We generalize the classical shadow tomography scheme to a broad class of finite-depth or finitetime local unitary ensembles, known as locally scrambled quantum dynamics, where the unitary ensemble is invariant under local basis transformations. In this case, the reconstruction map for the classical shadow tomography depends only on the average entanglement feature of classical snapshots. We provide an unbiased estimator of the quantum state as a linear combination of reduced classical snapshots in all subsystems, where the combination coefficients are solely determined by the entanglement feature. We also bound the number of experimental measurements required for the tomography scheme, so-called sample complexity, by formulating the operator shadow norm in the entanglement feature formalism. We numerically demonstrate our approach for finite-depth local unitary circuits and finite-time local-Hamiltonian generated evolutions. The shallow-circuit measurement can achieve a lower tomography complexity compared to the existing method based on Pauli or Clifford measurements. Our approach is also applicable to approximately locally scrambled unitary ensembles with a controllable bias that vanishes quickly. Surprisingly, we find a single instance of time-dependent local Hamiltonian evolution is sufficient to perform an approximate tomography as we numerically demonstrate it using a paradigmatic spin chain Hamiltonian modeled after trapped ion or Rydberg atom quantum simulators. Our approach significantly broadens the application of classical shadow tomography on near-term quantum devices.

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The neural ordinary differential equation (neural ODE) is a novel machine learning architecture whose weights are smooth functions of the continuous depth. We apply the neural ODE to holographic QCD by regarding the weight functions as a bulk metric, and train the machine with lattice QCD data of chiral condensate at finite temperature. The machine finds consistent bulk geometry at various values of temperature and discovers the emergent black hole horizon in the holographic bulk automatically. The holographic Wilson loops calculated with the emergent machine-learned bulk spacetime have consistent temperature dependence of confinement and Debye-screening behavior. In machine learning models with physically interpretable weights, the neural ODE frees us from discretization artifact leading to difficult ingenuity of hyperparameters, and improves numerical accuracy to make the model more trustworthy.

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Spontaneous time-reversal symmetry (TRS) breaking plays an important role in studying strongly correlated unconventional superconductors. When the superconducting gap functions with different pairing symmetries compete, an Ising (Z2) type symmetry breaking occurs due to the locking of the relative phase ∆θ12 via a second order Josephson coupling. The phase locking can take place even in the normal state in the phase fluctuation regime before the onset of superconductivity. If ∆θ12 = ± π 2 , then TRS is broken, otherwise, if ∆θ12 = 0, or, π, rotational symmetry is broken leading to a nematic state. In both cases, the order parameters possess a 4-fermion structure beyond the scope of mean-field theory. We employ an effective two-component XY -model assisted by a renormalization group analysis to address this problem. In addition, a quartetting, or, charge-"4e", superconductivity can also occur above Tc. Monte-Carlo simulations are performed and the results are in a good agreement with the renormalization group analysis. Our results provide useful guidance for studying novel symmetry breakings in strongly correlated superconductors.