## Analogue quantum chemistry simulation

Computing the electronic structure of molecules with high precision is a central challenge in the field of quantum chemistry. Despite the success of approximate methods, tackling this problem exactly with conventional computers remains a formidable task. Several theoretical and experimental attempts have been made to use quantum computers to solve chemistry problems, with early proof-of-principle realizations done digitally. An appealing alternative to the digital approach is analogue quantum simulation, which does not require a scalable quantum computer and has already been successfully applied to solve condensed matter physics problems. However, not all available or planned setups can be used for quantum chemistry problems, because it is not known how to engineer the required Coulomb interactions between them. Here we present an analogue approach to the simulation of quantum chemistry problems that relies on the careful combination of two technologies: ultracold atoms in optical lattices and cavity quantum electrodynamics. In the proposed simulator, fermionic atoms hopping in an optical potential play the role of electrons, additional optical potentials provide the nuclear attraction, and a single-spin excitation in a Mott insulator mediates the electronic Coulomb repulsion with the help of a cavity mode. We determine the operational conditions of the simulator and test it using a simple molecule. Our work opens up the possibility of efficiently computing the electronic structures of molecules with analogue quantum simulation.

## Cold atoms in twisted-bilayer optical potentials

The possibility of creating crystal bilayers twisted with respect to each other has led to the discovery of a wide range of novel electron correlated phenomena the full understanding of which is still under debate. Here we propose and analyze a method to simulate twisted bilayers using cold atoms in state-dependent optical lattices. Our proposed setup can be used as an alternative platform to explore twisted bilayers which allows one to control the inter- and intralayer coupling in a more flexible way than in the solid-state realizations. We focus on square geometries but also show how it can be extended to simulate other lattices which show Dirac-like physics. This setup opens a path to observe similar physics, e.g., band narrowing, with larger twist angles, to rule out some of the mechanisms to explain the observed strongly correlated effects, as well as to study other phenomena difficult to realize with crystals. As an example of the latter we explore the quantum optical consequences of letting emitters interact with twisted-bilayer reservoirs, and predict the appearance of unconventional radiation patterns and emitter interactions following the emergent Moiré geometry.

## Engineering and Harnessing Giant Atoms in High-Dimensional Baths: A Proposal for Implementation with Cold Atoms

Emitters coupled simultaneously to distant positions of a photonic bath, the so-called giant atoms, represent a new paradigm in quantum optics. When coupled to one-dimensional baths, as recently implemented with transmission lines or SAW waveguides, they lead to striking effects such as chiral emission or decoherence-free atomic interactions. Here, we show how to create giant atoms in dynamical state-dependent optical lattices, which offers the possibility of coupling them to structured baths in arbitrary dimensions. This opens up new avenues to a variety of phenomena and opportunities for quantum simulation. In particular, we show how to engineer unconventional radiation patterns, like multidirectional chiral emission, as well as collective interactions that can be used to simulate nonequilibrium many-body dynamics with no analog in other setups. Additionally, the recipes we provide to harness giant atoms in high dimensions can be exported to other platforms where such nonlocal couplings can be engineered.

## Ergodicity probes: using time-fluctuations to measure the Hilbert space dimension

Quantum devices, such as quantum simulators, quantum annealers, and quantum computers, may be exploited to solve problems beyond what is tractable with classical computers. This may be achieved as the Hilbert space available to perform such `calculations’ is far larger than that which may be classically simulated. In practice, however, quantum devices have imperfections, which may limit the accessibility to the whole Hilbert space. We thus determine that the dimension of the space of quantum states that are available to a quantum device is a meaningful measure of its functionality, though unfortunately this quantity cannot be directly experimentally determined. Here we outline an experimentally realisable approach to obtaining the required Hilbert space dimension of such a device to compute its time evolution, by exploiting the thermalization dynamics of a probe qubit. This is achieved by obtaining a fluctuation-dissipation theorem for high-temperature chaotic quantum systems, which facilitates the extraction of information on the Hilbert space dimension via measurements of the decay rate, and time-fluctuations.

## Floquet-engineered vibrational dynamics in a two-dimensional array of trapped ions

We demonstrate Floquet engineering in a basic yet scalable 2D architecture of individually trapped and controlled ions. Local parametric modulations of detuned trapping potentials steer the strength of long-range interion couplings and the related Peierls phase of the motional state. In our proof of principle, we initialize large coherent states and tune modulation parameters to control trajectories, directions, and interferences of the phonon flow. Our findings open a new pathway for future Floquet-based trapped-ion quantum simulators targeting correlated topological phenomena and dynamical gauge fields.

## From ergodic to non-ergodic chaos in Rosenzweig–Porter model

Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 47 (2019), arXiv: arXiv:1904.02716

The Rosenzweig–Porter model is a one-parameter family of random matrices with three different phases: ergodic, extended non-ergodic and localized. We characterize numerically each of these phases and the transitions between them. We focus on several quantities that exhibit non-analytical behaviour and show that they obey the scaling hypothesis. Based on this, we argue that non-ergodic chaotic and ergodic regimes are separated by a continuous phase transition, similarly to the transition between non-ergodic chaotic and localized phases.

## Mediator assisted cooling in quantum annealing

We show a significant reduction of errors for an architecture of quantum annealers (QA) where bosonic modes mediate the interaction between qubits. These systems have a large redundancy in the subspace of solutions, supported by arbitrarily large bosonic occupations. We explain how this redundancy leads to a mitigation of errors when the bosonic modes operate in the ultrastrong coupling regime. Numerical simulations also predict a large increase of qubit coherence for a specific annealing problem with mediated interactions. We provide evidences that noise reduction occurs in more general types of quantum computers with similar architectures.

## Modulated Continuous Wave Control for Energy-Efficient Electron-Nuclear Spin Coupling

We develop energy efficient, continuous microwave schemes to couple electron and nuclear spins, using phase or amplitude modulation to bridge their frequency difference. These controls have promising applications in biological systems, where microwave power should be limited, as well as in situations with high Larmor frequencies due to large magnetic fields and nuclear magnetic moments. These include nanoscale NMR where high magnetic fields achieves enhanced thermal nuclear polarization and larger chemical shifts. Our controls are also suitable for quantum information processors and nuclear polarization schemes.

## Non-ergodic quantum thermalization

Thermalization of generic closed quantum systems is well described by the Eigenstate Thermalization Hypothesis (ETH). One expects, however, that the presence of conservation laws may somewhat alter the adherence to the ETH. Here we see that in the presence of a single conservation law, for given physical initial states, thermalization occurs without the system fulfilling many predictions of the ETH. We find that certain local physical observables behave non-ergodically, even for non-integrable Hamiltonians, and yet an ETH-like relation, with non-random off-diagonals, is derived for observable matrix elements. This leads to a scaling law for equilibrium fluctuations that differs from that expected by the ETH. Further, we analytically compute the time-dependence of the decay to equilibrium, showing that it is proportional to the survival probability of the initial state. We further discuss (the lack of) scrambling of quantum information in this regime, and calculate the long-time limit of the out-of-time-ordered correlator. Relating our results to previous numerical observations of initial state dependent scrambling, we uncover the mechanism behind this feature.

## Quantum Simulation of Non-perturbative Cavity QED with Trapped Ions

We discuss the simulation of non-perturbative cavity-QED effects using systems of trapped ions. Specifically, we address the implementation of extended Dicke models with both collective dipole-field and direct dipole-dipole interactions, which represent a minimal set of models for describing light-matter interactions in the ultrastrong and deep-strong coupling regime. We show that this approach can be used in state-of-the-art trapped ion setups to investigate excitation spectra or the transition between sub- and superradiant ground states, which are currently not accessible in any other physical system. Our analysis also reveals the intrinsic difficulty of accessing this non-perturbative regime with larger numbers of dipoles, which makes the simulation of many-dipole cavity QED a particularly challenging test case for future quantum simulation platforms.

## Quantum chaotic fluctuation-dissipation theorem: Effective Brownian motion in closed quantum systems

We analytically describe the decay to equilibrium of generic observables of a non-integrable system after a perturbation in the form of a random matrix. We further obtain an analytic form for the time-averaged fluctuations of an observable in terms of the rate of decay to equilibrium. Our result shows the emergence of a Fluctuation-Dissipation theorem corresponding to a classical Brownian process, specifically, the Ornstein-Uhlenbeck process. Our predictions can be tested in quantum simulation experiments, thus helping to bridge the gap between theoretical and experimental research in quantum thermalization. We test our analytic results by exact numerical experiments in a spin-chain. We argue that our Fluctuation-Dissipation relation can be used to measure the density of states involved in the non-equilibrium dynamics of an isolated quantum system.

## Quantum metrology with one-dimensional superradiant photonic states

Photonic states with large and fixed photon numbers, such as Fock states, enable quantum-enhanced metrology but remain an experimentally elusive resource. A potentially simple, deterministic, and scalable way to generate these states consists of fully exciting N quantum emitters equally coupled to a common photonic reservoir, which leads to a collective decay known as Dicke superradiance. The emitted N-photon state turns out to be a highly entangled multimode state, and to characterize its metrological properties in this work we (i) develop theoretical tools to compute the quantum Fisher information of general multimode photonic states, (ii) use it to show that Dicke superradiant photons in one-dimensional waveguides achieve Heisenberg scaling, which can be saturated by a parity measurement, and (iii) study the robustness of these states to experimental limitations in state-of-the-art atom-waveguide QED setups.

## Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations

In this work we study the encoding of smooth, differentiable multivariate functions distributions in quantum registers, using quantum computers or tensor-network representations. We show that a large family of distributions can be encoded as low-entanglement states of the quantum register. These states can be efficiently created in a quantum computer, but they are also efficiently stored, manipulated and probed using Matrix-Product States techniques. Inspired by this idea, we present eight quantum-inspired numerical analysis algorithms, that include Fourier sampling, interpolation, differentiation and integration of partial derivative equations. These algorithms combine classical ideas—finite-differences, spectral methods—with the efficient encoding of quantum registers, and well known algorithms, such as the Quantum Fourier Transform. {When these heuristic methods work}, they provide an exponential speed-up over other classical algorithms, such as Monte Carlo integration, finite-difference and fast Fourier transforms (FFT). But even when they don’t, some of these algorithms can be translated back to a quantum computer to implement a similar task.

## Qubit-photon corner states in all dimensions

A single quantum emitter coupled to a one-dimensional photon field can perfectly trap a photon when placed close to a mirror. This occurs when the interference between the emitted and reflected light is completely destructive, leading to photon confinement between the emitter and the mirror. In higher dimensions, the spread of the light field in all directions hinders interference and, consequently, photon trapping by a single emitter is considered to be impossible. In this work, we show that is not the case by proving that a single emitter can indeed trap light in any dimension. We provide a constructive recipe based on judiciously coupling an emitter to a photonic crystal-like bath with properly designed open boundary conditions. The directional propagation of the photons in such baths enables perfect destructive interference, forming what we denote as\emph {qubit-photon corner states}. We characterize these states in all dimensions, showing that they are robust under fluctuations of the emitter’s properties, and persist also in the ultrastrong coupling regime.

## Single Photons by Quenching the Vacuum

Heisenberg’s uncertainty principle implies that the quantum vacuum is not empty but fluctuates. These fluctuations can be converted into radiation through nonadiabatic changes in the Hamiltonian. Here, we discuss how to control this vacuum radiation, engineering a single-photon emitter out of a two-level system (2LS) ultrastrongly coupled to a finite-band waveguide in a vacuum state. More precisely, we show the 2LS nonlinearity shapes the vacuum radiation into a non-Gaussian superposition of even and odd cat states. When the 2LS bare frequency lays within the band gaps, this emission can be well approximated by individual photons. This picture is confirmed by a characterization of the ground and bound states, and a study of the dynamics with matrix-product states and polaron Hamiltonian methods.

## Symmetries and conservation laws in quantum trajectories: Dissipative freezing

In driven-dissipative systems, the presence of a strong symmetry guarantees the existence of several steady states belonging to different symmetry sectors. Here we show that when a system with a strong symmetry is initialized in a quantum superposition involving several of these sectors, each individual stochastic trajectory will randomly select a single one of them and remain there for the rest of the evolution. Since a strong symmetry implies a conservation law for the corresponding symmetry operator on the ensemble level, this selection of a single sector from an initial superposition entails a breakdown of this conservation law at the level of individual realizations. Given that such a superposition is impossible in a classical stochastic trajectory, this is a a purely quantum effect with no classical analog. Our results show that a system with a closed Liouvillian gap may exhibit, when monitored over a single run of an experiment, a behavior completely opposite to the usual notion of dynamical phase coexistence and intermittency, which are typically considered hallmarks of a dissipative phase transition. We discuss our results on a coherently driven spin ensemble with a squeezed superradiant decay, a simple model that presents a wealth of nonergodic dynamics.

## Topological amplification in photonic lattices

We present a characterization of topological phases in photonic lattices. Our theory relies on a formal equivalence between the singular value decomposition of the non-Hermitian coupling matrix and the diagonalization of an effective Hamiltonian. By means of that mapping we unveil an application of topological band theory to the description of quantum amplification with non-reciprocal systems. We exemplify our ideas with an array of photonic cavities which can be mapped into a topological insulator Hamiltonian in the AIII symmetry class. We investigate stability properties and prove the existence of stable topologically non-trivial steady-state phases. Finally, we show numerically that the topological amplification process is robust against disorder in the lattice parameters.

## Ultrastrong coupling circuit QED in the radio-frequency regime

We study a circuit QED setup where multiple superconducting qubits are ultrastrongly coupled to a single radio-frequency resonator. In this extreme parameter regime of cavity QED the dynamics of the electromagnetic mode is very slow compared to all other relevant timescales and can be described as an effective particle moving in an adiabatic energy landscape defined by the qubits. The focus of this work is placed on settings with two or multiple qubits, where different types of symmetry-breaking transitions in the ground- and excited-state potentials can occur. Specifically, we show how the change in the level structure and the wave packet dynamics associated with these transition points can be probed via conventional excitation spectra and Ramsey measurements performed at GHz frequencies. More generally, this analysis demonstrates that state-of-the-art circuit QED systems can be used to access a whole range of particle-like quantum mechanical phenomena beyond the usual paradigm of coupled qubits and oscillators

## Ultrastrongly dissipative quantum Rabi model

We discuss both the spectrum and the dynamics of cavity QED in the presence of dissipation beyond the standard perturbative treatment of losses. Using the dynamical polaron ansatz and matrix-product state simulations, we discuss the case where both light-matter g coupling and system-bath interaction are in the ultra-strong-coupling regime. We provide a critical g for the onset of Rabi oscillations. Besides, we demonstrate that the qubit is dressed by the cavity and dissipation. Such a dressing governs the dynamics and, thus, it can be measured. Finally, we sketch an implementation for our theoretical ideas within circuit QED technology.

## Ultrastrongly dissipative quantum Rabi model

We discuss both the spectrum and the dynamics of cavity QED in the presence of dissipation beyond the standard perturbative treatment of losses. Using the dynamical polaron ansatz and matrix-product state simulations, we discuss the case where both light-matter g coupling and system-bath interaction are in the ultra-strong-coupling regime. We provide a critical g for the onset of Rabi oscillations. Besides, we demonstrate that the qubit is dressed by the cavity and dissipation. Such a dressing governs the dynamics and, thus, it can be measured. Finally, we sketch an implementation for our theoretical ideas within circuit QED technology.

## Unconventional quantum optics in topological waveguide QED

The discovery of topological materials has motivated recent developments to export topological concepts into photonics to make light behave in exotic ways. Here, we predict several unconventional quantum optical phenomena that occur when quantum emitters interact with a topological waveguide quantum electrodynamics bath, namely, the photonic analog of the Su-Schrieffer-Heeger model. When the emitters’ frequency lies within the topological bandgap, a chiral bound state emerges, which is located on just one side (right or left) of the emitter. In the presence of several emitters, this bound state mediates topological, tunable interactions between them, which can give rise to exotic many-body phases such as double Néel ordered states. Furthermore, when the emitters’ optical transition is resonant with the bands, we find unconventional scattering properties and different super/subradiant states depending on the band topology. Last, we propose several implementations where these phenomena can be observed with state-of-the-art technology.

## Unitary quantum perceptron as efficient universal approximator

We demonstrate that it is possible to implement a quantum perceptron with a sigmoid activation function as an efficient, reversible many-body unitary operation. When inserted in a neural network, the perceptron’s response is parameterized by the potential exerted by other neurons. We prove that such a quantum neural network is a universal approximator of continuous functions, with at least the same power as classical neural networks. While engineering general perceptrons is a challenging control problem—also defined in this work—the ubiquitous sigmoid-response neuron can be implemented as a quasi-adiabatic passage with an Ising model. In this construct, the scaling of resources is favorable with respect to the total network size and is dominated by the number of layers. We expect that our sigmoid perceptron will have applications also in quantum sensing or variational estimation of many-body Hamiltonians.