## Benchmarking quantum annealing dynamics: The spin-vector Langevin model

The classical spin-vector Monte Carlo (SVMC) model is a reference benchmark for the performance of a quantum annealer. Yet, as a Monte Carlo method, SVMC is unsuited for an accurate description of the annealing dynamics in real-time. We introduce the spin-vector Langevin (SVL) model as an alternative benchmark in which the time evolution is described by Langevin dynamics. The SVL model is shown to provide a more stringent test than the SVMC model for the identification of quantum signatures in the performance of quantum annealing devices, as we illustrate by describing the Kibble-Zurek scaling associated with the dynamics of symmetry breaking in the transverse field Ising model, recently probed using D-Wave machines. Specifically, we show that D-Wave data are reproduced by the SVL model.

## Bridging the gap between topological non-Hermitian physics and open quantum systems

We relate observables in open quantum systems with the topology of non-Hermitian models using the Keldysh path-integral method. This allows to extract an effective Hamiltonian from the Green’s function which contains all the relevant topological information and produces ω-dependent topological invariants, linked to the response functions at a given frequency. Then, we show how to detect a transition between different topological phases by measuring the response to local perturbations. Our formalism is exemplified in a one-dimensional Hatano-Nelson model, highlighting the difference between the bosonic and the fermionic case.

## Cold Atoms meet lattice gauge theories

The central idea of this review is to consider quantum field theory models relevant for particle physics and replace the fermionic matter in these models by a bosonic one. This is mostly motivated by the fact that bosons are more ‘accessible’ and easier to manipulate for experimentalists, but this ‘substitution’ also leads to new physics and novel phenomena. It allows us to gain new information about among other things confinement and the dynamics of the deconfinement transition. We will thus consider bosons in dynamical lattices corresponding to the bosonic Schwinger or Z2 Bose–Hubbard models. Another central idea of this review concerns atomic simulators of paradigmatic models of particle physics theory such as the Creutz–Hubbard ladder, or Gross–Neveu–Wilson and Wilson–Hubbard models. This article is not a general review of the rapidly growing field—it reviews activities related to quantum simulations for lattice field theories performed by the Quantum Optics Theory group at ICFO and their collaborators from 19 institutions all over the world. Finally, we will briefly describe our efforts to design experimentally friendly simulators of these and other models relevant for particle physics.

## Critical quantum metrology with fully-connected models: from Heisenberg to Kibble–Zurek scaling

## Decimation technique for open quantum systems: A case study with driven-dissipative bosonic chains

The unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (nonunitary) dynamics, which can be radically different from closed-system scenarios. Such open quantum system dynamics is generally described by Lindblad master equations, whose dynamical and steady-state properties are challenging to obtain, especially in the many-particle regime. Here, we introduce a method to deal with these systems based on the calculation of a (dissipative) lattice Green’s function with a real-space decimation technique. Compared to other methods, such a technique enables us to obtain compact analytical expressions for the dynamics and steady-state properties, such as asymptotic decays or correlation lengths. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity, including the Hatano-Nelson model. The latter is especially illustrative because its surface and bulk dissipative behavior are linked due to its nontrivial topology, which manifests in directional amplification.

## Devil’s staircase of topological Peierls insulators and Peierls supersolids

We consider a mixture of ultracold bosonic atoms on a one-dimensional lattice described by the XXZ-Bose-Hubbard model, where the tunneling of one species depends on the spin state of a second deeply trapped species. We show how the inclusion of antiferromagnetic interactions among the spin degrees of freedom generates a Devil’s staircase of symmetry-protected topological phases for a wide parameter regime via a bosonic analog of the Peierls mechanism in electron-phonon systems. These topological Peierls insulators are examples of symmetry-breaking topological phases, where long-range order due to spontaneous symmetry breaking coexists with topological properties such as fractionalized edge states. Moreover, we identify a region of supersolid phases that do not require long-range interactions. They appear instead due to a Peierls incommensurability mechanism, where competing orders modify the underlying crystalline structure of Peierls insulators, becoming superfluid. Our work show the possibilities that ultracold atomic systems offer to investigate strongly-correlated topological phenomena beyond those found in natural materials.

## Dispersive readout of molecular spin qudits

We study the physics of a magnetic molecule described by a “giant” spin with multiple (d>2) spin states interacting with the quantized cavity field produced by a superconducting resonator. By means of the input-output formalism, we derive an expression for the output modes in the dispersive regime of operation. It includes the effect of magnetic anisotropy, which makes different spin transitions addressable. We find that the measurement of the cavity transmission allows us to uniquely determine the spin state of the qudits. We discuss, from an effective Hamiltonian perspective, the conditions under which the qudit readout is a nondemolition measurement and consider possible experimental protocols to perform it. Finally, we illustrate our results with simulations performed for realistic models of existing magnetic molecules.

## Experimental validation of the Kibble-Zurek mechanism on a digital quantum computer

The Kibble-Zurek mechanism (KZM) captures the essential physics of nonequilibrium quantum phase transitions with symmetry breaking. KZM predicts a universal scaling power law for the defect density which is fully determined by the system’s critical exponents at equilibrium and the quenching rate. We experimentally tested the KZM for the simplest quantum case, a single qubit under the Landau-Zener evolution, on an open access IBM quantum computer (IBM-Q). We find that for this simple one-qubit model, experimental data validates the central KZM assumption of the adiabatic-impulse approximation for a well isolated qubit. Furthermore, we report on extensive IBM-Q experiments on individual qubits embedded in different circuit environments and topologies, separately elucidating the role of crosstalk between qubits and the increasing decoherence effects associated with the quantum circuit depth on the KZM predictions. Our results strongly suggest that increasing circuit depth acts as a decoherence source, producing a rapid deviation of experimental data from theoretical unitary predictions.

## Harnessing nonadiabatic excitations promoted by a quantum critical point: Quantum battery and spin squeezing

## Locality of spontaneous symmetry breaking and universal spacing distribution of topological defects formed across a phase transition

The crossing of a continuous phase transition results in the formation of topological defects with a density predicted by the Kibble-Zurek mechanism (KZM). We characterize the spatial distribution of pointlike topological defects in the resulting nonequilibrium state and model it using a Poisson point process in arbitrary spatial dimensions with KZM density. Numerical simulations in a one-dimensional ϕ4 theory unveil short-distance defect-defect corrections stemming from the kink excluded volume, while in two spatial dimensions, our model accurately describes the vortex spacing distribution in a strongly coupled superconductor indicating the suppression of defect-defect spatial correlations.

## Multiqudit interactions in molecular spins

We study photon-mediated interactions between molecular spin qudits in the dispersive regime of operation. We derive from a microscopic model the effective interaction between molecular spins, including their crystal field anisotropy (i.e., the presence of non-linear spin terms) and their multi-level structure. Finally, we calculate the long time dynamics for a pair of interacting molecular spins using the method of multiple scales analysis. This allows to find the set of 2-qudit gates that can be realized for a specific choice of molecular spins and to determine the time required for their implementation. Our results are relevant for the implementation of logical gates in general systems of qudits with unequally spaced levels or to determine an adequate computational subspace to encode and process the information.

## Phase Diagram of 1+1D Abelian-Higgs Model and Its Critical Point

We determine the phase diagram of the Abelian-Higgs model in one spatial dimension and time (1+ 1 D) on a lattice. We identify a line of first order phase transitions separating the Higgs region from the confined one. This line terminates in a quantum critical point above which the two regions are connected by a smooth crossover. We analyze the critical point and find compelling evidence for its description as the product of two noninteracting systems: a massless free fermion and a massless free boson. However, we find also some surprising results that cannot be explained by our simple picture, suggesting this newly discovered critical point is an unusual one.

## Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations

In this work we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schrödinger-type, Hamiltonian partial differential equations (PDEs), using space-efficient variational circuits, including the symmetries of the problem, and global and gradient-based optimizers. We use this algorithm to benchmark the performance of the representation techniques by means of the computation of the ground state in three PDEs, i.e., the one-dimensional quantum harmonic oscillator and the transmon and flux qubits, studying how they would perform in ideal and near-term quantum computers. With the Fourier methods developed here, we obtain low infidelities of order 10^{−4}–10^{−5} using only three to four qubits, demonstrating the high compression of information in a quantum computer. Practical fidelities are limited by the noise and the errors of the evaluation of the cost function in real computers, but they can also be improved through error mitigation techniques.

## Quenches to the critical point of the three-state Potts model: Matrix product state simulations and conformal field theory

Conformal field theories (CFTs) have been used extensively to understand the physics of critical lattice models at equilibrium. However, the applicability of CFT calculations to the behavior of the lattice systems in the out-of-equilibrium setting is not entirely understood. In this work, we compare the CFT results of the evolution of the entanglement spectrum after a quantum quench with numerical calculations of the entanglement spectrum of the three-state Potts model using matrix product state simulations. Our results lead us to conjecture that CFT does not describe the entanglement spectrum of the three-state Potts model at long times, contrary to what happens in the Ising model. We thus numerically simulate the out-of-equilibrium behavior of the Potts model according to the CFT protocol, i.e., by taking a particular product state and “cooling” it, then quenching to the critical point and finding that, in this case, the entanglement spectrum is indeed described by the CFT at long times.

## Role of boundary conditions in the full counting statistics of topological defects after crossing a continuous phase transition

In a scenario of spontaneous symmetry breaking in finite time, topological defects are generated at a density that scales with the driving time according to the Kibble-Zurek mechanism (KZM). Signatures of universality beyond the KZM have recently been unveiled: The number distribution of topological defects has been shown to follow a binomial distribution, in which all cumulants inherit the universal power-law scaling with the quench rate, with cumulant rations being constant. In this work, we analyze the role of boundary conditions in the statistics of topological defects. In particular, we consider a lattice system with nearest-neighbor interactions subject to soft antiperiodic, open, and periodic boundary conditions implemented by an energy penalty term. We show that for fast and moderate quenches, the cumulants of the kink number distribution present a universal scaling with the quench rate that is independent of the boundary conditions except for an additive term, which becomes prominent in the limit of slow quenches, leading to the breaking of power-law behavior. We test our theoretical predictions with a one-dimensional scalar theory on a lattice.

## Topology detection in cavity QED

We explore the physics of topological lattice models immersed in c-QED architectures for arbitrary coupling strength with the photon field. We propose the use of the cavity transmission as a topological marker and study its behavior. For this, we develop an approach combining the input–output formalism with a Mean-Field plus fluctuations description of the setup. We illustrate our results with the specific case of a fermionic Su–Schrieffer–Heeger (SSH) chain coupled to a single-mode cavity. Our findings confirm that the cavity can indeed act as a quantum sensor for topological phases, where the initial state preparation plays a crucial role. Additionally, we discuss the persistence of topological features when the coupling strength increases, in terms of an effective Hamiltonian, and calculate the entanglement entropy. Our approach can be applied to other fermionic systems, opening a route to the characterization of their topological properties in terms of experimental observables.

## Ultrastrong capcitive coupling of flux qubits

A flux qubit can interact strongly when it is capacitively coupled to other circuit elements. This interaction can be separated into two parts, one acting on the qubit subspaces and one in which excited states mediate the interaction. The first term dominates the interaction between the flux qubit and an LC-resonator, leading to ultrastrong couplings of the form σ_y(a+a†), which complement the inductive iσ_x(a†−a) coupling. However, when coupling two flux qubits capacitively, all terms need to be taken into account, leading to complex nonstoquastic ultrastrong interaction of types σ_yσ_y, σ_zσ_z, and σ_xσ_x. Our theory explains all these interactions, describing them in terms of general circuit properties—coupling capacitances, qubit gaps, inductive, Josephson and capacitive energies—that apply to a wide variety of circuits and flux qubit designs.