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 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.
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.
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.
Non-stabilizerness, also known as ``magic,'' quantifies how far a quantum state departs from the stabilizer set. It is a central resource behind quantum advantage and a useful probe of the complexity of quantum many-body states. Yet standard magic quantifiers, such as the stabilizer Rényi entropy (SRE) for qubits and the mana for qutrits, are costly to evaluate numerically, with the computational complexity growing rapidly with the number N of qudits. Here we introduce efficient, numerically exact algorithms that exploit the fast Hadamard transform to compute the SRE for qubits (d=2) and the mana for qutrits (d=3) for pure states given as state vectors. Our methods compute SRE and mana at cost O(Nd2N), providing an exponential improvement over the naive O(d3N) scaling, with substantial parallelism and straightforward GPU acceleration. We further show how to combine the fast Hadamard transform with Monte Carlo sampling to estimate the SRE of state vectors, and we extend the approach to compute the mana of mixed states. All algorithms are implemented in the open-source Julia package HadaMAG ( https://github.com/bsc-quantic/HadaMAG.jl/ ), which provides a high-performance toolbox for computing SRE and mana with built-in support for multithreading, MPI-based distributed parallelism, and GPU acceleration. The package, together with the methods developed in this work, offers a practical route to large-scale numerical studies of magic in quantum many-body systems.
Quantum magic, or nonstabilizerness, provides a crucial characterization of quantum systems, regarding the classical simulability with stabilizer states. In this work, we propose an alternative and efficient algorithm for computing stabilizer R\'enyi entropy, one of the measures for quantum magic, in spin systems with sign-problem free Hamiltonians. This algorithm is based on the quantum Monte Carlo simulation of the path integral of the work between two partition function ensembles and it applies to all spatial dimensions and temperatures. We demonstrate this algorithm on the one- and two-dimensional transverse field Ising model at both finite and zero temperatures and show the quantitative agreements with tensor-network based algorithms. We analyze the computational cost and provide analytical and numerical evidences for it to be polynomial in system size. This work also suggests a unifying framework for calculating various types of entropy quantities including entanglement R\'enyi entropy and entanglement R\'enyi negativity.
Tensor cross interpolation (TCI) is a powerful technique for learning a tensor train (TT) by adaptively sampling a target tensor based on an interpolation formula. However, when the tensor evaluations contain random noise, optimizing the TT is more advantageous than interpolating the noise. Here, we propose a new method that starts with an initial guess of TT and optimizes it using non-linear least-squares by fitting it to measured points obtained from TCI. We use quantics TCI (QTCI) in this method and demonstrate its effectiveness on sine and two-time correlation functions, with each evaluated with random noise. The resulting QTT exhibits increased robustness against noise compared to the QTCI method. Furthermore, we employ this optimized QTT of the correlation function in quantum simulation based on pseudo-imaginary-time evolution, resulting in ground-state energy with higher accuracy than the QTCI or Monte Carlo methods.
Characterizing a quantum state through the lens of quantum resources provides an information-theoretic perspective on many-body systems. While quantum entanglement serves as the paradigmatic example of a quantum resource, recent studies have shown that quantum magic, a resource for universal quantum computation, captures aspects of many-body states complementary to those described by entanglement. For instance, in spin systems, conformal field theory (CFT) analysis of the stabilizer Rényi entropy has revealed universal features of nonstabilizerness qualitatively distinct from entanglement. In bosonic and fermionic systems, however, a comparable formulation for their computational resource, non-Gaussianity, has yet to be established. In this work, we introduce a unified measure, the magic Rényi entropy (MRE), to quantify computational resources in spins, bosons, and fermions on an equal footing. We show that the MRE is a resource monotone under stabilizer and Gaussian protocols involving measurements and feedforward operations. The MRE reveals common universal aspects of nonstabilizerness and non-Gaussianity in critical many-body states. In particular, our CFT analysis shows that the universal contribution to the MRE appears as the size-independent term determined by the Affleck-Ludwig boundary entropy. We find that non-Gaussianity can continuously renormalize this universal contribution or drive a boundary transition through bulk-induced boundary renormalization-group flows. As a concrete example, we present a CFT analysis of non-Gaussianity in interacting spinless fermions described by the Tomonaga-Luttinger liquid, showing boundary transitions at the Luttinger parameters K=1/3 and K=3. Our field-theoretical predictions are confirmed by numerical calculations. These results provide a unified field-theoretical understanding of many-body magic across spins, bosons, and fermions.