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Gaussian processes (GPs) are Bayesian nonparametric models for function approximation with principled predictive uncertainty estimates. Deep Gaussian processes (DGPs) are multilayer generalizations of GPs that can represent complex marginal densities as well as complex mappings. As exact inference is either computationally prohibitive or analytically intractable in GPs and extensions thereof, some existing methods resort to variational inference (VI) techniques for tractable approximations. However, the expressivity of conventional approximate GP models critically relies on independent inducing variables that might not be informative enough for some problems. In this work we introduce amortized variational inference for DGPs, which learns an inference function that maps each observation to variational parameters. The resulting method enjoys a more expressive prior conditioned on fewer input dependent inducing variables and a flexible amortized marginal posterior that is able to model more complicated functions. We show with theoretical reasoning and experimental results that our method performs similarly or better than previous approaches at less computational cost.

Variational quantum algorithms (VQAs) have emerged as the leading strategy to obtain quantum advantage on the current noisy intermediate-scale devices. However, their entanglement-trainability correlation, as the major reason for the barren plateau (BP) phenomenon, poses a challenge to their applications. In this Letter, we suggest a gate-to-tensor (GTT) encoding method for parameterized quantum circuits (PQCs), with which two long short-term memory networks (L-G networks) are trained to predict both entanglement and trainability. The remarkable capabilities of the L-G networks afford a statistical way to delve into the entanglement-trainability correlation of PQCs within a dataset encompassing millions of instances. This machine-learning-driven method first confirms that the more entanglement, the more possible the BP problem. Then, we observe that there still exist PQCs with both high entanglement and high trainability. Furthermore, the trained L-G networks result in an impressive increase in time efficiency by about one million times when constructing a PQC with specific entanglement and trainability, demonstrating their practical applications in VQAs.

In this paper, we propose a general scheme to realize chiral TSCs through the "twistronics". Suppose we have a $D_n$-symmetric monolayer superconductor, which carries non-topological SC with pairing angular momentum $L=n/2$. Here we propose that we can obtain chiral TSC with the same $L$, by stacking two such monolayers with the largest twist angle $\pi/n$, forming a Moireless quasi-crystal (QC) structure, dubbed as the twist-bilayer QC (TB-QC) here. The chiral TSC in the TB-QC is driven by the interlay Josephson coupling between the pairing order parameters of the two layers. An argument based on the universal Ginzburg-Landau theory is provided to understand this proposal. One known example which fits our proposal is the $d+id$-chiral TSC in the 45$^\circ$-twisted bilayer cuprates. Here, based on the microscopic framework developed previously to treat with the electron-electron interactions in the TB-QC, we demonstrate the application of our proposal to a new example, i.e., the $f+if$-chiral TSC obtained by twisting two properly-doped honeycomb-Hubbard-model monolayers by the angle 30$^\circ$. This example is related to the newly synthesized 30$^\circ$-twisted bilayer graphene., Comment: 7 pages, 5 figures

We study the largest-angle twisted homo-bilayer (LA-THB) systems, hosting Moir\'eless quasi-crystal (QC) structure. We propose to use these materials to generate high-angular-momentum (HAM) topological superconductivities (TSCs) protected by their QC symmetries absent on conventional crystalline materials. This proposal is based on our universal Ginzburg-Landau theory based analysis which yields the general conclusion that, when each $D_n$-symmetric ($n$ is even) monolayer hosts SC with pairing angular momentum $l\le \frac{n}{2}$, the interlayer Josephson coupling will induce SC with pairing angular momentum $L=l$ or $L=n-l$ in the LA-THB, determined by microscopic details. The latter one is just the HAM TSC if $l>0$. Based on our revised perturbational-band theory, we develop general microscopic framework to study the QC LA-THBs involving electron-electron interactions, adopting which we study three examples, i.e. the 30$^\circ$- twisted bilayer graphene, the 30$^\circ$- twisted bilayer BC$_3$, and the 45$^\circ$- twisted bilayer cuprates. The $g+ig$- $h+ih$- and $d+id$- TSCs with HAM $L=4,5$ and $2$ can emerge in certain doping regimes in these systems, respectively., Comment: 4.2 pages, 3 figures, with Supplementary Materials. This is the generalized version of arXiv:2106.08542. In the old version, we only consider a special example, here we give a general theory

Based on our revised perturbational-band theory, we study possible pairing states driven by interaction in the electron-doped quasicrystal 30\degree-twisted bilayer graphene. Our mean-field study on the related t-J model predicts that, the beneath-van-Hove and beyond-van-Hove low doping regimes are covered by the chiral $d+id$ and $g+ig$ topological superconductivities (TSCs) respectively. The $g+ig$-TSC possesses a pairing angular momentum 4, and hence following each effective $C_{12}$- rotation by $\Delta\phi=n\pi/6$, the pairing phase changes $4\Delta\phi$. This intriguing TSC is novel, as it belongs to a special 2D $E_4$- irreducible representation of the effective $D_{12}$ point group unique to this quasicystal and absent on periodic lattices. The Ginzburg-Landau theory suggested that the $g+ig$- TSC originates from the Josephson coupling between the $d+id$ pairings on the two mono-layers., Comment: 4pages,3 figures, plus Supplementary Material

Exceptional points as branch singularities describe peculiar degeneracies of non-Hermitian systems that do not obey energy conservation. This work shows that exceptional points can emerge in a topological photonic system, for example, the topological photonic waveguide coupled with two degenerate counter-propagation topological whispering gallery modes. Such a photonic architecture is designed by crystal-symmetry-protected topological photonic insulators based on air rods in conventional dielectric materials. The relevant exceptional point reveals the breaking of the parity-time symmetry, reflected by the change of the transmission-dip number in the optical transmission spectra of the system. Achieving exceptional points in topological photonic systems possibly opens a new avenue toward robust optical devices with exceptional-point-based unique properties and functionalities., Comment: 8 pages, 4 figures

Rapidly developing photonics brings many interesting resonant optical phenomena, in which the Fano resonance (FR) always intrigues researchers because of its applications in optical switching and sensing. However, its sensitive dependence on environmental conditions makes it hard to implement in experiments. We in this work suggest a topologically-protected FR based on the photonic valley Hall insulators, immune to the system impurities. The topologically-protected FR is achieved by coupling the valley-dependent topological edge states (TESs) with one double-degenerate cavity. The $\delta$-type photonic transport theory we build reveals that this topological FR dates from the interference of the two transmissions that are attributed to the parity-odd cavity mode and the parity-even one. We confirm that the induced Fano line shape of the transmission spectra is robust against the bending domain walls and disorders. Our work may provoke exciting frontiers for manipulating the valley transport and pave a way for the topologically protected photonic devices such as optical switches, low-threshold lasers, and ultra-sensitive sensors., Comment: 10 pages, 4 figures

Starting from a realistic extended Hubbard model for a $p_{x,y}$-orbital tight-binding model on the Honeycomb lattice, we perform a thorough investigation on the possible electron instabilities in the MA-TBG near the van Hove (VH) dopings. Here we focus on the interplay between the approximate SU(2)$\times$SU(2) symmetry and the $D_3$ symmetry, which leads to intriguing quantum states relevant to recent experiments, as revealed by our systematic RPA based calculations followed by a succeeding mean-field energy minimization for the ground state energy. At the SU(2)$\times$SU(2) symmetric point, the degenerate inter-valley SDW and VDW are mixed into a new state of matter dubbed as the chiral SO(4) spin-valley DW. This state simultaneously hosts three 4-component vectorial spin-valley DW orders with each adopting one wave vector, and the polarization directions of the three DW orders are mutually perpendicular to one another. %in the $\mathbb{R}^4$ space. In the presence of a tiny inter-valley exchange interaction with coefficient $J_H\to 0^{-}$ which breaks the SU(2)$\times$SU(2) symmetry, a pure chiral SDW state is obtained. In the case of $J_H\to 0^{+}$, a nematic VDW+SDW state emerges which possesses a stripy distribution of the charge density, consistent with the recent STM observations. On the aspect of SC, while the triplet $p+ip$ and singlet $d+id$ topological SCs are degenerate at $J_H=0$ near the VH dopings, the former (latter) is favored for $J_H\to 0^{-}$ ($J_H\to 0^{+}$). In addition, the two asymmetric doping-dependent behaviors of the obtained pairing phase diagram are well consistent with experiments., Comment: Published version in PRB. 16 pages plus Appendices

Bulk superconductivity (SC) has recently been observed in the Al-Zn-Mg quasicrystal (QC). To settle several fundamental issues of the SC on the QC, we use an attractive Hubbard model to perform a systematic study on the Penrose lattice. The first issue is the Cooper instability of the QC, i.e., no Fermi surface under an infinitesimal attractive interaction. Starting from the two-electron problem outside the filled Fermi-sea, we analytically prove that an infinitesimal Hubbard attraction can lead to the Cooper instability as long as the density of state is nonzero at the Fermi level, which provides the basis of the SC on the QC. Our numerical results yield that the Cooper pairing always takes place between the two time-reversal states, satisfying the Anderson's theorem. On this theorem, we perform a mean-field (MF) study at both zero and finite temperatures. The MF study also shows that an arbitrarily weak attraction can lead to the pairing order, with the resulted pairing state being well described by the BCS theory and the thermal dynamic behaviors being well consistent with experimental results. The second issue is about the superfluid density on the QC without translational symmetry. It's clarified that although the normal state of the system locates at the critical point of the metal-insulator transition, the pairing state exhibits real SC, carrying finite superfluid density that can be verified by the Meissner effect, consistent with experiment also. These revealed properties of the SC on the Penrose lattice are universal for all QCs., Comment: 11 pages, 6 figures

The Kohn-Luttinger mechanism for unconventional superconductivity (SC) driven by weak repulsive electron-electron interactions on a periodic lattice is generalized to the quasicrystal (QC) via a real-space perturbative approach. The repulsive Hubbard model on the Penrose lattice is studied as an example, on which a classification of the pairing symmetries is performed and a pairing phase diagram is obtained. Two remarkable properties of these pairing states are revealed, due to the combination of the presence of the point-group symmetry and the lack of translation symmetry on this lattice. Firstly, the spin and spacial angular momenta of a Cooper pair is de-correlated: for each pairing symmetry, both spin-singlet and spin-triplet pairings are possible even in the weak-pairing limit. Secondly, the pairing states belonging to the 2D irreducible representations of the $D_5$ point group can be time-reversal-symmetry-breaking topological SCs carrying spontaneous bulk super current and spontaneous vortices. These two remarkable properties are general for the SCs on all QCs, and are rare on periodic lattices. Our work starts the new area of unconventional SCs driven by repulsive interactions on the QC., Comment: 4.5 pages, 4 figures, with Supplementary Material shown in the Appendix

Crystal-symmetry-protected photonic topological edge states (PTESs) based on air rods in conventional dielectric materials are designed as photonic topological waveguides (PTWs) coupled with side optical cavities. We demonstrate that the cavity coupled with the PTW can change the reflection-free transport of the PTESs, where the cavities with single mode and twofold degenerate modes are taken as examples. The single-mode cavities are able to perfectly reflect the PTESs at their resonant frequencies, forming a dip in the transmission spectra. The dip full width at half depth depends on the coupling strength between the cavity and PTW and thus on the cavity geometry and distance relative to the PTW. While the cavities with twofold degenerate modes lead to a more complex PTES transport whose transmission spectra can be in the Fano form. These effects well agree with the one-dimensional PTW-cavity transport theory we build, in which the coupling of the PTW with cavity is taken as $\delta$ or non-$\delta$ type. Such PTWs coupled with side cavities, combining topological properties and convenient tunability, have wide diversities for topological photonic devices., Comment: 9 pages, 9 figures

With coupled Gross-Piteavskii equations we study excitation of exciton-polariton vortices and antivortices in a pillar microcavity by a Gaussian pump beam. The structure of vortices and antivortices shows a strong dependence on the microcavity radius, pump geometry, and nonlinear exciton-exciton interaction. Due to the nonlinear interaction the strong Gaussian beam cannot excite more polariton vortices or antivortices with respect to the weak one. The calculation demonstrates that the weak Gaussian beam can excite vortex-antivortex pairs, vortices with high angular momentum, and superposition states of vortex and antivortex with high opposite angular momentum. The pump geometry for the Gaussian beam to excite these vortex structures are analyzed in detail, which holds a potential application for Sagnac interferometry and generating the optical beams with high angular momentum., Comment: 9 pages, 5 figures

With transfer matrix theory, we study the transmission of a distributed Bragg reflector (DBR) with two dielectric gratings on top and on the bottom. Owing to the diffraction of the two gratings, the transmission shows a comb-like spectrum which red shifts with increasing the grating period during the forbidden band of the DBR. The number density of the comb peaks increases with increasing the number of the DBR cells, while the ratio of the average full width at half maximum (FWHM) of the transmission peaks in the transmission comb to the corresponding average free spectral range, being about 0.04 and 0.02 for the TE and TM incident waves, is almost invariant. The average FWHM of the TM waves is about half of the TE waves, and both they could be narrower than 0.1 nm. In addition, the transmission comb peaks of the TE and TM waves can be fully separated during certain waveband. We further prove that the transmission comb is robust against the randomness of the heights of the DBR layers, even when a 15\% randomness is added to their heights. Therefore, the proposed structure is a candidate for a multichannel narrow-band filter or a multichannel polarizer., Comment: 4 pages, 5 figures

We theoretically study the transport of exciton polaritons under different applied photon potentials. The relation between the photon potentials and the thickness of the cavity layer is calculated by the finite element simulation. The theoretical analysis and numerical calculation indicate that the cavity photon potential is proportional to the thickness of the cavity layer with the coefficient being about $1.8$ meV/nm. Further, the periodic and linear photon potentials are considered to control the transport of the exciton polaritons in weak- and strong-field pump situations. In both situations the periodic potential cannot by itself effectively suppress the scatterings of the disorder potentials of the cavity photons and excitons and the nonlinear exciton-exciton interaction. When the linear potential is added to the cavity photons, the polariton transport exhibits the Bloch oscillation behavior. Importantly, the polariton Bloch oscillation can strongly suppress the space broadening due to the disorder potentials and nonlinear exciton-exciton interaction, which is beneficial for designing the polariton circuits., Comment: 7 pages, 8 figures

The nonlocal coupling effect between a one-dimensional waveguide (1DW) and a side optical cavity (SOC) is studied. We first find the real-space Hamiltonian of the nonlocal-coupling system of the 1DW and SOC, and then derive out an equation determining the energy of the hybridization state between the 1DW and SOC modes and an analytic formula for the single-photon transmission. Through them, we recognize that the single-photon transmission-dip position can be changed by adjusting the SOC size and the coupling strength between the 1DW and SOC. The transmission spectra strongly depends on the nonlocal-coupling function between the 1DW and SOC, and holds an asymmetry line shape. At last, we simulate the asymmetry of the single-photon transmission by a gold-based waveguide coupled with a rectangular SOC., Comment: 8 pages, 6 figures