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arXiv:2503.07468 · Physical Review Letters

Nonstabilizerness Dynamics in Many-Body Localized Systems

Pedro R. Nicácio Falcão, Piotr Sierant, Jakub Zakrzewski, Emanuele Tirrito

Published Nov 19, 2025• 14 citations
arXivHTMLPDF

Abstract

Nonstabilizerness, also known as "magic," quantifies the deviation of quantum states from stabilizer states, capturing the complexity necessary for quantum computational advantage. In this Letter, we investigate the dynamics of nonstabilizerness in disordered many-body localized (MBL) systems using the stabilizer Rényi entropy (SRE). Leveraging a phenomenological description based on the ℓ-bit model, we analytically and numerically demonstrate that interactions profoundly influence nonstabilizerness spreading, inducing a power-law growth of SRE that markedly contrasts with the rapid saturation observed in ergodic systems. We validate our theoretical predictions through numerical simulations of the disordered transverse-field Ising model, showing excellent agreement across various disorder strengths, system sizes, and initial states. Additionally, we uncover a universal relationship between SRE and entanglement entropy, revealing their common scaling in the MBL regime independent of disorder strength and system size. Our results offer critical insights into the interplay of disorder, interactions, and complexity in quantum many-body systems.

DOI
10.1103/xfp5-hhs4

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Nonstabilizerness, also known as "magic," quantifies the deviation of quantum states from stabilizer states, capturing the complexity necessary for quantum computational advantage. In this Letter, we investigate the dynamics of nonstabilizerness in disordered many-body localized (MBL) systems using the stabilizer Rényi entropy (SRE). Leveraging a phenomenological description based on the ℓ-bit model, we analytically and numerically demonstrate that interactions profoundly influence nonstabilizerness spreading, inducing a power-law growth of SRE that markedly contrasts with the rapid saturation observed in ergodic systems. We validate our theoretical predictions through numerical simulations of the disordered transverse-field Ising model, showing excellent agreement across various disorder strengths, system sizes, and initial states. Additionally, we uncover a universal relationship between SRE and entanglement entropy, revealing their common scaling in the MBL regime independent of disorder strength and system size. Our results offer critical insights into the interplay of disorder, interactions, and complexity in quantum many-body systems.
Disorder-free quantum many-body localization can strongly suppress transport while still enabling the dynamical buildup of computationally costly non-Clifford resources. In a tilted transverse-field Ising chain realizing disorder-free Stark many-body localization, we use the stabilizer Rényi entropy to quantify quantum magic (nonstabilizerness) and find that it remains finite and grows anomalously slowly over extended time windows before saturating to a size-dependent plateau deep in the strong-tilt regime, with pronounced initial-state selectivity. Upon increasing the Stark gradient, the long-time magic and half-chain entanglement exhibit consistent finite-size crossing behavior, indicating a crossover from ergodic dynamics to constrained localization. These results establish stabilizer-based magic as a practical complexity diagnostic of disorder-free ergodicity breaking and constrained dynamics, and provide an experimentally accessible route to benchmarking and designing near-term quantum simulators.
Magic, capturing the deviation of a quantum state from the stabilizer formalism, is a key resource underpinning the quantum advantage. The recently introduced stabilizer Rényi entropy (SRE) offers a tractable measure of magic, avoiding the complexity of conventional methods. We study SRE in translation-invariant matrix product states (MPS), deriving exact expressions for representative states and introducing a numerically stable algorithm, named bond-DMRG, to compute the SRE density in infinite systems. Applying this method, we obtain high-precision SRE densities for the ground state of the one-dimensional Ising model. We also analyze non-local SRE density, showing it is bounded by a universal function of entanglement entropy, and further prove that two-site mutual SRE vanishes asymptotically in injective MPS. Our work not only introduces a powerful method for extracting the SRE density in quantum many-body systems, but also numerically reveals a fundamental connection between magic and entanglement, thereby paving the way for deeper theoretical investigations into their interplay.
Understanding universal aspects of many-body systems is one of the central themes in modern physics. Recently, the stabilizer Rényi entropy (SRE) has emerged as a computationally tractable measure of nonstabilizerness, a crucial resource for fault-tolerant universal quantum computation. While numerical results suggested that the SRE in critical states can exhibit universal behavior, its comprehensive theoretical understanding has remained elusive. In this work, we develop a field-theoretical framework for the SRE in a ( 1 + 1 )-dimensional many-body system and elucidate its universal aspects by using boundary conformal field theory. We demonstrate that the SRE is equivalent to a participation entropy in the Bell basis of a doubled Hilbert space, which can be calculated from the partition function of a replicated field theory with the interlayer line defect created by the Bell-state measurements. This identification allows us to characterize the universal contributions to the SRE on the basis of the data of conformal boundary conditions imposed on the replicated theory. We find that the SRE of the entire system contains a universal size-independent term determined by the noninteger ground-state degeneracy known as the g factor. In contrast, we show that the mutual SRE exhibits the logarithmic scaling with a universal coefficient given by the scaling dimension of a boundary-condition-changing operator, which elucidates the origin of universality previously observed in numerical results. As a concrete demonstration, we present a detailed analysis of the Ising criticality, where we analytically derive the universal quantities at arbitrary Rényi indices and numerically validate them with high accuracy by employing tensor-network methods. These results establish a field-theoretical approach to understanding the universal features of nonstabilizerness in quantum many-body systems.
While nonstabilizerness (''magic'') is a key resource for universal quantum computation, its behavior in many-body quantum systems, especially near criticality, remains poorly understood. We develop a spectral transfer-matrix framework for the stabilizer Rényi entropy (SRE) in infinite matrix product states, showing that its spectrum contains universal subleading information. In particular, we identify an SRE correlation length -- distinct from the standard correlation length -- which diverges at continuous phase transitions and governs the spatial response of the SRE to local perturbations. We derive exact SRE expressions for the bond dimension χ=2χ=2χ=2 MPS ''skeleton'' of the cluster-Ising model, and we numerically probe its universal scaling along the Z2\mathbb{Z}_2Z2​ critical lines in the phase diagram. These results demonstrate that nonstabilizerness captures signatures of criticality and local perturbations, providing a new lens on the interplay between computational resources and emergent phenomena in quantum many-body systems.
Quantum spin liquids (QSLs) are long-range entangled phases of matter and a natural setting for exploring how quantum correlations generate quantum complexity. Motivated by the emergence of magic, or nonstabilizerness, as a diagnostic of many-body complexity beyond entanglement, we study magic of QSLs by computing the stabilizer Rényi entropy (SRE) of the Kitaev honeycomb model. We derive a correspondence between Pauli strings and products of itinerant Majorana operators in the model's free-fermion description; this enables the sampling of SRE using an optimized algorithm for Gaussian states in systems with thousands of spins. Our results show that magic is largest in the gapless phase, where subleading volume-law corrections indicate nonlocal contributions, while in the gapped phases, SRE decreases in quantitative agreement with a perturbative expansion that we develop in the anisotropic limit. At the topological phase transition, SRE exhibits universal scaling dictated by critical exponents.
Quantum state complexity metrics, such as anticoncentration and nonstabilizerness, offer key insights into many-body physics, information scrambling, and quantum computing. Anticoncentration and equilibration of magic resources under dynamics of random quantum circuits occur at times scaling logarithmically with system size. Here, we examine these phenomena in one-dimensional ergodic Floquet models and thermalizing Hamiltonian systems. Using participation and stabilizer entropies to probe anticoncentration and magic resources, we reveal significant differences between the two settings. Floquet systems align with random circuit predictions, exhibiting anticoncentration and magic saturation at timescales logarithmic in system size. In contrast, Hamiltonian dynamics deviate from the random circuit predictions and require times scaling approximately linearly with system size to achieve saturation of participation and stabilizer entropies, which remain smaller than that of the typical quantum states even in the long-time limit. Our findings establish the phenomenology of participation and stabilizer entropy growth in generic many-body systems and emphasize the role of conservation laws in constraining anticoncentration and magic dynamics.