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Representation by neural networks, in particular by restricted Boltzmann machines (RBM), has provided a powerful computational tool to solve quantum many-body problems. An important open question is how to characterize which class of quantum states can be efficiently represented with the RBM. Here, we show that the RBM can efficiently represent a wide class of many-body entangled states with rich exotic topological orders. This includes: (1) ground states of double semion and twisted quantum double models with intrinsic topological orders; (2) states of the AKLT model and 2D CZX model with symmetry protected topological order; (3) states of Haah code model with fracton topological order; (4) generalized stabilizer states and hypergraph states that are important for quantum information protocols. One twisted quantum double model state considered here harbors non-abelian anyon excitations. Our result shows that it is possible to study a variety of quantum models with exotic topological orders and rich physics using the RBM computational toolbox., 10 pages, 5 figures

A direct measurement of the density of Andreev bound states (ABSs) is experimentally investigated in a superconductor--quantum dot--superconductor hybrid nanowire system. A hard proximity-induced superconducting gap is observed, arising from superconducting correlations, in the transport spectrum of the hybrid system. Conductance peaks observed inside the superconducting gap reveal that the subgap states participate in the transport of the hybrid junction. We explore the evolution of low-energy Andreev bound states in a few-electron quantum dot (QD) coupled to superconductors, by probing the magnetic field dependence of exquisite detailed conductance spectra with a small bias voltage applied on the superconducting lead. In the presence of low magnetic fields, the resonance current is enhanced and broadened, attributed to the transport through Andreev bound states in QD, as the energy of ABSs reaches the threshold set by the applied bias voltage with the increase of the magnetic field. The simulated transport spectrum matches the experimentally observed evolution patterns of conductance, further implying the superconducting correlation nature of the observed electron transport. At high magnetic fields, the conductance maxes as the Fermi level reaches the degeneration point of Landau levels, leading to conductance peaks shown in the alternating narrow and wide patterns of Coulomb blockade oscillations.

Quantum entanglement, the essential resource for quantum information processing, has rich dynamics under different environments. Probing different entanglement dynamics typically requires exquisite control of complicated system-environment coupling in real experimental systems. Here, by a simple control of the effective solid-state spin bath in a diamond sample, we observe rich entanglement dynamics, including the conventional asymptotic decay as well as the entanglement sudden death, a term coined for the phenomenon of complete disappearance of entanglement after a short finite time interval. Furthermore, we observe counter-intuitive entanglement rebirth after its sudden death in the same diamond sample by tuning an experimental parameter, demonstrating that we can conveniently control the non-Markovianity of the system-environment coupling through a natural experimental knob. Further tuning of this experimental knob can make the entanglement dynamics completely coherent under the same environmental coupling. Probing of entanglement dynamics, apart from its fundamental interest, may find applications in quantum information processing through control of the environmental coupling.

We experimentally demonstrate a robust dynamical decoupling protocol with bounded controls using long soft pulses, eliminating a challenging requirement of strong control pulses in conventional implementations. This protocol is accomplished by designing the decoupling propagators to go through a Eulerian cycle of the coupler group [Phys. Rev. Lett. 90, 037901(2003)]. We demonstrate that this Eulerian decoupling scheme increases the coherence time by two orders of magnitude in our experiment under either dephasing or a universal noise environment., Comment: 7 pages, 4 figures

We propose an experimental scheme to observe non-abelian statistics with cold atoms in a two dimensional optical lattice. We show that the Majorana-Schockley modes associated with line defects obey non-abelian statistics and can be created, braided, and fused, all through adiabatic shift of the local chemical potentials. The detection of the topological qubit is transformed to local measurement of the atom number on a single lattice site. We demonstrate the robustness of the braiding operation by incorporating noise and experiential imperfections in numerical simulations, and show that the requirement fits well with the current experimental technology.

Protected by the chiral symmetry, three dimensional chiral topological insulators are characterized by an integer-valued topological invariant. How this invariant could emerge in physical observables is an important question. Here we show that the magneto-electric polarization can identify the integer-valued invariant if we gap the system without coating a quantum Hall layer on the surface. The quantized response is demonstrated to be robust against weak perturbations. We also study the topological properties by adiabatically coupling two nontrivial phases, and find that gapless states appear and are localized at the boundary region. Finally, an experimental scheme is proposed to realize the Hamiltonian and measure the quantized response with ultracold atoms in optical lattices., 8 pages, 6 figures, 2 Appendices

A remarkable discovery in recent years is that there exist various kinds of topological insulators and superconductors characterized by a periodic table according to the system symmetry and dimensionality. To physically realize these peculiar phases and study their properties, a critical step is to construct experimentally relevant Hamiltonians which support these topological phases. We propose a general and systematic method based on the quaternion algebra to construct the tight binding Hamiltonians for all the three-dimensional topological phases in the periodic table characterized by arbitrary integer topological invariants, which include the spin-singlet and the spin-triplet topological superconductors, the Hopf and the chiral topological insulators as particular examples. For each class, we calculate the corresponding topological invariants through both geometric analysis and numerical simulation., Comment: 7 pages (including supplemental material), 1 figure, 1 table

We use the recently developed tensor network algorithm based on infinite projected entangled pair states (iPEPS) to study the phase diagram of frustrated antiferromagnetic ${J}_{1}$-${J}_{2}$ Heisenberg model on a checkerboard lattice. The simulation indicates a N\'eel ordered phase when ${J}_{2}l0.88{J}_{1}$, a plaquette valence bond solid state when $0.88l{J}_{2}/{J}_{1}l1.11$, and a stripe phase when ${J}_{2}g1.11{J}_{1}$, with two first-order transitions across the phase boundaries. The calculation shows that the cross dimer state proposed before is unlikely to be the ground state of the model, although such a state indeed arises as a metastable state in some parameter region.