Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices

Suppressing nonperturbative gauge errors in the thermodynamic limit using local pseudogenerators

With recent progress in quantum simulations of lattice-gauge theories, it is becoming a pressing question how to reliably protect the gauge symmetry that defines such models. Recently, an experimentally feasible gauge-protection scheme has been proposed that is based on the concept of a local pseudogenerator, which is required to act identically to the full gauge-symmetry generator in the target gauge sector, but not necessarily outside of it. The scheme has been analytically and numerically shown to reliably stabilize lattice gauge theories in the presence of perturbative errors on finite-size analog quantum-simulation devices. In this work, through uniform matrix product state calculations, we demonstrate the efficacy of this scheme for nonperturbative errors in analog quantum simulators up to all accessible evolution times in the thermodynamic limit, where it is a priori neither established nor expected that this scheme will succeed. Our results indicate the presence of an emergent gauge symmetry in an adjusted gauge theory even in the thermodynamic limit, which is beyond our analytic predictions. Additionally, we show through quantum circuit model calculations that gauge protection with local pseudogenerators also successfully suppresses gauge violations on finite quantum computers that discretize time through Trotterization. Our results firm up the robustness and feasibility of the local pseudogenerator as a viable tool for enforcing gauge invariance in modern quantum simulators and noisy intermediate-scale quantum devices.

Mean Field Approaches to Lattice Gauge Theories: A Review

Due to their broad applicability, gauge theories (GTs) play a crucial role in various areas of physics, from high-energy physics to condensed matter. Their formulations on lattices, lattice gauge theories (LGTs), can be studied, among many other methods, with tools coming from statistical mechanics lattice models, such as mean field methods, which are often used to provide approximate results. Applying these methods to LGTs requires particular attention due to the intrinsic local nature of gauge symmetry, how it is reflected in the variables used to formulate the theory, and the breaking of gauge invariance when approximations are introduced. This issue has been addressed over the decades in the literature, yielding different conclusions depending on the formulation of the theory under consideration. In this article, we focus on the mean field theoretical approach to the analysis of GTs and LGTs, connecting both older and more recent results that, to the best of our knowledge, have not been compared in a pedagogical manner. After a brief overview of mean field theory in statistical mechanics and many-body systems, we examine its application to pure LGTs with a generic compact gauge group. Finally, we review the existing literature on the subject, discussing the results obtained so far and their dependence on the formulation of the theory.

Simulation of a Rohksar-Kivelson ladder on a NISQ device

We present a quantum-classical algorithm to study the dynamics of the Rohksar-Kivelson plaquette ladder on NISQ devices. We show that complexity is largely reduced using gauge invariance, additional symmetries, and a crucial property associated to how plaquettes are blocked against ring-exchange in the ladder geometry. This allows for an efficient simulation of sizable plaquette ladders with a small number of qubits, well suited for the capabilities of present NISQ devices. We illustrate the procedure for ladders with simulation of up to 8 plaquettes in an IBM-Q machine, employing scaled quantum gates.

Quantum Simulation of SU(3) Lattice Yang-Mills Theory at Leading Order in Large-N_{c} Expansion

Quantum simulations of the dynamics of QCD have been limited by the complexities of mapping the continuous gauge fields onto quantum computers. By parametrizing the gauge invariant Hilbert space in terms of plaquette degrees of freedom, we show how the Hilbert space and interactions can be expanded in inverse powers of N_{c}. At leading order in this expansion, the Hamiltonian simplifies dramatically, both in the required size of the Hilbert space as well as the type of interactions involved. Adding a truncation of the resulting Hilbert space in terms of local energy states we give explicit constructions that allow simple representations of SU(3) gauge fields on qubits and qutrits. This formulation allows a simulation of the real time dynamics of a SU(3) lattice gauge theory on a 5×5 and 8×8 lattice on ibm_torino with a CNOT depth of 113.

Temporal multilayer structures in discrete physical systems towards arbitrary-dimensional non-Abelian Aharonov-Bohm interferences

Temporal modulation recently draws great attentions in wave manipulations, with which one can introduce the concept of temporal multilayer structure, a temporal counterpart of spatially multilayer configurations. This kind of multilayer structure holds temporal interfaces in the time domain, which provides additional flexibility in temporal operations. Here we take this opportunity and propose to simulate a non-Abelian gauge field with a temporal multilayer structure in the discrete physical system. Two basic temporal operations, i.e., the folding/unfolding operation and the phase shift operation are used to design such a temporal multilayer structure, which hence can support noncommutative operations to realize the non-Abelian Aharonov-Bohm interference in the time domain. A two-/three-dimensional non-Abelian gauge field can be built, which may be further extended to higher dimensions. Our work therefore provides a unique platform enabling generalization of non-Abelian physics to arbitrary dimensions and offers a method for wave manipulations with temporal band engineering.

Local Readout and Control of Current and Kinetic Energy Operators in Optical Lattices

Quantum gas microscopes have revolutionized quantum simulations with ultracold atoms, allowing one to measure local observables and snapshots of quantum states. However, measurements so far were mostly carried out in the occupation basis. Here, we demonstrate how all kinetic operators, such as kinetic energy or current operators, can be measured and manipulated with single-bond resolution. Beyond simple expectation values of these observables, the single-shot measurements allow one to access full counting statistics and complex correlation functions. Our work paves the way for the implementation of efficient quantum state tomography and hybrid quantum computing protocols for itinerant particles on a lattice. In addition, we demonstrate how site-resolved programmable potentials enable a spatially selective, parallel readout in different bases as well as the engineering of arbitrary initial states.

Quantum advantage and stability to errors in analogue quantum simulators

Several quantum hardware platforms, while being unable to perform fully fault-tolerant quantum computation, can still be operated as analogue quantum simulators for addressing many-body problems. However, due to the presence of errors, it is not clear to what extent those devices can provide us with an advantage with respect to classical computers. In this work, we make progress on this problem for noisy analogue quantum simulators computing physically relevant properties of many-body systems both in equilibrium and undergoing dynamics. We first formulate a system-size independent notion of stability against extensive errors, which we prove for Gaussian fermion models, as well as for a restricted class of spin systems. Remarkably, for the Gaussian fermion models, our analysis shows the stability of critical models which have long-range correlations. Furthermore, we analyze how this stability may lead to a quantum advantage, for the problem of computing the thermodynamic limit of many-body models, in the presence of a constant error rate and without any explicit error correction.

High-Energy Collision of Quarks and Mesons in the Schwinger Model: From Tensor Networks to Circuit QED

With the aim of studying nonperturbative out-of-equilibrium dynamics of high-energy particle collisions on quantum simulators, we investigate the scattering dynamics of lattice quantum electrodynamics in 1+1 dimensions. Working in the bosonized formulation of the model and in the thermodynamic limit, we use uniform-matrix-product-state tensor networks to construct multiparticle wave-packet states, evolve them in time, and detect outgoing particles post collision. This facilitates the numerical simulation of scattering experiments in both confined and deconfined regimes of the model at different energies, giving rise to rich phenomenology, including inelastic production of quark and meson states, meson disintegration, and dynamical string formation and breaking. We obtain elastic and inelastic scattering cross sections, together with time-resolved momentum and position distributions of the outgoing particles. Furthermore, we propose an analog circuit-QED implementation of the scattering process that is native to the platform, requires minimal ingredients and approximations, and enables practical schemes for particle wave-packet preparation and evolution. This study highlights the role of classical and quantum simulation in enhancing our understanding of scattering processes in quantum field theories in real time.

Dynamical Localization Transition of String Breaking in Quantum Spin Chains

The fission of a string connecting two charges is an astounding phenomenon in confining gauge theories. The dynamics of this process have been studied intensively in recent years, with plenty of numerical results yielding a dichotomy: the confining string can decay relatively fast or persist up to extremely long times. Here, we put forward a dynamical localization transition as the mechanism underlying this dichotomy. To this end, we derive an effective string breaking description in the light-meson sector of a confined spin chain and show that the problem can be regarded as a dynamical localization transition in Fock space. Fast and suppressed string breaking dynamics are identified with delocalized and localized behavior, respectively. We then provide a further reduction of the dynamical string breaking problem onto a quantum impurity model, where the string is represented as an "impurity" immersed in a meson bath. It is shown that this model features a localization-delocalization transition, giving a general and simple physical basis to understand the qualitatively distinct string breaking regimes. These findings are directly relevant for a wider class of confining lattice models in any dimension and could be realized on present-day Rydberg quantum simulators.

Towards Quantum Computing Phase Diagrams of Gauge Theories with Thermal Pure Quantum States

The phase diagram of strong interactions in nature at finite temperature and chemical potential remains largely theoretically unexplored due to inadequacy of Monte-Carlo-based computational techniques in overcoming a sign problem. Quantum computing offers a sign-problem-free approach, but evaluating thermal expectation values is generally resource intensive on quantum computers. To facilitate thermodynamic studies of gauge theories, we propose a generalization of the thermal-pure-quantum-state formulation of statistical mechanics applied to constrained gauge-theory dynamics, and numerically demonstrate that the phase diagram of a simple low-dimensional gauge theory is robustly determined using this approach, including mapping a chiral phase transition in the model at finite temperature and chemical potential. Quantum algorithms, resource requirements, and algorithmic and hardware error analysis are further discussed to motivate future implementations. Thermal pure quantum states, therefore, may present a suitable candidate for efficient thermal simulations of gauge theories in the era of quantum computing.

Interrelated Thermalization and Quantum Criticality in a Lattice Gauge Simulator

Gauge theory and thermalization are both topics of essential importance for modern quantum science and technology. The recently realized atomic quantum simulator for lattice gauge theories provides a unique opportunity for studying thermalization in gauge theory, in which theoretical studies have shown that quantum thermalization can signal the quantum phase transition. Nevertheless, the experimental study remains a challenge to accurately determine the critical point and controllably explore the thermalization dynamics due to the lack of techniques for locally manipulating and detecting matter and gauge fields. We report an experimental investigation of the quantum criticality in the lattice gauge theory from both equilibrium and nonequilibrium thermalization perspectives, with the help of the single-site addressing and atom-number-resolved detection capabilities. We accurately determine the quantum critical point and observe that the Néel state thermalizes only in the critical regime. This result manifests the interplay between quantum many-body scars, quantum criticality, and symmetry breaking.

Non-Fermi Liquids from Kinetic Constraints in Tilted Optical Lattices

We study Fermi-Hubbard models with kinetically constrained dynamics that conserves both total particle number and total center of mass, a situation that arises when interacting fermions are placed in strongly tilted optical lattices. Through a combination of analytics and numerics, we show how the kinetic constraints stabilize an exotic non-Fermi liquid phase described by fermions coupled to a gapless bosonic field, which in several respects mimics a dynamical gauge field. This offers a novel route towards the study of non-Fermi liquid phases in the precision environments afforded by ultracold atom platforms.

Dynamical Gauge Fields with Bosonic Codes

The quantum simulation of dynamical gauge field theories offers the opportunity to study complex high-energy physics with controllable low-energy devices. For quantum computation, bosonic codes promise robust error correction that exploits multiparticle redundancy in bosons. Here, we demonstrate how bosonic codes can be used to simulate dynamical gauge fields. We encode both matter and dynamical gauge fields in a network of resonators that are coupled via three-wave mixing. The mapping to a Z_{2} dynamical lattice gauge theory is established when the gauge resonators operate as Schrödinger cat states. We explore the optimal conditions under which the system preserves the required gauge symmetries. Our findings promote realizing high-energy models using bosonic codes.

Prethermalization in one-dimensional quantum many-body systems with confinement

Unconventional nonequilibrium phases with restricted correlation spreading and slow entanglement growth have been proposed to emerge in systems with confined excitations, calling their thermalization dynamics into question. Here, we show that in confined systems the thermalization dynamics after a quantum quench instead exhibits multiple stages with well separated time scales. As an example, we consider the confined Ising spin chain, in which domain walls in the ordered phase form bound states reminiscent of mesons. The system first relaxes towards a prethermal state, described by a Gibbs ensemble with conserved meson number. The prethermal state arises from rare events in which mesons are created in close vicinity, leading to an avalanche of scattering events. Only at much later times a true thermal equilibrium is achieved in which the meson number conservation is violated by a mechanism akin to the Schwinger effect. The discussed prethermalization dynamics is directly relevant to generic one-dimensional, many-body systems with confined excitations.

Hardware Efficient Quantum Simulation of Non-Abelian Gauge Theories with Qudits on Rydberg Platforms

Non-Abelian gauge theories underlie our understanding of fundamental forces in nature, and developing tailored quantum hardware and algorithms to simulate them is an outstanding challenge in the rapidly evolving field of quantum simulation. Here we take an approach where gauge fields, discretized in spacetime, are represented by qudits and are time evolved in Trotter steps with multiqudit quantum gates. This maps naturally and hardware efficiently to an architecture based on Rydberg tweezer arrays, where long-lived internal atomic states represent qudits, and the required quantum gates are performed as holonomic operations supported by a Rydberg blockade mechanism. We illustrate our proposal for a minimal digitization of SU(2) gauge fields, demonstrating a significant reduction in circuit depth and gate errors in comparison to a traditional qubit-based approach, which puts simulations of non-Abelian gauge theories within reach of NISQ devices.

Sampling rare conformational transitions with a quantum computer

Structural rearrangements play a central role in the organization and function of complex biomolecular systems. In principle, Molecular Dynamics (MD) simulations enable us to investigate these thermally activated processes with an atomic level of resolution. In practice, an exponentially large fraction of computational resources must be invested to simulate thermal fluctuations in metastable states. Path sampling methods focus the computational power on sampling the rare transitions between states. One of their outstanding limitations is to efficiently generate paths that visit significantly different regions of the conformational space. To overcome this issue, we introduce a new algorithm for MD simulations that integrates machine learning and quantum computing. First, using functional integral methods, we derive a rigorous low-resolution spatially coarse-grained representation of the system's dynamics, based on a small set of molecular configurations explored with machine learning. Then, we use a quantum annealer to sample the transition paths of this low-resolution theory. We provide a proof-of-concept application by simulating a benchmark conformational transition with all-atom resolution on the D-Wave quantum computer. By exploiting the unique features of quantum annealing, we generate uncorrelated trajectories at every iteration, thus addressing one of the challenges of path sampling. Once larger quantum machines will be available, the interplay between quantum and classical resources may emerge as a new paradigm of high-performance scientific computing. In this work, we provide a platform to implement this integrated scheme in the field of molecular simulations.

Tuning Long-Range Fermion-Mediated Interactions in Cold-Atom Quantum Simulators

Engineering long-range interactions in cold-atom quantum simulators can lead to exotic quantum many-body behavior. Fermionic atoms in ultracold atomic mixtures can act as mediators, giving rise to long-range Ruderman-Kittel-Kasuya-Yosida-type interactions characterized by the dimensionality and density of the fermionic gas. Here, we propose several tuning knobs, accessible in current experimental platforms, that allow one to further control the range and shape of the mediated interactions, extending the existing quantum simulation toolbox. In particular, we include an additional optical lattice for the fermionic mediator, as well as anisotropic traps to change its dimensionality in a continuous manner. This allows us to interpolate between power-law and exponential decays, introducing an effective cutoff for the interaction range, as well as to tune the relative interaction strengths at different distances. Finally, we show how our approach allows one to investigate frustrated regimes that were not previously accessible, where symmetry-protected topological phases as well as chiral spin liquids emerge.

Realizing a 1D topological gauge theory in an optically dressed BEC

Topological gauge theories describe the low-energy properties of certain strongly correlated quantum systems through effective weakly interacting models^1,2. A prime example is the Chern-Simons theory of fractional quantum Hall states, where anyonic excitations emerge from the coupling between weakly interacting matter particles and a density-dependent gauge field³. Although in traditional solid-state platforms such gauge theories are only convenient theoretical constructions, engineered quantum systems enable their direct implementation and provide a fertile playground to investigate their phenomenology without the need for strong interactions⁴. Here, we report the quantum simulation of a topological gauge theory by realizing a one-dimensional reduction of the Chern-Simons theory (the chiral BF theory^5-7) in a Bose-Einstein condensate. Using the local conservation laws of the theory, we eliminate the gauge degrees of freedom in favour of chiral matter interactions^8-11, which we engineer by synthesizing optically dressed atomic states with momentum-dependent scattering properties. This allows us to reveal the key properties of the chiral BF theory: the formation of chiral solitons and the emergence of an electric field generated by the system itself. Our results expand the scope of quantum simulation to topological gauge theories and open a route to the implementation of analogous gauge theories in higher dimensions¹².

Design and construction of a quantum matter synthesizer

The quantum matter synthesizer (QMS) is a new quantum simulation platform in which individual particles in a lattice can be resolved and re-arranged into arbitrary patterns. The ability to spatially manipulate ultracold atoms and control their tunneling and interactions at the single-particle level allows full control of a many-body quantum system. We present the design and characterization of the QMS, which integrates into a single ultra-stable apparatus a two-dimensional optical lattice, a moving optical tweezer array formed by a digital micromirror device, and site-resolved atomic imaging. We demonstrate excellent mechanical stability between the lattice and tweezer array with relative fluctuations below 10 nm, diffraction-limited imaging at a resolution of 655 nm, and high-speed real-time control of the tweezer array at a 2.52 kHz refresh rate, which will be adopted to realize fast rearrangement of atoms. The QMS also features new technologies and schemes, such as nanotextured anti-reflective windows and all-optical long-distance transport of atoms.

Nonlinear localized modes in one-dimensional nanoscale dark-state optical lattices

Optical lattices (OLs) with conventional spatial periodic λ/2, formed by interfering the counterpropagating laser beams with wavelength λ, are versatile tools to study the dynamical and static properties of ultracold atoms. OLs with subwavelength spatial structure have been realized in recent quantum-gas experiment, offering new possibility for nonlinear and quantum control of ultracold atoms at the nano scale. Herein, we study theoretically and numerically the formation, property, and dynamics of matter-wave localized gap modes of Bose-Einstein condensates loaded in a one-dimensional nanoscale dark-state OL consisted of an array of optical subwavelength barriers. The nonlinear localized modes, in the forms of on- and off-site fundamental gap solitons, and dipole ones, are demonstrated; and we uncover that, counterintuitively, these modes exhibit always a cusplike (side peaks) mode even for a deeply subwavelength adiabatic lattice, contrary to the previously reported results in conventional deep OLs where the localized gap modes are highly confined in a single lattice cell. The (in)stability features of all the predicted localized modes are verified through the linear-stability analysis and direct perturbed simulations. Our predicted results are attainable in current ultracold atoms experiments with the cutting-edge technique, pushing the nonlinear control of ultracold atoms with short-period OLs as an enabling technology into subwavelength structures.

Improved Hamiltonians for Quantum Simulations of Gauge Theories

Quantum simulations of lattice gauge theories for the foreseeable future will be hampered by limited resources. The historical success of improved lattice actions in classical simulations strongly suggests that Hamiltonians with improved discretization errors will reduce quantum resources, i.e., require ≳2^{d} fewer qubits in quantum simulations for lattices with d-spatial dimensions. In this work, we consider O(a^{2})-improved Hamiltonians for pure gauge theories and design the corresponding quantum circuits for its real-time evolution in terms of primitive gates. An explicit demonstration for Z_{2} gauge theory is presented including exploratory tests using the ibm_perth device.

From Asymptotic Freedom to θ Vacua: Qubit Embeddings of the O(3) Nonlinear σ Model

Conventional lattice formulations of θ vacua in the 1+1-dimensional O(3) nonlinear sigma model suffer from a sign problem. Here, we construct the first sign-problem-free regularization for arbitrary θ. Using efficient lattice Monte Carlo algorithms, we demonstrate how a Hamiltonian model of spin-1/2 degrees of freedom on a two-dimensional spatial lattice reproduces both the infrared sector for arbitrary θ, as well as the ultraviolet physics of asymptotic freedom. Furthermore, as a model of qubits on a two-dimensional square lattice with only nearest-neighbor interactions, it is naturally suited for studying the physics of θ vacua and asymptotic freedom on near-term quantum devices. Our construction generalizes to θ vacua in all CP(N-1) models, solving a long-standing sign problem.

Large-S and Tensor-Network Methods for Strongly-Interacting Topological Insulators

The study of correlation effects in topological phases of matter can benefit from a multidisciplinary approach that combines techniques drawn from condensed matter, high-energy physics and quantum information science. In this work, we exploit these connections to study the strongly-interacting limit of certain lattice Hubbard models of topological insulators, which map onto four-Fermi quantum field theories with a Wilson-type discretisation and have been recently shown to be at reach of cold-atom quantum simulators based on synthetic spin-orbit coupling. We combine large-S and tensor-network techniques to explore the possible spontaneous symmetry-breaking phases that appear when the interactions of the topological insulators are sufficiently large. In particular, we show that varying the Wilson parameter r of the lattice discretisations leads to a novel Heisenberg–Ising compass model with critical lines that flow with the value of r.

Observing Dynamical Currents in a Non-Hermitian Momentum Lattice

We report on the experimental realization and detection of dynamical currents in a spin-textured lattice in momentum space. Collective tunneling is implemented via cavity-assisted Raman scattering of photons by a spinor Bose-Einstein condensate into an optical cavity. The photon field inducing the tunneling processes is subject to cavity dissipation, resulting in effective directional dynamics in a non-Hermitian setting. We observe that the individual tunneling events are superradiant in nature and locally resolve them in the lattice by performing real-time, frequency-resolved measurements of the leaking cavity field. The results can be extended to a regime exhibiting a cascade of currents and simultaneous coherences between multiple lattice sites, where numerical simulations provide further understanding of the dynamics. Our observations showcase dynamical tunneling in momentum-space lattices and provide prospects to realize dynamical gauge fields in driven-dissipative settings.

Stochastic dynamics of a few sodium atoms in presence of a cold potassium cloud

Single particle resolution is a requirement for numerous experimental protocols that emulate the dynamics of small systems in a bath. Here, we accurately resolve through atom counting the stochastic dynamics of a few sodium atoms in presence of a cold potassium cloud. This capability enables us to rule out the effect of inter-species interaction on sodium atom number dynamics, at very low atomic densities present in these experiments. We study the noise sources for sodium and potassium in a common framework. Thereby, we assign the detection limits to 4.3 atoms for potassium and 0.2 atoms (corresponding to 96% fidelity) for sodium. This opens possibilities for future experiments with a few atoms immersed in a quantum degenerate gas.

Emergent gauge symmetries: making symmetry as well as breaking it

Gauge symmetries play an essential role in determining the interactions of particle physics. Where do they come from? Might the gauge symmetries of the Standard Model unify in the ultraviolet or might they be emergent in the infrared, below some large scale close to the Planck scale? Emergent gauge symmetries are important in quantum many-body systems in quantum phases associated with long range entanglement and topological order, e.g. they arise in high temperature superconductors, with string-net condensation and in the A-phase of superfluid ³He. String-nets and superfluid ³He exhibit emergent properties similar to the building blocks of particle physics. Emergent gauge symmetries also play an important role in simulations of quantum field theories. This article discusses recent thinking on possible emergent gauge symmetries in particle physics, commenting also on Higgs phenomena and the vacuum energy or cosmological constant puzzle in emergent gauge systems. This article is part of the theme issue 'Quantum technologies in particle physics'.

Quantum simulation of lattice gauge theories in more than one space dimension-requirements, challenges and methods

Over recent years, the relatively young field of quantum simulation of lattice gauge theories, aiming at implementing simulators of gauge theories with quantum platforms, has gone through a rapid development process. Nowadays, it is not only of interest to the quantum information and technology communities. It is also seen as a valid tool for tackling hard, non-perturbative gauge theory problems by particle and nuclear physicists. Along the theoretical progress, nowadays more and more experiments implementing such simulators are being reported, manifesting beautiful results, but mostly on [Formula: see text] dimensional physics. In this article, we review the essential ingredients and requirements of lattice gauge theories in more dimensions and discuss their meanings, the challenges they pose and how they could be dealt with, potentially aiming at the next steps of this field towards simulating challenging physical problems in analogue, or analogue-digital ways. This article is part of the theme issue 'Quantum technologies in particle physics'.

Loop-free tensor networks for high-energy physics

This brief review introduces the reader to tensor network methods, a powerful theoretical and numerical paradigm spawning from condensed matter physics and quantum information science and increasingly exploited in different fields of research, from artificial intelligence to quantum chemistry. Here, we specialize our presentation on the application of loop-free tensor network methods to the study of high-energy physics problems and, in particular, to the study of lattice gauge theories where tensor networks can be applied in regimes where Monte Carlo methods are hindered by the sign problem. This article is part of the theme issue 'Quantum technologies in particle physics'.

Cold atoms meet lattice gauge theory

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 [Formula: see text] 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. This article is part of the theme issue 'Quantum technologies in particle physics'.

From quantum link models to D-theory: a resource efficient framework for the quantum simulation and computation of gauge theories

Quantum link models provide an extension of Wilson's lattice gauge theory in which the link Hilbert space is finite-dimensional and corresponds to a representation of an embedding algebra. In contrast to Wilson's parallel transporters, quantum links are intrinsically quantum degrees of freedom. In D-theory, these discrete variables undergo dimensional reduction, thus giving rise to asymptotically free theories. In this way [Formula: see text] [Formula: see text] models emerge by dimensional reduction from [Formula: see text] [Formula: see text] quantum spin ladders, the [Formula: see text] confining [Formula: see text] gauge theory emerges from the Abelian Coulomb phase of a [Formula: see text] quantum link model, and [Formula: see text] QCD arises from a non-Abelian Coulomb phase of a [Formula: see text] [Formula: see text] quantum link model, with chiral quarks arising naturally as domain wall fermions. Thanks to their finite-dimensional Hilbert space and their economical mechanism of reaching the continuum limit by dimensional reduction, quantum link models provide a resource efficient framework for the quantum simulation and computation of gauge theories. This article is part of the theme issue 'Quantum technologies in particle physics'.

Domain-wall dynamics in Bose-Einstein condensates with synthetic gauge fields

Interactions in many-body physical systems, from condensed matter to high-energy physics, lead to the emergence of exotic particles. Examples are mesons in quantum chromodynamics and composite fermions in fractional quantum Hall systems, which arise from the dynamical coupling between matter and gauge fields^1,2. The challenge of understanding the complexity of matter-gauge interaction can be aided by quantum simulations, for which ultracold atoms offer a versatile platform via the creation of artificial gauge fields. An important step towards simulating the physics of exotic emergent particles is the synthesis of artificial gauge fields whose state depends dynamically on the presence of matter. Here we demonstrate deterministic formation of domain walls in a stable Bose-Einstein condensate with a gauge field that is determined by the atomic density. The density-dependent gauge field is created by simultaneous modulations of an optical lattice potential and interatomic interactions, and results in domains of atoms condensed into two different momenta. Modelling the domain walls as elementary excitations, we find that the domain walls respond to synthetic electric field with a charge-to-mass ratio larger than and opposite to that of the bare atoms. Our work offers promising prospects to simulate the dynamics and interactions of previously undescribed excitations in quantum systems with dynamical gauge fields.

Qubit Regularization and Qubit Embedding Algebras

Qubit regularization is a procedure to regularize the infinite dimensional local Hilbert space of bosonic fields to a finite dimensional one, which is a crucial step when trying to simulate lattice quantum field theories on a quantum computer. When the qubit-regularized lattice quantum fields preserve important symmetries of the original theory, qubit regularization naturally enforces certain algebraic structures on these quantum fields. We introduce the concept of qubit embedding algebras (QEAs) to characterize this algebraic structure associated with a qubit regularization scheme. We show a systematic procedure to derive QEAs for the (N) lattice spin models and the SU(N) lattice gauge theories. While some of the QEAs we find were discovered earlier in the context of the D-theory approach, our method shows that QEAs are far richer. A more complete understanding of the QEAs could be helpful in recovering the fixed points of the desired quantum field theories.

Photon-Mediated Stroboscopic Quantum Simulation of a Z_{2} Lattice Gauge Theory

Quantum simulation of lattice gauge theories, aiming at tackling nonperturbative particle and condensed matter physics, has recently received a lot of interest and attention, resulting in many theoretical proposals as well as several experimental implementations. One of the current challenges is to go beyond 1+1 dimensions, where four-body (plaquette) interactions, not contained naturally in quantum simulating devices, appear. In this Letter, we propose a method to obtain them based on a combination of stroboscopic optical atomic control and the nonlocal photon-mediated interactions appearing in nanophotonic or cavity QED setups. We illustrate the method for a Z_{2} lattice gauge theory. We also show how to prepare the ground state and measure Wilson loops using state-of-the-art techniques in atomic physics.

SU(2) hadrons on a quantum computer via a variational approach

Quantum computers have the potential to create important new opportunities for ongoing essential research on gauge theories. They can provide simulations that are unattainable on classical computers such as sign-problem afflicted models or time evolutions. In this work, we variationally prepare the low-lying eigenstates of a non-Abelian gauge theory with dynamically coupled matter on a quantum computer. This enables the observation of hadrons and the calculation of their associated masses. The SU(2) gauge group considered here represents an important first step towards ultimately studying quantum chromodynamics, the theory that describes the properties of protons, neutrons and other hadrons. Our calculations on an IBM superconducting platform utilize a variational quantum eigensolver to study both meson and baryon states, hadrons which have never been seen in a non-Abelian simulation on a quantum computer. We develop a hybrid resource-efficient approach by combining classical and quantum computing, that not only allows the study of an SU(2) gauge theory with dynamical matter fields on present-day quantum hardware, but further lays out the premises for future quantum simulations that will address currently unanswered questions in particle and nuclear physics.

Confinement and Mott Transitions of Dynamical Charges in One-Dimensional Lattice Gauge Theories

Confinement is an ubiquitous phenomenon when matter couples to gauge fields, which manifests itself in a linear string potential between two static charges. Although gauge fields can be integrated out in one dimension, they can mediate nonlocal interactions which in turn influence the paradigmatic Luttinger liquid properties. However, when the charges become dynamical and their densities finite, understanding confinement becomes challenging. Here we show that confinement in 1D Z_{2} lattice gauge theories, with dynamical matter fields and arbitrary densities, is related to translational symmetry breaking in a nonlocal basis. The exact transformation to this string-length basis leads us to an exact mapping of Luttinger parameters reminiscent of a Luther-Emery rescaling. We include the effects of local, but beyond contact, interactions between the matter particles, and show that confined mesons can form a Mott-insulating state when the deconfined charges cannot. While the transition to the Mott state cannot be detected in the Green's function of the charges, we show that the metallic state is characterized by hidden off-diagonal quasi-long-range order. Our predictions provide new insights to the physics of confinement of dynamical charges, and can be experimentally addressed in Rydberg-dressed quantum gases in optical lattices.

Breaking of Gauge Symmetry in Lattice Gauge Theories

We study perturbations that break gauge symmetries in lattice gauge theories. As a paradigmatic model, we consider the three-dimensional Abelian-Higgs (AH) model with an N-component scalar field and a noncompact gauge field, which is invariant under U(1) gauge and SU(N) transformations. We consider gauge-symmetry breaking perturbations that are quadratic in the gauge field, such as a photon mass term and determine their effect on the critical behavior of the gauge-invariant model, focusing mainly on the continuous transitions associated with the charged fixed point of the AH field theory. We discuss their relevance and compute the (gauge-dependent) exponents that parametrize the departure from the critical behavior (continuum limit) of the gauge-invariant model. We also address the critical behavior of lattice AH models with broken gauge symmetry, showing an effective enlargement of the global symmetry, from U(N) to O(2N), which reflects a peculiar cyclic renormalization-group flow in the space of the lattice AH parameters and of the photon mass.

Lattice gauge theories in the presence of a linear gauge-symmetry breaking

We study the effects of gauge-symmetry breaking (GSB) perturbations in three-dimensional lattice gauge theories with scalar fields. We study this issue at transitions in which gauge correlations are not critical and the gauge symmetry only selects the gauge-invariant scalar degrees of freedom that become critical. A paradigmatic model in which this behavior is realized is the lattice CP^{1} model or, more generally, the lattice Abelian-Higgs model with two-component complex scalar fields and compact gauge fields. We consider this model in the presence of a linear GSB perturbation. The gauge symmetry turns out to be quite robust with respect to the GSB perturbation: the continuum limit is gauge invariant also in the presence of a finite small GSB term. We also determine the phase diagram of the model. It has one disordered phase and two phases that are tensor and vector ordered, respectively. They are separated by continuous transition lines, which belong to the O(3), O(4), and O(2) vector universality classes, and which meet at a multicritical point. We remark that the behavior at the CP^{1} gauge-symmetric critical point substantially differs from that at transitions in which gauge correlations become critical, for instance at transitions in the noncompact lattice Abelian-Higgs model that are controlled by the charged fixed point: in this case, the behavior is extremely sensitive to GSB perturbations.

Polymer Physics by Quantum Computing

Sampling equilibrium ensembles of dense polymer mixtures is a paradigmatically hard problem in computational physics, even in lattice-based models. Here, we develop a formalism based on interacting binary tensors that allows for tackling this problem using quantum annealing machines. Our approach is general in that properties such as self-avoidance, branching, and looping can all be specified in terms of quadratic interactions of the tensors. Microstates' realizations of different lattice polymer ensembles are then seamlessly generated by solving suitable discrete energy-minimization problems. This approach enables us to capitalize on the strengths of quantum annealing machines, as we demonstrate by sampling polymer mixtures from low to high densities, using the D-Wave quantum annealer. Our systematic approach offers a promising avenue to harness the rapid development of quantum machines for sampling discrete models of filamentous soft-matter systems.

Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks

Gauge theories are of paramount importance in our understanding of fundamental constituents of matter and their interactions. However, the complete characterization of their phase diagrams and the full understanding of non-perturbative effects are still debated, especially at finite charge density, mostly due to the sign-problem affecting Monte Carlo numerical simulations. Here, we report the Tensor Network simulation of a three dimensional lattice gauge theory in the Hamiltonian formulation including dynamical matter: Using this sign-problem-free method, we simulate the ground states of a compact Quantum Electrodynamics at zero and finite charge densities, and address fundamental questions such as the characterization of collective phases of the model, the presence of a confining phase at large gauge coupling, and the study of charge-screening effects.

Confinement and entanglement dynamics on a digital quantum computer

Confinement describes the phenomenon when the attraction between two particles grows with their distance, most prominently found in quantum chromodynamics (QCD) between quarks. In condensed matter physics, confinement can appear in quantum spin chains, for example, in the one dimensional transverse field Ising model (TFIM) with an additional longitudinal field, famously observed in the quantum material cobalt niobate or in optical lattices. Here, we establish that state-of-the-art quantum computers have reached capabilities to simulate confinement physics in spin chains. We report quantitative confinement signatures of the TFIM on an IBM quantum computer observed via two distinct velocities for information propagation from domain walls and their mesonic bound states. We also find the confinement induced slow down of entanglement spreading by implementing randomized measurement protocols for the second order Rényi entanglement entropy. Our results are a crucial step for probing non-perturbative interacting quantum phenomena on digital quantum computers beyond the capabilities of classical hardware.

Vector solitons in nonlocal optical media with pseudo spin-orbit-coupling

We numerically investigate the existence and stability of nonlocal vector solitons with pseudo spin-orbit-coupling (SOC). The pseudo SOC is realized by a framework based on the spatial-domain copropagation of two beams with mutually orthogonal polarizations and opposite transverse components of the carrier wave vectors in nonlocal optical media. The numerical results show that there are two kinds of solutions for vector solitons, one is central symmetric, and the other is noncentral symmetric. The solitons may exist below a certain threshold value of the effective SOC strength in the system.

Observation of gauge invariance in a 71-site Bose-Hubbard quantum simulator

The modern description of elementary particles, as formulated in the standard model of particle physics, is built on gauge theories¹. Gauge theories implement fundamental laws of physics by local symmetry constraints. For example, in quantum electrodynamics Gauss's law introduces an intrinsic local relation between charged matter and electromagnetic fields, which protects many salient physical properties, including massless photons and a long-ranged Coulomb law. Solving gauge theories using classical computers is an extremely arduous task², which has stimulated an effort to simulate gauge-theory dynamics in microscopically engineered quantum devices^3-6. Previous achievements implemented density-dependent Peierls phases without defining a local symmetry^7,8, realized mappings onto effective models to integrate out either matter or electric fields^9-12, or were limited to very small systems^13-16. However, the essential gauge symmetry has not been observed experimentally. Here we report the quantum simulation of an extended U(1) lattice gauge theory, and experimentally quantify the gauge invariance in a many-body system comprising matter and gauge fields. These fields are realized in defect-free arrays of bosonic atoms in an optical superlattice of 71 sites. We demonstrate full tunability of the model parameters and benchmark the matter-gauge interactions by sweeping across a quantum phase transition. Using high-fidelity manipulation techniques, we measure the degree to which Gauss's law is violated by extracting probabilities of locally gauge-invariant states from correlated atom occupations. Our work provides a way to explore gauge symmetry in the interplay of fundamental particles using controllable large-scale quantum simulators.

Reliability of Lattice Gauge Theories

Currently, there are intense experimental efforts to realize lattice gauge theories in quantum simulators. Except for specific models, however, practical quantum simulators can never be fine-tuned to perfect local gauge invariance. There is thus a strong need for a rigorous understanding of gauge-invariance violation and how to reliably protect against it. As we show through analytic and numerical evidence, in the presence of a gauge invariance-breaking term the gauge violation accumulates only perturbatively at short times before proliferating only at very long times. This proliferation can be suppressed up to infinite times by energetically penalizing processes that drive the dynamics away from the initial gauge-invariant sector. Our results provide a theoretical basis that highlights a surprising robustness of gauge-theory quantum simulators.

Confined Phases of One-Dimensional Spinless Fermions Coupled to Z_{2} Gauge Theory

We investigate a quantum many-body lattice system of one-dimensional spinless fermions interacting with a dynamical Z_{2} gauge field. The gauge field mediates long-range attraction between fermions resulting in their confinement into bosonic dimers. At strong coupling we develop an exactly solvable effective theory of such dimers with emergent constraints. Even at generic coupling and fermion density, the model can be rewritten as a local spin chain. Using the density matrix renormalization group the system is shown to form a Luttinger liquid, indicating the emergence of fractionalized excitations despite the confinement of lattice fermions. In a finite chain we observe the doubling of the period of Friedel oscillations which paves the way towards experimental detection of confinement in this system. We discuss the possibility of a Mott phase at the commensurate filling 2/3.

A scalable realization of local U(1) gauge invariance in cold atomic mixtures

In the fundamental laws of physics, gauge fields mediate the interaction between charged particles. An example is the quantum theory of electrons interacting with the electromagnetic field, based on U(1) gauge symmetry. Solving such gauge theories is in general a hard problem for classical computational techniques. Although quantum computers suggest a way forward, large-scale digital quantum devices for complex simulations are difficult to build. We propose a scalable analog quantum simulator of a U(1) gauge theory in one spatial dimension. Using interspecies spin-changing collisions in an atomic mixture, we achieve gauge-invariant interactions between matter and gauge fields with spin- and species-independent trapping potentials. We experimentally realize the elementary building block as a key step toward a platform for quantum simulations of continuous gauge theories.

Multiparticle Interactions for Ultracold Atoms in Optical Tweezers: Cyclic Ring-Exchange Terms

Dominant multiparticle interactions can give rise to exotic physical phases with anyonic excitations and phase transitions without local order parameters. In spin systems with a global SU(N) symmetry, cyclic ring-exchange couplings constitute the first higher-order interaction in this class. In this Letter, we propose a protocol showing how SU(N)-invariant multibody interactions can be implemented in optical tweezer arrays. We utilize the flexibility to rearrange the tweezer configuration on short timescales compared to the typical lifetimes, in combination with strong nonlocal Rydberg interactions. As a specific example, we demonstrate how a chiral cyclic ring-exchange Hamiltonian can be implemented in a two-leg ladder geometry. We study its phase diagram using density-matrix renormalization group simulations and identify phases with dominant vector chirality, a ferromagnet, and an emergent spin-1 Haldane phase. We also discuss how the proposed protocol can be utilized to implement the strongly frustrated J-Q model, a candidate for hosting a deconfined quantum critical point.

Scattering-Based Geometric Shaping of Photon-Photon Interactions

We construct an effective Hamiltonian of interacting bosons, based on scattered radiation off vibrational modes of designed molecular architectures. Making use of the infinite yet countable set of spatial modes representing the scattering of light, we obtain a variable photon-photon interaction in this basis. The effective Hamiltonian Hermiticity is controlled by a geometric factor set by the overlaps of spatial modes. Using this mapping, we relate intensity measurements of the light to correlation functions of the interacting bosons evolving according to the effective Hamiltonian, rendering local as well as nonlocal observables accessible. This architecture may be used to simulate the dynamics of interacting bosons, as well as a designing tool for multiqubit photonic gates in quantum computing applications. Variable hopping, interaction, and confinement of the active space of the bosons are demonstrated on a model system.

Coupling ultracold matter to dynamical gauge fields in optical lattices: From flux attachment to ℤ2 lattice gauge theories

From the standard model of particle physics to strongly correlated electrons, various physical settings are formulated in terms of matter coupled to gauge fields. Quantum simulations based on ultracold atoms in optical lattices provide a promising avenue to study these complex systems and unravel the underlying many-body physics. Here, we demonstrate how quantized dynamical gauge fields can be created in mixtures of ultracold atoms in optical lattices, using a combination of coherent lattice modulation with strong interactions. Specifically, we propose implementation of ℤ2 lattice gauge theories coupled to matter, reminiscent of theories previously introduced in high-temperature superconductivity. We discuss a range of settings from zero-dimensional toy models to ladders featuring transitions in the gauge sector to extended two-dimensional systems. Mastering lattice gauge theories in optical lattices constitutes a new route toward the realization of strongly correlated systems, with properties dictated by an interplay of dynamical matter and gauge fields.