Tuning quantum magic of pure quantum chaotic states with a gravity dual

Abstract: Quantum magic is a fundamental resource that quantifies to what extent quantum states can be efficiently simulated on a classical computer. We study it for states constructed from the Sachdev-Ye-Kitaev (SYK) Hamiltonian with $N$ Majoranas by the fermionic anti-flatness (FAF). We show analytically that, in the large $N$ limit, the quantum magic of pure Kourkoulou-Maldacena (KM) states, dual to a quantum black hole with an end-of-world particle behind the horizon, is linear in $N$ with a slope, depending on the black hole temperature, that can be tuned between zero and $1/2$. By contrast, the FAF of Gaussian states evolved in real time with the SYK Hamitonian approaches $\approx N/2$ exponentially at a rate given by a multiple of the leading Ruelle-Pollicot resonance. Subleading corrections in $N$ for SYK energy eigenstates, computed numerically for $N \leq 54$ by combining Krylov subspace with GPU acceleration techniques, decay exponentially with $N$, but power-law if the SYK couplings are sparsified, and are order of magnitude larger for states close to the ground state, a region with an established gravity analogue. Our results offer new insights about the relation between quantum information, quantum chaos and low-dimension quantum gravity.

• The paper shows that quantum magic, measured by fermionic anti-flatness, scales linearly with system size and is tunable via the inverse temperature parameter.
• It employs both analytic derivations and large-scale numerical simulations (up to N=54) to capture the dynamics and saturation behavior of quantum magic in SYK states.
• The findings offer new insights into quantum gravity and holography, providing a diagnostic for chaos, scrambling, and computational complexity in many-body systems.

Tuning Quantum Magic in SYK States via Gravity Duals

Introduction

The quantification of "quantum magic"—that is, the non-stabilizerness or intrinsic non-Gaussian character of quantum states—is central to understanding the classical simulability and computational complexity of quantum many-body systems. In this work ["Tuning quantum magic of pure quantum chaotic states with a gravity dual", (2607.01930)], the authors present a comprehensive analytic and numerical investigation of quantum magic in pure states constructed from the Sachdev-Ye-Kitaev (SYK) model. They leverage the fermionic anti-flatness (FAF) as a practical and sensitive measure of magic. The study not only clarifies how quantum chaoticity generates magic in systems with known holographic duals, but also systematically explores the tunability of magic in pure states corresponding to gravity dual configurations, such as quantum black holes in nearly-AdS$_2$ backgrounds.

Fermionic Anti-Flatness and Quantum Magic

The FAF is introduced as a robust, efficiently computable magic monotone for systems of $N$ Majorana fermions, distinguishing Gaussian (classically simulable) states from highly complex, Haar-random states. Specifically, for a state $|\psi\rangle$, the FAF is defined in terms of the second moment of the commutator of Majorana operators and vanishes for Gaussian states, while for Haar-random states it scales as $N/2$. This scaling is directly connected to the logarithm of the Hilbert space dimension, establishing FAF as an extensive measure in chaotic regimes.

Analytical Results for Pure KM States and Time Evolution

A central result is the linear scaling of FAF with system size $N$ for Kourkoulou-Maldacena (KM) pure states, which interpolate from Gaussian product states to strongly correlated, low-energy states upon Euclidean evolution under the SYK Hamiltonian. In the large $N$ limit, the quantum magic of these states is shown to be proportional to $N$ with a slope that can be continuously tuned via the inverse temperature parameter $\beta$. Specifically, the slope varies from zero (Gaussian, no magic) to $1/2$ (maximal magic), revealing a direct and smooth "knob" for controlling magic in this many-body setting. This tunability is mirrored by the dual gravity interpretation, where varying $\beta$ corresponds to adjusting the temperature of the associated black hole configuration.

The time evolution of FAF is derived analytically for Gaussian initial states evolved under the SYK Hamiltonian. The approach to saturation at $N$0 occurs exponentially and, importantly, with a timescale independent of $N$1, governed instead by the leading Ruelle-Pollicott resonance of the underlying dynamics. This finding is in contrast to other many-body systems, where finite-size effects typically induce an $N$2-dependent relaxation behavior.

Numerical Verification and Subleading Corrections

Analytic predictions are strongly supported by large-scale numerics. Krylov subspace methods with GPU acceleration enable the exact computation of FAF for $N$3 up to 54. For SYK energy eigenstates, the $N$4 corrections to the leading $N$5 scaling decay exponentially in $N$6 for dense couplings (SYK standard ensemble), but transition to a power law in sparse variants. Notably, states near the ground state exhibit subleading corrections that are orders of magnitude larger than for high-energy eigenstates, reflecting known features of the gravity dual in the low-temperature regime governed by Schwarzian dynamics.

In time-dependent settings, numerics display excellent agreement with the analytic solution of the Schwinger-Dyson equations after simple finite-size extrapolation. The time to reach magic saturation is again found to be $N$7-independent. On Heisenberg timescales, FAF displays the characteristic ramp-plateau structure predicted by random matrix theory, establishing a correspondence with the spectral form factor and random matrix theory predictions for quantum chaos.

Implications for Quantum Gravity and Holography

The ability to tune quantum magic by varying parameters in KM states has direct implications for holographic duality. In the large $N$8 (low temperature) regime, these states are dual to nearly-AdS$N$9 black holes with end-of-world particles behind the horizon, providing a direct probe of magic in quantum gravity configurations. The behavior of FAF in these pure and eigenstates thus provides a diagnostic for the quantum information complexity of states with gravity duals.

The identification of exponential versus power-law subleading corrections depending on sparsity and spectral position has implications for the nature of chaos and scrambling in strongly interacting systems and may guide the classification of holographic versus non-holographic regimes in random all-to-all models.

Future Prospects

This work opens several avenues. The observed $|\psi\rangle$0-independent timescale for magic saturation suggests a possible universality class for certain quantum chaotic systems with gravity duals and motivates similar studies in other models with established (or conjectured) holographic correspondences. The distinct subleading scaling near the spectral edges versus bulk also invites detailed studies of quantum error correction capabilities and complexity growth in these regimes. Moreover, the analytic tractability of the FAF in the large-$|\psi\rangle$1 SYK limit sets the stage for analytical studies in more general random circuit and integrable models.

Conclusion

The systematic investigation of quantum magic via FAF in chaotic SYK states reveals a direct correspondence between the tunability of quantum informational complexity and the structure of the gravity dual. Analytical and numerical evidence confirm a linear scaling of magic with a temperature-tunable slope and an $|\psi\rangle$2-independent approach to saturation in time evolution. The interplay between chaos, magic, and holography elucidated in this study is likely to inform both foundational questions in quantum gravity and practical research into quantum computational complexity in strongly interacting systems.