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arXiv:2609.09545

Sub-extensive non-stabilizerness in the Dyck-Fredkin spin chain

Yasunori Lee

quant-phcond-mat.stat-mechSubmitted Sep 9, 2026• Announced Sep 10, 2026• 0 citations
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Abstract

The stabilizer Rényi entropy is a quantitative measure of non-stabilizerness, or magic, and has typically been found to scale extensively with system size $N$ (i.e., $Θ(N)$) for a variety of many-body quantum states. In this note, we study the stabilizer Rényi entropy of the ground state of the spin-$\frac{1}{2}$ Dyck-Fredkin chain and its $t$-deformation, a local frustration-free model with unusual spectral-gap scaling. Exploiting the combinatorial structure, we carry out numerically exact finite-size calculations, which indicate asymptotic behavior depending on $t$: $Θ(N)$ for $t<1$, $Θ(\log N)$ at $t=1$, and $Θ(1)$ for $t>1$. The scaling at $t=1$ could be another manifestation of the unconventional criticality of the model, while the contrast with the behavior of the entanglement entropy suggests that non-stabilizerness might provide a new window into quantum many-body systems.

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The stabilizer Rényi entropy is a quantitative measure of non-stabilizerness, or magic, and has typically been found to scale extensively with system size NNN (i.e., Θ(N)Θ(N)Θ(N)) for a variety of many-body quantum states. In this note, we study the stabilizer Rényi entropy of the ground state of the spin-12\frac{1}{2}21​ Dyck-Fredkin chain and its ttt-deformation, a local frustration-free model with unusual spectral-gap scaling. Exploiting the combinatorial structure, we carry out numerically exact finite-size calculations, which indicate asymptotic behavior depending on ttt: Θ(N)Θ(N)Θ(N) for t<1t<1t<1, Θ(log⁡N)Θ(\log N)Θ(logN) at t=1t=1t=1, and Θ(1)Θ(1)Θ(1) for t>1t>1t>1. The scaling at t=1t=1t=1 could be another manifestation of the unconventional criticality of the model, while the contrast with the behavior of the entanglement entropy suggests that non-stabilizerness might provide a new window into quantum many-body systems.
We advance the characterization of complexity in quantum many-body systems by examining W W -states embedded in a spin chain. Such states show an amount of non-stabilizerness or “magic”, measured as the Stabilizer Rényi Entropy, that grows logarithmically with the number of qubits/spins. We focus on systems whose Hamiltonian admits a classical point with extensive degeneracy. Near these points, a Clifford circuit can convert the ground state into a W W -state, while in the rest of the phase to which the classical point belongs, it is dressed with local quantum correlations. Topological frustrated quantum spin-chains host phases with the desired phenomenology, and we show that their ground state’s Stabilizer Rényi Entropy is the sum of that of the W W -states plus an extensive local contribution. Our work reveals that W W -states/frustrated ground states display a non-local degree of complexity that can be harvested as a quantum resource and has no counterpart in GHZ states/non-frustrated systems.
Stabilizer Rényi entropy has emerged as a probe of nonstabilizerness in quantum many-body systems, but its scaling structure beyond critical points remains poorly understood compared with entanglement entropy. Recent field-theory approaches indicate that stabilizer entropy contains universal critical data and boundary-sensitive terms, raising the question of how these structures extend into massive and crossover regimes. We address this problem for a broad class of finite-range spin chains at Rényi index one-half. We derive exact finite-size formulas for both full periodic chains and finite intervals of the infinite chain, making the universal crossover from critical to noncritical behavior analytically accessible. In periodic geometry, the entropy obeys a volume law away from criticality and exhibits a universal finite-size crossover controlled by the competition between system size and correlation length. We also show that the large-scale SRE density develops a cusp across the field-tuned critical line, while the XX endpoint is governed by a distinct scaling regime associated with the saturation point. In the subsystem geometry, the interval entropy separates bulk critical behavior from boundary contributions generated by the way the finite region cuts the infinite chain. The crossover from critical to massive behavior is then encoded in boundary constants and universal functions controlled by the correlation length. Through exact stabilizer-entropy correspondences, the scaling theory extends to internal XY reductions, Finite-range spin chains, and Cluster--Ising representatives. Our results provide an exact lattice benchmark for the emerging QFT description of stabilizer entropy beyond isolated conformal points.
Non-stabilizerness – commonly known as magic – measures the extent to which a quantum state deviates from stabilizer states and is a fundamental resource for achieving universal quantum computation. In this work, we investigate the behavior of non-stabilizerness around criticality in quantum spin chains. To quantify non-stabilizerness, we employ a monotone called mana, based on the negativity of the discrete Wigner function. This measure captures non-stabilizerness for both pure and mixed states. We introduce Rényi generalizations of mana, which are also measures of non-stabilizerness for pure states, and utilize it to compute mana in large quantum systems. We consider the three-state Potts model and its non-integrable extension and we provide strong evidence that the mutual mana exhibits universal logarithmic scaling with distance in conformal field theory, as is the case for entanglement.
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.
Ground states of quantum many-body systems are both entangled and possess a kind of quantum complexity, as their preparation requires universal resources that go beyond the Clifford group and stabilizer states. These resources---sometimes described as magic---are also the crucial ingredient for quantum advantage. We study the behavior of the stabilizer R\'enyi entropy in the integrable transverse field Ising spin chain. We show that the locality of interactions results in a localized stabilizer R\'enyi entropy in the gapped phase, thus making this quantity computable in terms of local quantities in the gapped phase, while measurements involving LLL spins are necessary at the critical point to obtain an error scaling with $O({L}^{\ensuremath{-}1})$.
Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an “area law” which means that the entanglement entropy is proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a logarithm over two decades. Recent discovery of Motzkin and Fredkin spin chain models is striking, since these models provide significant violation of the entanglement beyond the belief, growing as a square root of the volume in spite of local interactions. In this paper, we first analytically compute the Rényi entropy of the Motzkin and Fredkin models by careful treatment of asymptotic analysis. The Rényi entropy is an important quantity, since the whole spectrum of an entangled subsystem is reconstructed once the Rényi entropy is known as a function of its parameter. We find nonanalytic behavior of the Rényi entropy with respect to the parameter, which is a novel phase transition never seen in any other spin chain studied so far. Interestingly, similar behavior is seen in the Rényi entropy of Rokhsar–Kivelson states in two dimensions.