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arXiv:2007.00668 · PRX Quantum

Gauge-Symmetry Protection Using Single-Body Terms

Jad C. Halimeh, Haifeng Lang, Julius Mildenberger, Jiang Zhang, Philipp Hauke

Published Oct 18, 2021• 80 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/prxquantum.2.040311

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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.
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.
Gauge invariance is central to modern physics and underpins quantum simulations of lattice gauge theories (LGTs). Existing quantum simulation approaches employ Gauss's law either to energetically suppress gauge-violating processes in analog platforms or to detect and discard gauge-violating outcomes in digital devices. Here we introduce a third paradigm, in which Gauss's law is used to dynamically generate the gauge theory itself from a substantially simpler Hamiltonian. Starting from a readily programmable three-body XXX model, we employ experimentally efficient single-qubit U(1) gauge symmetry-generator terms that induce the dynamics of a U(1) LGT. We implement this approach using 101 qubits on a 156-qubit IBM quantum processor and observe real-time dynamics in quantitative agreement with the target LGT while reducing the entangling-gate depth per Trotter step by a factor of five compared with a direct implementation. Our results establish gauge protection as a resource for Hamiltonian engineering rather than merely symmetry preservation, opening a scalable resource-efficient route towards digital quantum simulations of increasingly complex gauge theories in higher spatial dimensions.
We present a strategy for the quantum simulation of many-body lattice models with constrained Hilbert spaces. We focus on lattice gauge theories (LGTs), which underlie a wide range of phenomena in particle physics, condensed matter, and quantum information. In present-day quantum computing platforms, perfect restrictions of the Hilbert space to the desired gauge sectors are beyond reach: for LGTs, violations of the local constraint are unavoidable, posing a formidable challenge for the emulation of the underlying physics. Here, we develop a Floquet-engineering framework that restructures departures from a target sector such that a series of emergent local symmetries occurs hierarchically in time and in a controllable way. This leads to a set of approximate dynamical selection rules that strongly restrict inter-sector couplings, resulting in a pronounced, symmetry-controlled hierarchy of lifetimes for the state population to spread among sectors. Concretely, this protects U(1)U(1)U(1) LGTs against violations of the {defining} local symmetry. While some sectors remain very long-lived, others are destabilized on shorter timescales. We numerically verify our theory for the one-dimensional U(1)U(1)U(1) quantum link model. In addition, we reveal that `defects', whose movement accounts for violations of the gauge constraint, are kinetically constrained, becoming mobile only through the assistance of intra-sector dynamics, which we describe using an effective quantum marble model. Our results can thus be used to extend the lifetime, in the spirit of passive error correction, of quantum simulations of complex many-body problems when emergent or desired local symmetries are only implemented approximately.
The postulate of gauge invariance in nature does not lend itself directly to implementations of lattice gauge theories in modern setups of quantum synthetic matter. Unavoidable gauge-breaking errors in such devices require gauge invariance to be enforced for faithful quantum simulation of gauge-theory physics. This poses major experimental challenges, in large part due to the complexity of the gauge-symmetry generators. Here, we show that gauge invariance can be reliably stabilized by employing simplified local pseudogenerators designed such that within the physical sector they act identically to the actual local generator. Dynamically, they give rise to emergent exact gauge theories up to time scales polynomial and even exponential in the protection strength. This obviates the need for implementing often complex multibody full gauge symmetries, thereby further reducing experimental overhead in physical realizations. We showcase our method in the Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theory, and discuss experimental considerations for its realization in modern ultracold-atom setups.
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-1/21/21/2 quantum link lattice gauge theories that upon introducing an energy-penalty term of sufficiently large strength VVV, 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 SSS. 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{V}_{0}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, VVV 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.
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.