Isospectral Twirling and Quantum Chaos

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).

Transitions in Entanglement Complexity in Random Circuits

“…In quantum mechanics, the characterization of chaos in quantum systems is an ongoing enterprise that is central in the field of quantum information and quantum many-body theory [5][6][7][8][9][10]. For example, studies in random matrix theory [5,[11][12][13][14][15][16][17] have provided analytical means of identifying chaos in quantum Hamiltonians and systems of entangled qubits. Indeed, it is well known that the emergence of chaos in quantum systems goes hand in hand with entanglement [9,[18][19][20][21][22][23][24], a fact that has brought the problem of creating sufficiently complex entanglement to the forefront of quantum computing [25,26] as physicists endeavour to recreate the information scrambling properties of nature's most chaotic systems such as black holes [10,13,[26][27][28][29][30][31][32].…”

Entanglement complexity of the Rokhsar-Kivelson-sign wavefunctions

“…Nonetheless, the complexity of entanglement is generally thought of as stemming from the conjunction of a volume-law scaling for entanglement entropies, and a finite magic. Indeed these properties, along with a universal entanglement spectrum statistics, have been shown to be at the root of the onset of quantum chaos [43,[45][46][47]; the hardness of disentangling algorithms [32,33,44]; the universal behavior of out-of-time-order correlation functions (OTOCs); [34,48,49]; and the hardness of simulatability of quantum many-body systems [37,50].…”

Nonstabilizerness determining the hardness of direct fidelity estimation

“…We show that the sample complexity, i.e. the number of uses of a given U, is quantified by multi-points out of time order correlators (OTOCs) associated with the target unitary operator U. OTOCs are conventionally employed to probe quantum chaos: a quantum evolution is commonly considered to be chaotic in terms of attaining the Haar value for general OTOCs [97,[100][101][102], that is, the value that would be reached by a random unitary operator. We claim: the closer these correlators are to the Haar value, the more chaotic is the evolution [100] and the more inefficient is the quantum verification.…”