On the stability of the spectrum of a chaotic quantum system

Gong-ou Xu, Quanlin Jie

Surajit Bera, Marco Schirò

We study the non-stabilizerness or quantum magic of the Sachdev-Ye-Kitaev ( \mathrm{SYK} S Y K ) model, a prototype example of maximally chaotic quantum matter. We show that the Majorana spectrum of its ground state, encoding the spreading of the state in the Majorana basis, displays a Gaussian distribution as expected for chaotic quantum many-body systems. We compare our results with the case of the \mathrm{SYK_2} S Y K 2 model, describing non-chaotic random free fermions, and show that the Majorana spectrum is qualitatively different in the two cases, featuring an exponential Laplace distribution for the \mathrm{SYK_2} S Y K 2 model rather than a Gaussian. From the spectrum we extract the Stabilizer Renyi Entropy (SRE) and show that for both models it displays a linear scaling with system size, with a prefactor that is larger for the SYK model, which has therefore higher magic. Finally, we discuss the spreading of quantun magic under unitary dynamics, as described by the evolution of the Majorana spectrum and the Stabilizer Renyi Entropy starting from a stabilizer state. We show that the SRE for the \mathrm{SYK_2} S Y K 2 model equilibrates rapidly, but that in the steady-state the interacting chaotic SYK model has more magic than the simple \mathrm{SYK_2} S Y K 2 . Our results suggest that the Majorana spectrum is qualitatively distinct in chaotic and non-chaotic many-body systems.

Stefano Cusumano, Gianluca Esposito, Alioscia Hamma

Chaotic behavior of quantum systems can be characterized by the adherence of the expectation values of given probes to moments of the Haar distribution. In this work, we analyze the behavior of several probes of chaos using a technique known as Isospectral Twirling [1]. This consists in fixing the spectrum of the Hamiltonian and picking its eigenvectors at random. Here, we study the transition from stabilizer bases to random bases according to the Haar measure by T-doped random quantum circuits. We then compute the average value of the probes over ensembles of random spectra from Random Matrix Theory, the Gaussian Diagonal Ensemble and the Gaussian Unitary Ensemble, associated with non-chaotic and chaotic behavior respectively. We also study the behavior of such probes over the Toric Code Hamiltonian.

T. S. Monteiro

The influence of quantum chaos on small atomic systems is reviewed. It now seems clear that chaos in the usually understood sense (e.g. exponential sensitivity to perturbations) is not found in isolated quantum systems. However, there are phenomena which appear only when the corresponding classical system is chaotic. The stability of the classical dynamics, in other words whether it is regular or chaotic, has a profound effect on the character of the corresponding quantum spectrum and wavefunctions.

Holger Frahm, H.-J. Mikeska

Soumyadeep Sarma, Tobias Haug, John Preskill +1 more

Thermalization conventionally describes local properties of isolated many-body systems at equilibrium. Magic, or nonstabilizerness $\unicode{x2013}$ the resource enabling universal quantum computation $\unicode{x2013}$ is by contrast encoded in the global structure of the many-body wavefunction. We show that, despite its global nature, the magic of equilibrium pure states of chaotic many-body systems, including late-time evolved states and energy eigenstates, is universal: it is captured by the thermal Scrooge ensemble, the minimally informative ensemble of pure states consistent with the Gibbs state at the same effective temperature. Therefore, for systems with no conserved quantities other than the total energy, equilibrium magic is a function of temperature alone, independent of the initial state and other microscopic features of the equilibrium state. This yields concrete universal predictions for the stabilizer Rényi entropies (SREs). At infinite temperature, the SRE is set by Haar-like fluctuations of the Pauli spectrum, while at finite temperature energy conservation induces a volume-law thermodynamic correction controlled by the thermal Pauli spectrum. We support these predictions with analytical arguments and extensive numerical simulations. We further show that chaotic many-body systems at high temperatures possess long-range magic and entanglement that cannot be removed by finite-depth local quantum circuits. Our results establish magic as a thermodynamic property of chaotic many-body systems and suggest that Scrooge ensembles may provide a unified framework for quantum many-body resources.