Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Complexity in the Quantum Simulation of the Sawtooth Map: Entanglement, Non-stabilizerness and Anti-flatness

VIRGILIO GANGA

Published Sep 16, 2026• 0 citations
OpenAlex

Abstract

The search for quantum advantage, understood as the point at which quantum computation surpasses classical computation, has led to the study of the emergence of quantum complexity, and consequently to the investigation of what makes a quantum algorithm hard to simulate on a classical computer. Quantum complexity can be characterized in different ways: the first is given by the entanglement present within the state of the system, which characterizes the non-classical correlations among its subsystems. In general, entanglement provides a measure of how physically complex the system is, but it is not informative about the complexity of simulating the system. This latter kind of complexity is quantified by non-stabilizerness, which is generated, in particular, by all the operations forming a quantum algorithm that do not belong to the Clifford group. Recently, the anti-flatness has been introduced to provide a bridge between the entanglement present in a bipartition of the system and non-stabilizerness, a global property of the system's state. This thesis considers these three complementary quantities in the context of the simulation of a specific quantum algorithm: the quantum sawtooth map, for which an efficient implementation on a quantum computer exists. The dynamics of the sawtooth map is governed by a kicking strength parameter $k$ and the period of the kicking $T$; depending on their values, the system exhibits a rich phenomenology, including integrable, diffusive, and localized regimes. Among them, this thesis focuses on dynamical localization, a phenomenon strictly analogous to Anderson localization in disordered systems, in which the evolution of momentum eigenstates suppresses momentum diffusion, causing the wavefunction to remain localized in the momentum basis around a given eigenvalue, over a characteristic localization length $\xi$. The study investigates how dynamical localization affects bipartite entanglement, non-stabilizerness, and anti-flatness, and how these quantities vary as a function of the system size, the parameters characterizing the sawtooth map, and the choice of initial state. The main result is that, as the system is driven out of the localized regime and into the ergodic one, the onset of ergodicity is clearly and consistently marked by all three quantities, each approaching the corresponding theoretical prediction for Haar-random states. This behavior makes entanglement, non-stabilizerness, and anti-flatness reliable and complementary probes for distinguishing the two regimes: entanglement and anti-flatness capture the correlations that arise during the application of the algorithm, including features that depend on the logical position of the qubits within the register due to the encoding, while non-stabilizerness offers a basis-independent, global measure of computational complexity. Furthermore, this work promotes the use of the sawtooth map as a test and benchmarking algorithm for quantum hardware.

Related papers

Entanglement and Quantum Complexity in Monitored Quantum Many-Body Systems

Alessio Paviglianiti

Out-of-equilibrium quantum many-body systems stand at the forefront of modern theoretical physics, addressing fundamental questions on thermalization, transport, and universal phenomena. These fields, already well established in condensed matter, statistical physics, quantum optics, and quantum information theory, are progressively gaining even greater relevance with the rapid development of quantum technologies and simulation, which inherently operate in dynamical regimes. In recent years, the traditional paradigm of unitary quantum evolution has been expanded to include measurements, opening new directions in out-of-equilibrium physics. At the core of these advances lie measurement-induced phase transitions (MIPTs), which have emerged as a new class of dynamical critical phenomena characterizing the general behavior of monitored quantum dynamics. When external monitoring intertwines with unitary evolution, many-body quantum correlations change their structure, giving rise to distinct entanglement phases of matter. This discovery has sparked enormous interest in MIPTs, leading to significant advances in open quantum systems, entanglement theory, and more broadly quantum complexity. Despite much progress, a full understanding of monitored many-body dynamics is far from complete, leaving several open questions on the nature of MIPTs, their experimental observability, and the possibilities offered by measurements to enhance control over synthetic quantum matter. These issues persist due to the intrinsic complexity of the problem and the lack of efficient tools to study it, mainly caused by the stochastic character of monitored evolution. This thesis addresses these challenges by expanding the investigation of measurement-induced phenomena in new settings and introducing innovative probes of entanglement and many-body quantum complexity for MIPTs. A core question we investigate is the role of symmetries, non-ergodicity, and especially integrability in measurement-induced criticality, which dramatically affect the non-equilibrium phases. We further explore how these phenomena extend beyond bipartite quantum correlations to multipartite entanglement and quantum non-stabilizerness, highlighting the non-trivial interplay between measurements and complexity notions rooted in quantum information theory. Finally, we focus on the compelling problem of decoherence, modeling how noise spoils entanglement structures. These findings, supported by advanced numerical simulations and theoretical analysis, deepen the current understanding of entanglement, complexity, and integrability in monitored quantum many-body systems, offering new perspectives on their rich behavior. In parallel, we address the experimental problem of dissipation in MIPTs, which is of key relevance for practical implementations. We anticipate the present investigation to foster future research on the nature of monitored dynamical critical phenomena and, more broadly, the applications of measurements in quantum state evolution.

•Sep 15, 2025•0 citations

Quantum Complexity and Stabilizer-State Expansions: From Pairing to Deformation in Fermionic Systems

S. Momme Hengstenberg, Caroline E. P. Robin

Understanding how collective phenomena such as pairing and deformation emerge in interacting fermionic systems, and how the correlations underlying them shape the resources required for their simulation, is a central challenge in many-body physics. We investigate various aspects of quantum complexity in a fermionic system exhibiting two main forms of collectivity, specifically deformation and pairing, through entanglement, non-stabilizerness, and fermionic non-Gaussianity. In the deformed phase, high entanglement coexists with low non-stabilizerness, while the superfluid pairing phase, by contrast, exhibits substantial multipartite entanglement and non-stabilizerness, together with maximal fermionic anti-flatness. In a second step we examine how to describe the phase diagram of the system in a classically scalable and symmetry-preserving way, by combining stabilizer-state expansion and variational basis optimization. A physics-guided decomposition of the Hamiltonian identifies the relevant stabilizer states, which can be constructed from graph states. A single stabilizer state captures the deformed phase to a large extent, and, although the number of stabilizer states required to describe the pairing phase grows exponentially with system size, a symmetry-preserving basis rotation yields a high-fidelity approximation using far fewer states. This rotated state can then serve as a practical starting point for quantum refinement.

quant-pharXiv:2610.31792•Oct 26, 2026•0 citations

Entanglement entropy in quantum many particle systems and their simulation via ansatz states

Thomas Barthel, Ulrich Schollwöck

In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.

•Jan 1, 2009•0 citations

Quantum and classical aspects of complexity in open many-body dynamics

Luca Lumia

The notion of complexity pervades physics, with manifestations rang- ing from classical networks to quantum many-body dynamics. This thesis investigates the complexity of open quantum systems, focus- ing on the interplay between unitary evolution and dissipation, with particular emphasis on measurement processes. Two complementary perspectives are developed: an approach based on quantum resource theories and a data-driven analysis of quantum trajectories. In the first part, quantum complexity is characterized in terms of resources such as entanglement, non-stabilizerness, and non-Gaussianity. These resources capture the non-classical features that enable quantum com- putational advantage. We analyze the generation of resources in mon- itored systems, where the competition between coherent dynamics and measurements gives rise to novel phase transitions. Through semiclassical analyses, tensor networks, and Gaussian and stabilizer techniques, we identify distinct dynamical regimes and clarify the interplay between different resources under repeated quantum mea- surements. The second part focuses on a classical complexity measure inspired by non-parametric machine learning methods, known as the intrinsic dimension. We employ this measure to quantify the com- plexity of ensembles of quantum trajectories, demonstrating that it is sensitive to structural properties such as integrability, fragmenta- tion, and the decoupling of correlation functions, thereby witnessing the complexity of the underlying dynamics. By combining resource- theoretic and unsupervised learning approaches, this thesis provides a wide perspective on the complexity of monitored quantum systems, bridging statistical physics, quantum information, and data science.

•Sep 22, 2025•0 citations

Quantum Computation, Complexity, and Many-Body Physics

Rolando D. Somma

Recently developed quantum algorithms suggest that quantum computers can solve certain problems and perform certain tasks more efficiently than conventional computers. Among other reasons, this is due to the possibility of creating non-classical correlations, or quantum entanglement, which is a phenomena hard or impossible to reproduce by classical-information methods. In this thesis I first investigate the simulation of quantum systems on a quantum computer constructed of two-level quantum elements or qubits. For this purpose, I present algebra mappings that allow one to obtain physical properties and compute correlation functions of fermionic, anyonic, and bosonic systems with such a computer. The results obtained show that the complexity of preparing a quantum state which contains the desired information for the computation is crucial. Second, I present a wide class of quantum computations, which could involve entangled states, that can be simulated with the same efficiency on both types of computers. The notion of generalized quantum entanglement then naturally emerges. This generalization of entanglement is based on the idea that entanglement is an observer-dependent concept, that is, relative to a set of preferred observables.

arXiv:quant-ph/0512209•Dec 22, 2005•19 citations

Non-stabilizerness Entanglement Entropy: a measure of hardness in the classical simulation of quantum many-body systems

Jiale Huang, Xiangjian Qian, Mingpu Qin

Classical and quantum states can be distinguished by entanglement entropy, which can be viewed as a measure of quantum resources. Entanglement entropy also plays a pivotal role in understanding computational complexity in simulating quantum systems. However, stabilizer states formed solely by Clifford gates can be efficiently simulated with the tableau algorithm according to the Gottesman-Knill theorem, although they can host large entanglement entropy. In this work, we introduce the concept of non-stabilizerness entanglement entropy which is basically the minimum residual entanglement entropy for a quantum state by excluding the contribution from Clifford circuits. It can serve as a new practical and better measure of difficulty in the classical simulation of quantum many-body systems. We discuss why it is a better criterion than previously proposed metrics such as Stabilizer Rényi Entropy. We also show numerical results of non-stabilizerness entanglement entropy with concrete quantum many-body models. The concept of non-stabilizerness entanglement entropy expands our understanding of the ``hardness`` in the classical simulation of quantum many-body systems.

arXiv:2409.16895•Sep 25, 2024•0 citations
The search for quantum advantage, understood as the point at which quantum computation surpasses classical computation, has led to the study of the emergence of quantum complexity, and consequently to the investigation of what makes a quantum algorithm hard to simulate on a classical computer. Quantum complexity can be characterized in different ways: the first is given by the entanglement present within the state of the system, which characterizes the non-classical correlations among its subsystems. In general, entanglement provides a measure of how physically complex the system is, but it is not informative about the complexity of simulating the system. This latter kind of complexity is quantified by non-stabilizerness, which is generated, in particular, by all the operations forming a quantum algorithm that do not belong to the Clifford group. Recently, the anti-flatness has been introduced to provide a bridge between the entanglement present in a bipartition of the system and non-stabilizerness, a global property of the system's state. This thesis considers these three complementary quantities in the context of the simulation of a specific quantum algorithm: the quantum sawtooth map, for which an efficient implementation on a quantum computer exists. The dynamics of the sawtooth map is governed by a kicking strength parameter kkk and the period of the kicking TTT; depending on their values, the system exhibits a rich phenomenology, including integrable, diffusive, and localized regimes. Among them, this thesis focuses on dynamical localization, a phenomenon strictly analogous to Anderson localization in disordered systems, in which the evolution of momentum eigenstates suppresses momentum diffusion, causing the wavefunction to remain localized in the momentum basis around a given eigenvalue, over a characteristic localization length ξ\xiξ. The study investigates how dynamical localization affects bipartite entanglement, non-stabilizerness, and anti-flatness, and how these quantities vary as a function of the system size, the parameters characterizing the sawtooth map, and the choice of initial state. The main result is that, as the system is driven out of the localized regime and into the ergodic one, the onset of ergodicity is clearly and consistently marked by all three quantities, each approaching the corresponding theoretical prediction for Haar-random states. This behavior makes entanglement, non-stabilizerness, and anti-flatness reliable and complementary probes for distinguishing the two regimes: entanglement and anti-flatness capture the correlations that arise during the application of the algorithm, including features that depend on the logical position of the qubits within the register due to the encoding, while non-stabilizerness offers a basis-independent, global measure of computational complexity. Furthermore, this work promotes the use of the sawtooth map as a test and benchmarking algorithm for quantum hardware.
Out-of-equilibrium quantum many-body systems stand at the forefront of modern theoretical physics, addressing fundamental questions on thermalization, transport, and universal phenomena. These fields, already well established in condensed matter, statistical physics, quantum optics, and quantum information theory, are progressively gaining even greater relevance with the rapid development of quantum technologies and simulation, which inherently operate in dynamical regimes. In recent years, the traditional paradigm of unitary quantum evolution has been expanded to include measurements, opening new directions in out-of-equilibrium physics. At the core of these advances lie measurement-induced phase transitions (MIPTs), which have emerged as a new class of dynamical critical phenomena characterizing the general behavior of monitored quantum dynamics. When external monitoring intertwines with unitary evolution, many-body quantum correlations change their structure, giving rise to distinct entanglement phases of matter. This discovery has sparked enormous interest in MIPTs, leading to significant advances in open quantum systems, entanglement theory, and more broadly quantum complexity. Despite much progress, a full understanding of monitored many-body dynamics is far from complete, leaving several open questions on the nature of MIPTs, their experimental observability, and the possibilities offered by measurements to enhance control over synthetic quantum matter. These issues persist due to the intrinsic complexity of the problem and the lack of efficient tools to study it, mainly caused by the stochastic character of monitored evolution. This thesis addresses these challenges by expanding the investigation of measurement-induced phenomena in new settings and introducing innovative probes of entanglement and many-body quantum complexity for MIPTs. A core question we investigate is the role of symmetries, non-ergodicity, and especially integrability in measurement-induced criticality, which dramatically affect the non-equilibrium phases. We further explore how these phenomena extend beyond bipartite quantum correlations to multipartite entanglement and quantum non-stabilizerness, highlighting the non-trivial interplay between measurements and complexity notions rooted in quantum information theory. Finally, we focus on the compelling problem of decoherence, modeling how noise spoils entanglement structures. These findings, supported by advanced numerical simulations and theoretical analysis, deepen the current understanding of entanglement, complexity, and integrability in monitored quantum many-body systems, offering new perspectives on their rich behavior. In parallel, we address the experimental problem of dissipation in MIPTs, which is of key relevance for practical implementations. We anticipate the present investigation to foster future research on the nature of monitored dynamical critical phenomena and, more broadly, the applications of measurements in quantum state evolution.
Understanding how collective phenomena such as pairing and deformation emerge in interacting fermionic systems, and how the correlations underlying them shape the resources required for their simulation, is a central challenge in many-body physics. We investigate various aspects of quantum complexity in a fermionic system exhibiting two main forms of collectivity, specifically deformation and pairing, through entanglement, non-stabilizerness, and fermionic non-Gaussianity. In the deformed phase, high entanglement coexists with low non-stabilizerness, while the superfluid pairing phase, by contrast, exhibits substantial multipartite entanglement and non-stabilizerness, together with maximal fermionic anti-flatness. In a second step we examine how to describe the phase diagram of the system in a classically scalable and symmetry-preserving way, by combining stabilizer-state expansion and variational basis optimization. A physics-guided decomposition of the Hamiltonian identifies the relevant stabilizer states, which can be constructed from graph states. A single stabilizer state captures the deformed phase to a large extent, and, although the number of stabilizer states required to describe the pairing phase grows exponentially with system size, a symmetry-preserving basis rotation yields a high-fidelity approximation using far fewer states. This rotated state can then serve as a practical starting point for quantum refinement.
In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.
The notion of complexity pervades physics, with manifestations rang- ing from classical networks to quantum many-body dynamics. This thesis investigates the complexity of open quantum systems, focus- ing on the interplay between unitary evolution and dissipation, with particular emphasis on measurement processes. Two complementary perspectives are developed: an approach based on quantum resource theories and a data-driven analysis of quantum trajectories. In the first part, quantum complexity is characterized in terms of resources such as entanglement, non-stabilizerness, and non-Gaussianity. These resources capture the non-classical features that enable quantum com- putational advantage. We analyze the generation of resources in mon- itored systems, where the competition between coherent dynamics and measurements gives rise to novel phase transitions. Through semiclassical analyses, tensor networks, and Gaussian and stabilizer techniques, we identify distinct dynamical regimes and clarify the interplay between different resources under repeated quantum mea- surements. The second part focuses on a classical complexity measure inspired by non-parametric machine learning methods, known as the intrinsic dimension. We employ this measure to quantify the com- plexity of ensembles of quantum trajectories, demonstrating that it is sensitive to structural properties such as integrability, fragmenta- tion, and the decoupling of correlation functions, thereby witnessing the complexity of the underlying dynamics. By combining resource- theoretic and unsupervised learning approaches, this thesis provides a wide perspective on the complexity of monitored quantum systems, bridging statistical physics, quantum information, and data science.
Recently developed quantum algorithms suggest that quantum computers can solve certain problems and perform certain tasks more efficiently than conventional computers. Among other reasons, this is due to the possibility of creating non-classical correlations, or quantum entanglement, which is a phenomena hard or impossible to reproduce by classical-information methods. In this thesis I first investigate the simulation of quantum systems on a quantum computer constructed of two-level quantum elements or qubits. For this purpose, I present algebra mappings that allow one to obtain physical properties and compute correlation functions of fermionic, anyonic, and bosonic systems with such a computer. The results obtained show that the complexity of preparing a quantum state which contains the desired information for the computation is crucial. Second, I present a wide class of quantum computations, which could involve entangled states, that can be simulated with the same efficiency on both types of computers. The notion of generalized quantum entanglement then naturally emerges. This generalization of entanglement is based on the idea that entanglement is an observer-dependent concept, that is, relative to a set of preferred observables.
Classical and quantum states can be distinguished by entanglement entropy, which can be viewed as a measure of quantum resources. Entanglement entropy also plays a pivotal role in understanding computational complexity in simulating quantum systems. However, stabilizer states formed solely by Clifford gates can be efficiently simulated with the tableau algorithm according to the Gottesman-Knill theorem, although they can host large entanglement entropy. In this work, we introduce the concept of non-stabilizerness entanglement entropy which is basically the minimum residual entanglement entropy for a quantum state by excluding the contribution from Clifford circuits. It can serve as a new practical and better measure of difficulty in the classical simulation of quantum many-body systems. We discuss why it is a better criterion than previously proposed metrics such as Stabilizer Rényi Entropy. We also show numerical results of non-stabilizerness entanglement entropy with concrete quantum many-body models. The concept of non-stabilizerness entanglement entropy expands our understanding of the ``hardness`` in the classical simulation of quantum many-body systems.