Isospectral twirling and quantum chaos

We show that the most important measures of quantum chaos like 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 be always 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, by exploiting random matrix theory, these measures of quantum chaos clearly distinguish the finite time profiles 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). On the other hand, we show that the asymptotic values do depend on the eigenvectors of the Hamiltonian. We see that the isospectral twirling of Hamiltonians with eigenvectors 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. Finally, we show a crossover in the OTOC behavior between a class of integrable models and quantum chaos.Comment: Updated version with several new result

Isospectral twirling and quantum chaos

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 quantum circuits by measurements

Random Clifford circuits doped with non Clifford gates exhibit transitions to universal entanglement spectrum statistics[1] and quantum chaotic behavior. In [2] we proved that the injection of $O(n)$ non Clifford gates into a $n$-qubit Clifford circuit drives the transition towards the universal value of the purity fluctuations. In this paper, we show that doping a Clifford circuit with $O(n)$ single qubit non Clifford measurements is both necessary and sufficient to drive the transition to universal fluctuations of the purity

Transitions in entanglement complexity in random quantum circuits by measurements

Random Clifford circuits doped with non Clifford gates exhibit transitions to universal entanglement spectrum statistics [1] and quantum chaotic behavior. In [2] we proved that the injection of Ω(n) non Clifford gates into a n-qubit Clifford circuit drives the transition towards the universal value of the purity fluctuations. In this paper, we show that doping a Clifford circuit with Ω(n) single qubit non Clifford measurements is both necessary and sufficient to drive the transition to universal fluctuations of the purity

Learning t-doped stabilizer states

In this paper, we present a learning algorithm aimed at learning states obtained from computational basis states by Clifford circuits doped with a finite number t of non-Clifford gates. To tackle this problem, we introduce a novel algebraic framework for t-doped stabilizer states by utilizing tools from stabilizer entropy. Leveraging this new structure, we develop an algorithm that uses sampling from the distribution obtained by squaring expectation values of Pauli operators that can be obtained by Bell sampling on the state and its conjugate in the computational basis. The algorithm requires resources of complexity O(\exp(t)\poly(n)) and exhibits an exponentially small probability of failure.Comment: L.L. and S.O. contributed equally to this wor

Quantum chaos is quantum

It is well known that a quantum circuit on N qubits composed of Clifford gates with the addition of k non Clifford gates can be simulated on a classical computer by an algorithm scaling as poly(N)exp (k)[1]. We show that, for a quantum circuit to simulate quantum chaotic behavior, it is both necessary and sufficient that k = Θ(N). This result implies the impossibility of simulating quantum chaos on a classical computer

Optimizing the relaxivity of MRI probes at high magnetic field strengths with binuclear GdIIIComplexes

The key criteria to optimize the relaxivity of a Gd(III) contrast agent at high fields (defined as the region 65 1.5 T) can be summarized as follows: (i) the occurrence of a rotational correlation time \u3c4R in the range of ca. 0.2\u20130.5 ns; (ii) the rate of water exchange is not critical, but a \u3c4M < 100 ns is preferred; (iii) a relevant contribution from water molecules in the second sphere of hydration. In addition, the use of macrocycle-based systems ensures the formation of thermodynamically and kinetically stable Gd(III) complexes. Binuclear Gd(III) complexes could potentially meet these requirements. Their efficiency depends primarily on the degree of flexibility of the linker connecting the two monomeric units, the absence of local motions and the presence of contribution from the second sphere water molecules. With the aim to maximize relaxivity (per Gd) over a wide range of magnetic field strengths, two binuclear Gd(III) chelates derived from the well-known macrocyclic systems DOTA-monopropionamide and HPDO3A (Gd2L1 and Gd2L2, respectively) were synthesized through a multistep synthesis. Chemical Exchange Saturation Transfer (CEST) experiments carried out on Eu2L2 at different pH showed the occurrence of a CEST effect at acidic pH that disappears at neutral pH, associated with the deprotonation of the hydroxyl groups. Then, a complete 1H and 17O NMR relaxometric study was carried out in order to evaluate the parameters that govern the relaxivity associated with these complexes. The relaxivities of Gd2L1 and Gd2L2 (20 MHz, 298 K) are 8.7 and 9.5 mM 121 s 121, respectively, +77% and +106% higher than the relaxivity values of the corresponding mononuclear GdDOTAMAP-En and GdHPDO3A complexes. A significant contribution of second sphere water molecules was accounted for the strong relaxivity enhancement of Gd2L2. MR phantom images of the dinuclear complexes compared to GdHPDO3A, recorded at 7 T, confirmed the superiority of Gd2L2. Finally, ab initio (DFT) calculations were performed to obtain information about the solution structure of the dinuclear complexes

Stability of topological purity under random local unitaries

In this work, we provide an analytical proof of the robustness of topological entanglement under a model of random local perturbations. We define a notion of average topological subsystem purity and show that, in the context of quantum double models, this quantity does detect topological order and is robust under the action of a random quantum circuit of shallow depth.Comment: Added new reference