Recent years have seen strong progress in quantum simulation of gauge-theory dynamics using ultracold-atom experiments. A principal challenge in these efforts is the certification of gauge invariance, which has recently been realized in [B.~Yang et al., arXiv:2003.08945]. One major but poorly investigated experimental source of gauge-invariance violation is an imperfect preparation of the initial state. Using the time-dependent density-matrix renormalization group, we analyze the robustness of gauge-invariant dynamics against potential preparation defects in the above ultracold-atom implementation of a $\mathrm{U}(1)$ gauge theory. We find defects related to an erroneous initialization of matter fields to be innocuous, as the associated gauge-invariance violation remains strongly localized throughout the time evolution. A defect due to faulty initialization of the gauge field leads to a mild proliferation of the associated violation. Furthermore, we characterize the influence of immobile and mobile defects by monitoring the spread of entanglement entropy. Overall, our results indicate that the aforementioned experimental realization exhibits a high level of fidelity in the gauge invariance of its dynamics at all evolution times. Our work provides strong evidence that ultracold-atom setups can serve as an extremely reliable framework for the quantum simulation of gauge-theory dynamics.
Recent years have seen strong progress in quantum simulation of gauge-theory dynamics using ultracold-atom experiments. A principal challenge in these efforts is the certification of gauge invariance, which has recently been realized in [B.~Yang et al., arXiv:2003.08945]. One major but poorly investigated experimental source of gauge-invariance violation is an imperfect preparation of the initial state. Using the time-dependent density-matrix renormalization group, we analyze the robustness of gauge-invariant dynamics against potential preparation defects in the above ultracold-atom implementation of a U(1) gauge theory. We find defects related to an erroneous initialization of matter fields to be innocuous, as the associated gauge-invariance violation remains strongly localized throughout the time evolution. A defect due to faulty initialization of the gauge field leads to a mild proliferation of the associated violation. Furthermore, we characterize the influence of immobile and mobile defects by monitoring the spread of entanglement entropy. Overall, our results indicate that the aforementioned experimental realization exhibits a high level of fidelity in the gauge invariance of its dynamics at all evolution times. Our work provides strong evidence that ultracold-atom setups can serve as an extremely reliable framework for the quantum simulation of gauge-theory dynamics.
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.
Although gauge invariance is a postulate in fundamental theories of nature such as quantum electrodynamics, in quantum-simulation implementations of gauge theories it is compromised by experimental imperfections. In a recent paper [Halimeh and Hauke, Phys. Rev. Lett. 125, 030503 (2020)], it has been shown in finite-size spin-1/2 quantum link lattice gauge theories that upon introducing an energy-penalty term of sufficiently large strength V, unitary gauge-breaking errors at strength $\ensuremath{\lambda}$ are suppressed $\ensuremath{\propto}{\ensuremath{\lambda}}^{2}/{V}^{2}$ up to all accessible evolution times. Here, we show numerically that this result extends to quantum link models in the thermodynamic limit and with larger spin S. As we show analytically, the dynamics at short times is described by an adjusted gauge theory up to a timescale that is at earliest ${\ensuremath{\tau}}_{\text{adj}}\ensuremath{\propto}\sqrt{V/{V}_{0}^{3}}$, with V0 an energy factor. Moreover, our analytics predicts that a renormalized gauge theory dominates at intermediate times up to a timescale ${\ensuremath{\tau}}_{\text{ren}}\ensuremath{\propto}exp(V/{V}_{0})/{V}_{0}$. In both emergent gauge theories, V is volume independent and scales at worst $\ensuremath{\sim}{S}^{2}$. Furthermore, we numerically demonstrate that robust gauge invariance is also retained through a single-body gauge-protection term, which is experimentally straightforward to implement in ultracold-atom setups and NISQ devices.
Quantum simulations of high-energy physics, and especially of gauge theories, is an emerging and exciting direction in quantum simulations. However, simulations of such theories, compared to simulations of condensed matter physics, must satisfy extra restrictions, such as local gauge invariance and relativistic structure. In this paper we discuss these special requirements, and present a method for quantum simulation of lattice gauge theories using ultracold atoms. This method allows us to include local gauge invariance as a fundamental symmetry of the atomic Hamiltonian, arising from natural atomic interactions and conservation laws (and not as a property of a low-energy sector). This allows us to implement elementary gauge invariant interactions for three lattice gauge theories: U(1) (compact QED), ZN and SU(N) (Yang-Mills), which can be used to build quantum simulators in 1+1 dimensions. We also present a loop method, which uses the elementary interactions as building blocks in the effective construction of quantum simulations for d+1 dimensional lattice gauge theories (d>1), but unlike in previous proposals, here gauge invariance and Gauss's law are natural symmetries, which do not have to be imposed as a constraint. We discuss in detail the quantum simulation of 2+1 dimensional compact QED and provide a numerical proof of principle. The simplicity of the already gauge-invariant elementary interactions of this model suggests it may be useful for future experimental realizations.
Abstract Protection of gauge invariance in experimental realizations of lattice gauge theories based on energy-penalty schemes has recently stimulated impressive efforts both theoretically and in setups of quantum synthetic matter. A major challenge is the reliability of such schemes in non-abelian gauge theories where local conservation laws do not commute. Here, we show through exact diagonalization (ED) that non-abelian gauge invariance can be reliably controlled using gauge-protection terms that energetically stabilize the target gauge sector in Hilbert space, suppressing gauge violations due to unitary gauge-breaking errors. We present analytic arguments that predict a volume-independent protection strength V, which when sufficiently large leads to the emergence of an adjusted gauge theory with the same local gauge symmetry up to least a timescale ∝ V / V 0 3 . Thereafter, a renormalized gauge theory dominates up to a timescale ∝exp(V/V 0)/V 0 with V 0 a volume-independent energy factor, similar to the case of faulty abelian gauge theories. Moreover, we show for certain experimentally relevant errors that single-body protection terms robustly suppress gauge violations up to all accessible evolution times in ED, and demonstrate that the adjusted gauge theory emerges in this case as well. These single-body protection terms can be readily implemented with fewer engineering requirements than the ideal gauge theory itself in current ultracold-atom setups and noisy intermediate-scale quantum (NISQ) devices.
Abstract 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 devices3–6. Previous achievements implemented density-dependent Peierls phases without defining a local symmetry7, 8, realized mappings onto effective models to integrate out either matter or electric fields9–12, or were limited to very small systems13–16. 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.
Quantum-simulator hardware promises new insights into problems from particle and nuclear physics.A major challenge is to reproduce gauge invariance, as violations of this quintessential property of lattice gauge theories can have dramatic consequences, e.g., the generation of a photon mass in quantum electrodynamics.Here, we introduce an experimentally friendly method to protect gauge invariance in U(1) lattice gauge theories against coherent errors in a controllable way.Our method employs only single-body energy-penalty terms, thus enabling practical implementations.As we derive analytically, some sets of penalty coefficients render undesired gauge sectors inaccessible by unitary dynamics for exponentially long times.Further, for few-body error terms, we show numerically that this is achieved with resources exhibiting little dependence on system size.These findings constitute an exponential improvement over previously known results from energy-gap protection or perturbative treatments.In our method, the gaugeinvariant subspace is protected by an emergent global symmetry, meaning it can be immediately applied to other symmetries.In our numerical benchmarks for continuous-time and digital quantum simulations, gauge protection holds for all calculated evolution times (up to t > 10 10 /J for continuous time, with J the relevant energy scale).Crucially, our gauge-protection technique is simpler to realize than the associated ideal gauge theory, and can thus be readily implemented in current ultracold-atom analog simulators as well as digital noisy intermediate-scale quantum devices.