Abstract
We show that the most important measures of quantum chaos, such as frame potentials, scrambling, Loschmidt echo and out-of-time-order correlators (OTOCs), can be described by the unified framework of the isospectral twirling, namely the Haar average of a k-fold unitary channel. We show that such measures can then always be cast in the form of an expectation value of the isospectral twirling. In literature, quantum chaos is investigated sometimes through the spectrum and some other times through the eigenvectors of the Hamiltonian generating the dynamics. We show that thanks to this technique, we can interpolate smoothly between integrable Hamiltonians and quantum chaotic Hamiltonians. The isospectral twirling of Hamiltonians with eigenvector stabilizer states does not possess chaotic features, unlike those Hamiltonians whose eigenvectors are taken from the Haar measure. As an example, OTOCs obtained with Clifford resources decay to higher values compared with universal resources. By doping Hamiltonians with non-Clifford resources, we show a crossover in the OTOC behavior between a class of integrable models and quantum chaos. Moreover, exploiting random matrix theory, we show that these measures of quantum chaos clearly distinguish the finite time behavior of probes to quantum chaos corresponding to chaotic spectra given by the Gaussian Unitary Ensemble (GUE) from the integrable spectra given by Poisson distribution and the Gaussian Diagonal Ensemble (GDE).
Highlights
Pseudorandomness is characterized by the frame potential, which describes the adherence to moments of the Haar measure [46,47]; the complexity of entanglement is characterized by the adherence to the random matrix theory distribution of the gaps in the entanglement spectrum [22,23]
One could take the behavior of order correlation functions (OTOCs), entanglement, frame potentials and other probes under a time evolution induced by a chaotic Hamiltonian, e.g., a random Hamiltonian from Gaussian Unitary Ensemble (GUE), and define it as the characteristic behavior of these quantities for quantum chaos [54,55,56]
Though we agree with the heuristics of this approach, it would be important to compare the time behavior of these probes in systems that are not characterized by a spectrum given by a random matrix, or on the other hand, by Hamiltonians whose eigenvectors are not a random basis according to the Haar measure, e.g., Hamiltonians with eigenvectors that, possessing high entanglement, do not contain any magic, that is, they are stabilizer states
Summary
The onset of chaotic dynamics is at the center of many important phenomena in quantum many-body systems, from thermalization in a closed system [1,2,3,4,5,6,7,8,9,10,11] to scrambling of information in quantum channels [12,13,14,15,16,17], black hole dynamics [18,19,20,21], entanglement complexity [22,23], to pseudorandomness in quantum circuits [24,25,26,27,28], and the complexity of quantum evolutions [29,30,31,32]. The main results of this paper are: (i) the isospectral twirling unifies all the fundamental probes P used to describe quantum chaos in the form of PO G = tr[T OR (2k)], where Tis a rescaled permutation operator, O characterizes the probe, · G is the Haar average and (ii) by considering the isospectral twirling associated to a k−doped Clifford group C(d), we show that the asymptotic temporal behavior of the OTOCs interpolates between a class of integrable models and quantum chaos and does not depend on the specific spectrum of the Hamiltonian; (iii) by computing the isospectral twirling for the spectra corresponding to the chaotic Hamiltonians in GUE and integrable ones—Poisson, Gaussian Diagonal Ensemble (GDE)—the isospectral twirling can distinguish chaotic from non-chaotic behavior in the temporal profile of the probes, though all the spectra lead to the same asymptotic behavior—a sign of the fact that chaos is not solely determined by the spectrum of the Hamiltonian and by its eigenvectors
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