A large set of recent experiments has been exploring topological transport in bosonic systems,e.g. of photons or phonons. In the vast majority, time-reversal symmetry is preserved, and bandstructures are engineered by a suitable choice of geometry, to produce topologically nontrivialbandgaps in the vicinity of high-symmetry points. However, this leaves open the possibility oflarge-quasimomentum backscattering, destroying the topological protection. Up to now, it has beenunclear what precisely are the conditions where this effect can be sufficiently suppressed. In thepresent work, we introduce a comprehensive semiclassical theory of tunneling transitions in momen-tum space, describing backscattering for one of the most important system classes, based on thevalley Hall effect. We predict that even for a smooth domain wall effective scattering centres developat locations determined by both the local slope of the wall and the energy. Moreover, our theoryprovides a quantitative analysis of the exponential suppression of the overall reflection amplitudewith increasing domain wall smoothness.

A large number of applications in classical and quantum photonics require the capability of implementing arbitrary linear unitary transformations on a set of optical modes. In a seminal work by Reck et al. it was shown how to build such multiport universal interferometers with a mesh of beam splitters and phase shifters, and this design became the basis for most experimental implementations in the last decades. However, the design of Reck et al. is difficult to scale up to a large number of modes, which would be required for many applications. Here we present a constructive proof that it is possible to realize a multiport universal interferometer on N modes with a succession of 6N Fourier transforms and 6N+1 phase masks, for any even integer N. Furthermore, we provide an algorithm to find the correct succesion of Fourier transforms and phase masks to realize a given arbitrary unitary transformation. Since Fourier transforms and phase masks are routinely implemented in several optical setups and they do not suffer from the scalability issues associated with building extensive meshes of beam splitters, we believe that our design can be useful for many applications in photonics.

In recent years, dynamical phase transitions and out-of-equilibrium criticality have been at the forefront of ultracold gases and condensed matter research. Whereas universality and scaling are established topics in equilibrium quantum many-body physics, out-of-equilibrium extensions of such concepts still leave much to be desired. Using exact diagonalization and the time-dependent variational principle in uniform matrix product states, we calculate the time evolution of the local order parameter and Loschmidt return rate in transverse-field Ising chains with antiferromagnetic power law-decaying interactions, and map out the corresponding rich dynamical phase diagram. Anomalous cusps in the return rate, which are ubiquitous at small quenches within the ordered phase in the case of ferromagnetic long-range interactions, are absent within the accessible timescales of our simulations in the antiferromagnetic case, showing that long-range interactions are not a sufficient condition for their appearance. We attribute this to much weaker domain-wall binding in the antiferromagnetic case. For quenches across the quantum critical point, regular cusps appear in the return rate and connect to the local order parameter changing sign, indicating the concurrence of two major concepts of dynamical phase transitions. Our results consolidate conclusions of previous works that a necessary condition for the appearance of anomalous cusps in the return rate after quenches within the ordered phase is for topologically trivial local spin flips to be the energetically dominant excitations in the spectrum of the quench Hamiltonian. Our findings are readily accessible in modern trapped-ion setups and we outline the associated experimental considerations.

Discord, originally notable as a signature of bipartite quantum correlation, in fact can be nonzero<br>classically, i.e. arising from noisy measurements by one of the two parties. Here we redefine<br>classical discord to quantify channel distortion, in contrast to the previous restriction of classical<br>discord to a state, and we then show a monotonic relationship between classical (channel) discord<br>and channel distortion. We show that classical discord is equivalent to (doubly stochastic) channel<br>distortion by numerically discovering a monotonic relation between discord and total-variation<br>distance for a bipartite protocol with one party having a noiseless channel and the other party<br>having a noisy channel. Our numerical method includes randomly generating doubly stochastic<br>matrices for noisy channels and averaging over a uniform measure of input messages. Connecting<br>discord with distortion establishes discord as a signature of classical, not quantum, channel<br>distortion.

Dielectric spheres of various sizes may sustain electromagnetic whispering-gallery modes resonating at optical frequencies with very narrow linewidths. Arbitrary small deviations from the spherical shape typically shift and broaden such resonances. Our goal is to determine these shifted and broadened resonances. A boundary-condition perturbation theory for the acoustic vibrations of nearly circular membranes was developed by Rayleigh more than a century ago. We extend this theory to describe the electromagnetic excitations of nearly spherical dielectric cavities. This approach permits us to avoid dealing with decaying quasinormal modes. We explicitly find the frequencies and the linewidths of the optical resonances for arbitrarily deformed nearly spherical dielectric cavities, as power series expansions by a small parameter, up to and including second-order terms. We thoroughly discuss the physical conditions for the applicability of perturbation theory.

We propose a platform that combines the fields of cavity optomagnonics and levitated optome-<br>chanics in order to control and probe the coupled spin-mechanics of magnetic dielectric particles. We theoretically study the dynamics of a levitated Faraday-active dielectric microsphere serving as an optomagnonic cavity, placed in an external magnetic field and driven by an external laser. We find that the optically driven magnetization dynamics induces angular oscillations of the particle with low associated damping. Further, we show that the magnetization and angular motion dynamics<br>can be probed via the power spectrum of the outgoing light. Namely, the characteristic frequencies attributed to the angular oscillations and the spin dynamics are imprinted in the light spectrum by two main resonance peaks. Additionally, we demonstrate that a ferromagnetic resonance setup with an oscillatory perpendicular magnetic field can enhance the resonance peak corresponding to<br>the spin oscillations and induce fast rotations of the particle around its anisotropy axis.

Shortcuts to adiabaticity (STA) is a general methodology for speeding up adiabatic quantumprotocols, and has many potential applications in quantum information processing. Unfortunately,analytically constructing STAs for systems having complex interactions and more than a few levelsis a challenging task. This is usually overcome by assuming an idealized Hamiltonian (e.g., only alimited subset of energy levels are retained, and the rotating-wave approximation (RWA) is made).Here, we develop ananalyticapproach that allows one to go beyond these limitations. Our methodis general and results in analytically-derived pulse shapes that correct both non-adiabatic errorsas well as non-RWA errors. We also show that our approach can yield pulses requiring a smallerdriving power than conventional non-adiabatic protocols. We show in detail how our ideas can beused to analytically design high-fidelity single-qubit “tripod” gates in a realistic superconductingfluxonium qubit.

The design of periodic nanostructures allows to tailor the transport of photons, phonons, and matter waves for specific applications. Recent years have seen a further expansion of this field by engineering topological properties. However, what is missing currently are efficient ways to rapidly explore and optimize band structures and to classify their topological characteristics for arbitrary unit-cell geometries. In this work, we show how deep learning can address this challenge. We introduce an approach where a neural network first maps the geometry to a tight-binding model. The tight-binding model encodes not only the band structure but also the symmetry properties of the Bloch waves. This allows us to rapidly categorize a large set of geometries in terms of their band representations, identifying designs for fragile topologies. We demonstrate that our method is also suitable to calculate strong topological invariants, even when (like the Chern number) they are not symmetry indicated. Engineering of domain walls and optimization are accelerated by orders of magnitude. Our method directly applies to any passive linear material, irrespective of the symmetry class and space group. It is general enough to be extended to active and nonlinear metamaterials.

These brief lecture notes cover the basics of neural networks and deep learning as well as their applications in the quantum domain, for physicists without prior knowledge. In the first part, we describe training using back-propagation, image classification, convolutional networks and autoencoders.The second part is about advanced techniques like reinforcement learning (for discovering control strategies), recurrent neural networks (for analyzing timetraces), and Boltzmann machines (for learning probability distributions). In the third lecture, we discuss first recent applications to quantum physics, with an emphasis on quantum information processing machines. Finally, the fourth lecture is devoted to the promise of using quantum effects to accelerate machine learning.

We derive a well-defined renormalized version of mutual information that allows to estimate the dependence between continuous random variables in the important case when one is deterministically dependent on the other. This is the situation relevant for feature extraction, where the goal is to produce a low-dimensional effective description of a high-dimensional system. Our approach enables the discovery of collective variables in physical systems, thus adding to the toolbox of artificial scientific discovery, while also aiding the analysis of information flow in artificial neural networks.

Incorporating protection against quantum errors into adiabatic quantum computing (AQC) is an important task due to the inevitable presence of decoherence. Here, we investigate an error-protected encoding of the AQC Hamiltonian, where qubit ensembles are used in place of qubits. Our Hamiltonian only involves total spin operators of the ensembles, offering a simpler route towards error-corrected quantum computing. Our scheme is particularly suited to neutral atomic gases where it is possible to realize large ensemble sizes and produce ensemble-ensemble entanglement. We identify a critical ensemble size Nc where the nature of the first excited state becomes a single particle perturbation of the ground state, and the gap energy is predictable by mean-field theory. For ensemble sizes larger than Nc, the ground state becomes protected due to the presence of logically equivalent states and the AQC performance improves with N, as long as the decoherence rate is sufficiently low.

Open quantum systems can display periodic dynamics at the classical level either due to external periodic modulations or to self-pulsing phenomena typically following a Hopf bifurcation. In both cases, the quantum fluctuations around classical solutions do not reach a quantum-statistical stationary state, which prevents adopting the simple and reliable methods used for stationary quantum systems. Here we put forward a general and efficient method to compute two-time correlations and corresponding spectral densities of time-periodic open quantum systems within the usual linearized (Gaussian) approximation for their dynamics. Using Floquet theory, we show how the quantum Langevin equations for the fluctuations can be efficiently integrated by partitioning the time domain into one-period duration intervals, and relating the properties of each period to the first one. Spectral densities, like squeezing spectra, are computed similarly, now in a two-dimensional temporal domain that is treated as a chessboard with one-period × one-period cells. This technique avoids cumulative numerical errors as well as efficiently saving computational time. As an illustration of the method, we analyze the quantum fluctuations of a damped parametrically driven oscillator (degenerate parametric oscillator) below threshold and far away from rotating-wave approximation conditions, which is a relevant scenario for modern low-frequency quantum oscillators. Our method reveals that the squeezing properties of such devices are quite robust against the amplitude of the modulation or the low quality of the oscillator, although optimal squeezing can appear for parameters that are far from the ones predicted within the rotating-wave approximation.

Understanding how to tailor quantum dynamics to achieve a desired evolution is a crucial problemin almost all quantum technologies. We present a very general method for designing high-efficiencycontrol sequences that are always fully compatible with experimental constraints on available inter-actions and their tunability. Our approach reduces in the end to finding control fields by solvinga set of time-independent linear equations. We illustrate our method by applying it to a numberof physically-relevant problems: the strong-driving limit of a two-level system, fast squeezing in aparametrically driven cavity, the leakage problem in transmon qubit gates, and the acceleration ofSNAP gates in a qubit-cavity system.

Continuous-variable codes are an expedient solution for quantum information processing and quantum communication involving optical networks. Here we characterize the squeezed comb, a finite superposition of equidistant squeezed coherent states on a line, and its properties as a continuous-variable encoding choice for a logical qubit. The squeezed comb is a realistic approximation to the ideal code proposed by Gottesman et al. [D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001)], which is fully protected against errors caused by the paradigmatic types of quantum noise in continuous-variable systems: damping and diffusion. This is no longer the case for the code space of finite squeezed combs, and noise robustness depends crucially on the encoding parameters. We analyze finite squeezed comb states in phase space, highlighting their complicated interference features and characterizing their dynamics when exposed to amplitude damping and Gaussian diffusion noise processes. We find that squeezed comb states are more suitable and less error prone when exposed to damping, which speaks against standard error-correction strategies that employ linear amplification to convert damping into easier-to-describe isotropic diffusion noise.