constant gain regime, where disentangling is volume-independent, and a log-gain regime, where disentanglement increases with volume, characterized by a reduced effective central charge. In both cases, disentangling efficiency correlates with the target state magic, with larger magic leading to more effective cooling. The dichotomy between the two cases stems from mutual stabilizer Rényi entropy, which influences the entanglement cooling process. We provide an analytical understanding of such effect in the context of cluster Ising models, that feature disentangling global Clifford operations. Our findings indicate that matrix product states possess subclasses based on the relationship between entanglement and magic, and clarifying the potential of new classes of variational states embedding Clifford dynamics within matrix product states.

Recent breakthrough experiments have demonstrated how it is now possible to explore the dynamics of quantum Hall states interacting with quantum electromagnetic cavity fields. While the impact of strongly coupled nonlocal cavity modes on integer quantum Hall physics has been recently addressed, the effects on fractional quantum Hall (FQH) liquids - and, more generally, fractionalized states of matter - remain largely unexplored. In this work, we develop a theoretical framework for the understanding of FQH states coupled to quantum light. In particular, combining analytical arguments with tensor network simulations, we study the dynamics of a ν=1/3 Laughlin state in a single-mode cavity with finite electric field gradients. We find that the topological signatures of the FQH state remain robust against the nonlocal cavity vacuum fluctuations, as indicated by the endurance of the quantized Hall resistivity. The entanglement spectra, however, carry direct fingerprints of light-matter entanglement and topology, revealing peculiar polaritonic replicas of the U(1) counting. As a further response to cavity fluctuations, we also find a squeezed FQH geometry, encoded in long-wavelength correlations. By exploring the low-energy excited spectrum inside the FQH phase, we identify a new neutral quasiparticle, the graviton polariton, arising from the hybridization between quadrupolar FQH collective excitations (known as gravitons) and light. Pushing the light-matter interaction to ultrastrong-coupling regimes, we find other two important effects, a cavity vacuum-induced Stark shift for charged quasiparticles and a potential instability toward a density modulated stripe phase, competing against the phase separation driven by the Stark shift. Finally, we discuss the experimental implications of our findings and possible extension of our results to more complex scenarios.

We present a thorough investigation of nonstabilizerness - a fundamental quantum resource that quantifies state complexity within the framework of quantum computing - in a one-dimensional U(1) lattice gauge theory including matter fields. We show how nonstabilizerness is always extensive with volume, and has no direct relation to the presence of critical points. However, its derivatives typically display discontinuities across the latter: This indicates that nonstabilizerness is strongly sensitive to criticality, but in a manner that is very different from entanglement (which, typically, is maximal at the critical point). Our results indicate that error-corrected simulations of lattice gauge theories close to the continuum limit have similar computational costs to those at finite correlation length and provide rigorous lower bounds for quantum resources of such quantum computations.

We perform a principal component analysis (PCA) of two one-dimensional lattice models belonging to distinct nonequilibrium universality classes - directed bond percolation and branching and annihilating random walks with an even number of offspring. We find that the uncentered PCA of data sets storing various system's configurations can be successfully used to determine the critical properties of these nonequilibrium phase transitions. In particular, in both cases, we obtain good estimates of the critical point and the dynamical critical exponent of the models. For directed bond percolation, we are furthermore able to extract critical exponents associated with the correlation length and the order parameter. We discuss the relation of our analysis with low-rank approximations of data sets.

Nonstabilizerness describes the distance of a quantum state to its closest stabilizer state. It is - like entanglement - a necessary resource for a quantum advantage over classical computing. We study nonstabilizerness, quantified by stabilizer entropy, in a hybrid quantum circuit with projective measurements and a controlled injection of non-Clifford resources. We discover a phase transition between a power law and constant scaling of nonstabilizerness with system size controlled by the rate of measurements. The same circuit also exhibits a phase transition in entanglement that appears, however, at a different critical measurement rate. This mechanism shows how, from the viewpoint of a quantum advantage, hybrid circuits can host multiple distinct transitions where not only entanglement, but also other nonlinear properties of the density matrix come into play.

We study nonequilibrium quantum dynamics of spin chains by employing principal component analysis on data sets of wave function snapshots and examine how information propagates within these data sets. The quantities we employ are derived from the spectrum of the sample second moment matrix, built directly from data sets. Our investigations on several interacting spin chains featuring distinct spin or energy transport reveal that the growth of data information spreading follows the same dynamical exponents as that of the underlying quantum transport of spin or energy. Specifically, our approach enables an easy, data-driven, and, importantly, interpretable diagnostic to track energy transport with a limited number of samples, which is usually challenging without any assumption on the Hamiltonian form. These observations are obtained at a modest finite-size and evolution time, which aligns with experimental and numerical constraints. Our framework directly applies to experimental quantum simulator data sets of dynamics in higher-dimensional systems, where classical simulation methods usually face significant limitations and apply equally to both near- and far-from-equilibrium quenches.

In this paper, we investigate the relationship between entanglement and nonstabilizerness (also known as magic) in matrix product states (MPSs). We study the relation between magic and the bond dimension used to approximate the ground state of a many-body system in two different contexts: full state of magic and mutual magic (the nonstabilizer analog of mutual information, thus free of boundary effects) of spin-1 anisotropic Heisenberg chains. Our results indicate that obtaining converged results for nonstabilizerness is typically considerably easier than entanglement. For full state magic at critical points and at sufficiently large volumes, we observe convergence with 1/χ2, with χ being the MPS bond dimension. At small volumes, magic saturation is so quick that, within error bars, we cannot appreciate any finite-χ correction. Mutual magic also shows a fast convergence with bond dimension, whose specific functional form is however hindered by sampling errors. As a byproduct of our study, we show how Pauli-Markov chains (originally formulated to evaluate magic) resets the state of the art in terms of computing mutual information for MPS. We illustrate this last fact by verifying the logarithmic increase of mutual information between connected partitions at critical points. By comparing mutual information and mutual magic, we observe that, for connected partitions, the latter is typically scaling much slower - if at all - with the partition size, while for disconnected partitions, both are constant in size.

Nonstabilizerness, also known as "magic,"stands as a crucial resource for achieving a potential advantage in quantum computing. Its connection to many-body physical phenomena is poorly understood at present, mostly due to a lack of practical methods to compute it at large scales. We present a novel approach for the evaluation of nonstabilizerness within the framework of matrix product states (MPSs), based on expressing the MPS directly in the Pauli basis. Our framework provides a powerful tool for efficiently calculating various measures of nonstabilizerness, including stabilizer Rényi entropies, stabilizer nullity, and Bell magic, and enables the learning of the stabilizer group of an MPS. We showcase the efficacy and versatility of our method in the ground states of Ising and XXZ spin chains, as well as in circuits dynamics that has recently been realized in Rydberg atom arrays, where we provide concrete benchmarks for future experiments on logical qubits up to twice the sizes already realized.

Network science provides very powerful tools for extracting information from interacting data. Although recently the unsupervised detection of phases of matter using machine learning has raised significant interest, the full prediction power of network science has not yet been systematically explored in this context. Here we fill this gap by providing an in-depth statistical, combinatorial, geometrical, and topological characterization of 2D Ising snapshot networks (IsingNets) extracted from Monte Carlo simulations of the 2D Ising model at different temperatures, going across the phase transition. Our analysis reveals the complex organization properties of IsingNets in both the ferromagnetic and paramagnetic phases and demonstrates the significant deviations of the IsingNets with respect to randomized null models. In particular percolation properties of the IsingNets reflect the existence of the symmetry between configurations with opposite magnetization below the critical temperature and the very compact nature of the two emerging giant clusters revealed by our persistent homology analysis of the IsingNets. Moreover, the IsingNets display a very broad degree distribution and significant degree-degree correlations and weight-degree correlations demonstrating that they encode relevant information present in the configuration space of the 2D Ising model. The geometrical organization of the critical IsingNets is reflected in their spectral properties deviating from the one of the null model. This work reveals the important insights that network science can bring to the characterization of phases of matter. The set of tools described hereby can be applied as well to numerical and experimental data.

Nonstabilizerness, also colloquially referred to as magic, is a resource for advantage in quantum computing and lies in the access to non-Clifford operations. Developing a comprehensive understanding of how nonstabilizerness can be quantified and how it relates to other quantum resources is crucial for studying and characterizing the origin of quantum complexity. In this work, we establish a direct connection between nonstabilizerness and entanglement spectrum flatness for a pure quantum state. We show that this connection can be exploited to efficiently probe nonstabilizerness even in the presence of noise. Our results reveal a direct connection between nonstabilizerness and entanglement response, and define a clear experimental protocol to probe nonstabilizerness in cold atom and solid-state platforms.

Programmable quantum devices are now able to probe wave functions at unprecedented levels. This is based on the ability to project the many-body state of atom and qubit arrays onto a measurement basis which produces snapshots of the system wave function. Extracting and processing information from such observations remains, however, an open quest. One often resorts to analyzing low-order correlation functions - that is, discarding most of the available information content. Here, we introduce wave-function networks - a mathematical framework to describe wave-function snapshots based on network theory. For many-body systems, these networks can become scale-free - a mathematical structure that has found tremendous success and applications in a broad set of fields, ranging from biology to epidemics to Internet science. We demonstrate the potential of applying these techniques to quantum science by introducing protocols to extract the Kolmogorov complexity corresponding to the output of a quantum simulator and implementing tools for fully scalable cross-platform certification based on similarity tests between networks. We demonstrate the emergence of scale-free networks analyzing experimental data obtained with a Rydberg quantum simulator manipulating up to 100 atoms. Our approach illustrates how, upon crossing a phase transition, the simulator complexity decreases while correlation length increases - a direct signature of buildup of universal behavior in data space. Comparing experiments with numerical simulations, we achieve cross-certification at the wave-function level up to timescales of 4 μs with a confidence level of 90% and determine experimental calibration intervals with unprecedented accuracy. Our framework is generically applicable to the output of quantum computers and simulators with in situ access to the system wave function and requires probing accuracy and repetition rates accessible to most currently available platforms.

Topology plays a fundamental role in our understanding of many-body physics, from vortices and solitons in classical field theory to phases and excitations in quantum matter. Topological phenomena are intimately connected to the distribution of information content that, differently from ordinary matter, is now governed by nonlocal degrees of freedom. However, a precise characterization of how topological effects govern the complexity of a many-body state, i.e., its partition function, is presently unclear. In this paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely present a theory that shows how the Kolmogorov complexity of a classical partition function sampling carries unique, distinctive features depending on the presence of topological excitations in the system. We confront two-dimensional Ising, Heisenberg, and XY models on several topologies and study the corresponding samplings as high-dimensional manifolds in configuration space, quantifying their complexity via the intrinsic dimension. While for the Ising and Heisenberg models the intrinsic dimension is independent of the real-space topology, for the XY model it depends crucially on temperature: across the Berezkinskii-Kosterlitz-Thouless (BKT) transition, complexity becomes topology dependent. In the BKT phase, it displays a characteristic dependence on the homology of the real-space manifold, and, for g-torii, it follows a scaling that is solely genus dependent. We argue that this behavior is intimately connected to the emergence of an order parameter in data space, the conditional connectivity, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.

Quantum simulators offer powerful means to investigate strongly correlated quantum matter. However, interpreting measurement outcomes in such systems poses significant challenges. Here, we present a theoretical framework for information extraction in synthetic quantum matter, illustrated for the case of a quantum quench in a spinor Bose-Einstein condensate experiment. Employing nonparametric unsupervised learning tools that provide different measures of information content, we demonstrate a theory-agnostic approach to identify dominant degrees of freedom. This enables us to rank operators according to their relevance, akin to effective field theory. To characterize the corresponding effective description, we then explore the intrinsic dimension of data sets as a measure of the complexity of the dynamics. This reveals a simplification of the data structure, which correlates with the emergence of time-dependent universal behavior in the studied system. Our assumption-free approach can be immediately applied in a variety of experimental platforms.