Abstract The evolution of statistical physics from its mid-twentieth-century roots to its modern interplay with quantum information (QI) has reshaped how complex many-body systems are modeled and quantified. Classical tools now coexist with quantum-mechanical concepts like entanglement, coherence, and non-stabilizerness, which function as resources in the operational framework of quantum resource theories. As a case study, we present a recent numerical approach based on tensor cross interpolation (TCI) for computing resource measures that conceptually consolidates ideas that underlie modern computational techniques. Motivated by this, we survey numerical methods for evaluating quantum resources in many-body systems, emphasizing what scales to large sizes in practice. We organize the discussion around three families: subspace decomposition (Krylov methods, tensor networks (TNs)), stochastic sampling (Monte Carlo), and learning-based models (neural quantum states), and outline how each supports the evaluation of entanglement, non-stabilizerness (magic), and coherence. We provide a concise comparison that summarizes available measures and highlights current open problems and perspectives.
Abstract The evolution of statistical physics from its mid-twentieth-century roots to its modern interplay with quantum information (QI) has reshaped how complex many-body systems are modeled and quantified. Classical tools now coexist with quantum-mechanical concepts like entanglement, coherence, and non-stabilizerness, which function as resources in the operational framework of quantum resource theories. As a case study, we present a recent numerical approach based on tensor cross interpolation (TCI) for computing resource measures that conceptually consolidates ideas that underlie modern computational techniques. Motivated by this, we survey numerical methods for evaluating quantum resources in many-body systems, emphasizing what scales to large sizes in practice. We organize the discussion around three families: subspace decomposition (Krylov methods, tensor networks (TNs)), stochastic sampling (Monte Carlo), and learning-based models (neural quantum states), and outline how each supports the evaluation of entanglement, non-stabilizerness (magic), and coherence. We provide a concise comparison that summarizes available measures and highlights current open problems and perspectives.
Quantum information quantifiers are indispensable tools for analyzing strongly correlated systems. Consequently, developing efficient and robust numerical methods for their computation is crucial. We propose a general procedure based on the family of Tensor Cross Interpolation (TCI) algorithms to address this challenge in a fully general framework, independent of the system or the quantifier under consideration. To substantiate our approach, we compute the non-stabilizerness Rényi entropy (SRE) and Relative Entropy of Coherence (REC) considering the 1D and 2D ferromagnetic Ising models with minimal modifications to the numerical procedure. This method not only demonstrates its versatility, but also provides a generic framework for exploring other quantum information quantifiers in complex systems.
Quantum Monte Carlo (QMC) methods are among the central numerical tools for studying strongly correlated quantum many-body systems, particularly in higher dimensions. As quantum information has introduced new information-theoretic perspectives and diagnostics into many-body physics, QMC methods have accordingly been extended beyond the measurement of conventional linear observables. This review summarizes recent progress in adapting QMC to many-body quantum-information, focusing on qubit or spin-1/2 systems as a concrete setting while keeping the discussion broadly applicable to qudit and bosonic systems. We present a unified perspective on the extraction of nonlinear diagnostics, including entanglement entropies and entanglement spectra, Rényi negativities for mixed-state entanglement, stabilizer entropies for quantum magic, and decoherence-driven phenomena such as the interplay between imaginary-time evolution and decoherence and strong-to-weak spontaneous symmetry breaking.
Abstract Variational techniques have long been at the heart of atomic, solid-state, and many-body physics. They have recently extended to quantum and classical machine learning, providing a basis for representing quantum states via neural networks. These methods generally aim to minimize the energy of a given ansatz, though open questions remain about the expressivity of quantum and classical variational ansätze. The connection between variational techniques and quantum computing, through variational quantum algorithms, offers opportunities to explore the quantum complexity of classical methods. We demonstrate how the concept of non-stabilizerness, or magic, can create a bridge between quantum information and variational techniques and we show that energy accuracy is a necessary but not always sufficient condition for accuracy in non-stabilizerness. Through systematic benchmarking of neural network quantum states, matrix product states, and variational quantum methods, we show that while classical techniques are more accurate in non-stabilizerness, not accounting for the symmetries of the system can have a severe impact on this accuracy. Our findings form a basis for a universal expressivity characterization of both quantum and classical variational methods.
A defining feature of quantum many-body systems is the exponential scaling of the Hilbert space with the number of degrees of freedom. This exponential complexity naïvely renders a complete state characterization, for instance via the complete set of bipartite Renyi entropies for all disjoint regions, a challenging task. Recently, a compact way of storing subregions' purities by encoding them as amplitudes of a fictitious quantum wave function, known as entanglement feature, was proposed. Notably, the entanglement feature can be a simple object even for highly entangled quantum states. However the complexity and practical usage of the entanglement feature for general quantum states has not been explored. In this work, we demonstrate that the entanglement feature can be efficiently learned using only a polynomial amount of samples in the number of degrees of freedom through the so-called tensor cross interpolation (TCI) algorithm, assuming it is expressible as a finite bond dimension MPS. We benchmark this learning process on Haar and random MPS states, confirming analytic expectations. Applying the TCI algorithm to quantum eigenstates of various one dimensional quantum systems, we identify cases where eigenstates have entanglement feature learnable with TCI. We conclude with possible applications of the learned entanglement feature, such as quantifying the distance between different entanglement patterns and finding the optimal one-dimensional ordering of physical indices in a given state, highlighting the potential utility of the proposed purity interpolation method.
The field of quantum chemistry has long been defined by the central computational challenge of the quantum many-body fermionic system problem, main the electronic Schrödinger equation. This exponential scaling makes it computationally intractable to solve, the exact governing electronic Schrödinger equation. Approximation methods have been developed, ranging from the widely used Density Functional Theory (DFT) to Coupled Cluster (CC) theory. While powerful, these conventional approaches often involve controlled truncations which caused a trade-off between accuracy, leaving many systems—particularly those with strong electron correlation—beyond the reach of ab initio simulation. Furthermore, the famous fermionic sign oscillation problem is particularly severe in stochastic methods like Quantum Monte Carlo (QMC). 1 Traditional Ansätze are also frequently engineered specifically for particular systems, which results in a fragmentation of results and fundamentally limits the transferability. Recent research demonstrates that with carefully designed architectures, ANNs can not only represent ground states of model systems with high precision but also tackle the sign problem, offering a promising path forward for strongly correlated electronic structure calculations.2,3 This review charts the paradigm shift driven by Neural Network Quantum States (NNQS), which leverage the representational power of deep learning to overcome these barriers. We detail the architectural evolution from Restricted Boltzmann Machines to autoregressive models like RNNs and Transformers, which enable exact, uncorrelated sampling and bypass critical bottlenecks.2,4,5Furthermore, we explore the geometric foundation of high-efficiency optimization via the neural networks, which decouples optimization cost from model complexity.6 The frontier of the field is marked by the integration of operator learning and hybrid quantum-classical frameworks, such as Reinforcement Learning for contracted quantum eigen-solvers and Quantum-Enhanced Neural Networks, which leverage quantum processors to enhance expressivity for hardware-efficient Ansätze.7,8 By framing the problem through the lens of computational resource allocation—specifically, trading exponential memory demands for polynomialcomplexity optimization and sampling—this review elucidates how recent advances at the intersection of artificial neural networks and quantum computing have evolved. The emergence of those methods not merely progress into a powerful simulation-tools, but also a hybrid computational interface.9 This convergence offers a versatile and systematically improvable framework for potentially addressing some of the most challenging problems in quantum physics and chemistry.
We introduce a methodology to estimate nonstabilizerness or “magic,” a key resource for quantum complexity, with neural quantum states (NQS). Our framework relies on two schemes based on Monte Carlo sampling to quantify nonstabilizerness via stabilizer Rényi entropy (SRE) in arbitrary variational wave functions. When combined with NQS, this approach is effective for systems with strong correlations and in dimensions larger than one, unlike tensor network methods. First, we study the magic content in an ensemble of random NQS, demonstrating that neural network parametrizations of the wave function capture finite nonstabilizerness besides large entanglement. Second, we investigate the nonstabilizerness in the ground state of the J 1 − J 2 Heisenberg model. In one dimension, we find that the SRE vanishes at the Majumdar-Ghosh point J 2 = J 1 / 2 , consistent with a stabilizer ground state. In two dimensions, a dip in the SRE is observed near maximum frustration around J 2 / J 1 ≈ 0.6 , suggesting a valence bond solid between the two antiferromagnetic phases.