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Simulating Lattice Gauge Theories within Quantum Technologies
M. C. Bañuls, R. Blatt, J. Catani, A. Celi, J. I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C. A. Muschik, B. Reznik, E. Rico, L. Tagliacozzo, K. Van Acoleyen, F. Verstraete, U. -J. Wiese, M. Wingate, J. Zakrzewski, P. Zoller
TL;DR
Lattice gauge theories motivate quantum technologies because Monte Carlo methods cannot reliably address important sign-problem regimes, including finite-density and real-time dynamics. This review synthesizes tensor-network methods and quantum-simulation implementations across several hardware platforms, highlighting advanced trapped-ion simulations and broader directions toward quantum technologies.
Problem
Monte Carlo lattice calculations are limited by sign problems in finite-density, real-time, and topological-term scenarios, while higher-dimensional tensor-network applications remain challenging.
Method
The review surveys tensor-network approaches and digital or analog lattice-gauge-theory quantum simulators using trapped ions, Rydberg atoms, and superconducting circuits.
Results
The review presents the most advanced quantum simulation of a lattice gauge theory reported there: digital ion-trap variational optimization for the Schwinger-model ground state.
Takeaways & Limitations
Lattice gauge theories provide a framework for interdisciplinary development of quantum-simulation hardware and software, with QCD real-time evolution and finite-density physics remaining long-term goals.
Takeaways & Limitations
Gaussian tensor-network states cannot describe correlations between gauge and matter fields in interacting lattice gauge theories.
Abstract
from arXiv · showhide
Lattice gauge theories, which originated from particle physics in the context of Quantum Chromodynamics (QCD), provide an important intellectual stimulus to further develop quantum information technologies. While one long-term goal is the reliable quantum simulation of currently intractable aspects of QCD itself, lattice gauge theories also play an important role in condensed matter physics and in quantum information science. In this way, lattice gauge theories provide both motivation and a framework for interdisciplinary research towards the development of special purpose digital and analog quantum simulators, and ultimately of scalable universal quantum computers. In this manuscript, recent results and new tools from a quantum science approach to study lattice gauge theories are reviewed. Two new complementary approaches are discussed: first, tensor network methods are presented - a classical simulation approach - applied to the study of lattice gauge theories together with some results on Abelian and non-Abelian lattice gauge theories. Then, recent proposals for the implementation of lattice gauge theory quantum simulators in different quantum hardware are reported, e.g., trapped ions, Rydberg atoms, and superconducting circuits. Finally, the first proof-of-principle trapped ions experimental quantum simulations of the Schwinger model are reviewed.
1 Introduction
Lattice gauge theories connect quantum information technologies with problems across high-energy, condensed-matter, and quantum information science. The review surveys tensor-network methods, quantum-simulator proposals, and early trapped-ion experiments.
- Lattice gauge theories describe phenomena spanning fundamental interactions, condensed matter, and some computationally hard problems.
- Quantum information approaches include quantum simulators that reproduce target models and tensor networks for strongly correlated many-body systems.
- Monte Carlo methods face severe sign or complex-action problems for high-density QCD and real-time dynamics, motivating quantum technologies.
- Tensor-network and matrix-product-state methods can validate quantum simulators quantitatively in some out-of-equilibrium, especially lower-dimensional, settings.
- The review covers Abelian and non-Abelian studies, digital and analog implementations, and first experimental realizations within the QTFLAG program.
2 Lattice Field Theory background
Lattice field theory provides a first-principles framework for gauge theories but encounters sign problems in important regimes. Tensor networks and finite-dimensional quantum link models offer complementary alternatives, with higher-dimensional simulation remaining difficult.
- Wilson’s lattice formulation enabled non-perturbative lattice QCD calculations and broader studies of gauge-field phenomena.
- Monte Carlo path-integral methods are limited by sign problems involving finite baryon density, real-time evolution, and topological terms.
- Tensor networks avoid the sign problem and can study one-dimensional mass spectra, temperatures, chemical potentials, topological terms, real-time dynamics, and entanglement.
- Higher-dimensional tensor-network applications remain challenging despite formulations such as projected entangled pair states.
- Quantum link models use finite-dimensional quantum degrees of freedom, such as spins for U(1) transporters, and can approach the continuum limit by increasing spin.
3 Quantum Science and Technologies tools
Quantum simulation engineers controllable quantum systems to reproduce target models and extract their properties through measurements. Dedicated platforms such as cold atoms, trapped ions, and superconducting circuits extend access to otherwise inaccessible physical systems.
- Purpose-based quantum simulators engineer a target Hamiltonian in a controllable quantum system and retrieve information through repeated measurements.
- Cold atoms, trapped ions, and superconducting circuits are prominent candidate platforms because they offer control and high-fidelity measurements.
- Quantum simulation can investigate experimentally inaccessible systems and physical properties of unreal systems mapped onto controllable simulators.
4 Quantum information techniques
This section reviews tensor-network methods for lattice gauge theories, emphasizing MPS studies of the Schwinger model and extensions to non-Abelian, finite-density, and real-time settings.
- Tensor-network studies cover Abelian and non-Abelian lattice gauge theories, including one-dimensional SU(2) and SU(3) models.
- Matrix product states for gauge field theories: Continuum chiral-condensate calculations subtract the free-theory ultraviolet divergence and account for lattice corrections dominated by a log a.
- Tensor networks provide controlled low-energy calculations and error estimates, while avoiding Monte Carlo sign problems in finite-density and real-time scenarios.Their classical advantage is strongest in low dimensions, whereas higher-dimensional extensions remain difficult.
- Matrix product states for gauge field theories: MPS simulations reach the continuum limit by approaching the critical point and performing extrapolations at decreasing lattice spacing.Simulations reached a ≈1/(30g), with correlation lengths ξ/a ≈15−35 depending on m/g.
- Matrix product states for gauge field theories: MPS accurately determine Schwinger-model ground-state and excitation properties, including vector and scalar particles and their dispersion relations.Three particles were identified: two vector particles with C = −1 and one scalar particle with C = +1.
- Dynamical evolution: Real-time tensor-network simulations study string breaking, scattering, and Schwinger pair creation, including electric-field back-reaction and damped oscillations.
5 Quantum computation and digital quantum simulation
The section reviews quantum algorithms and digital simulators for quantum field and lattice gauge theories, spanning hybrid variational methods, trapped ions, superconducting circuits, and Rydberg-atom platforms. It reports applications ranging from nuclear binding energies and scattering probabilities to particle production and non-Abelian gauge dynamics.
- Quantum Algorithms for Quantum Field Theories: Quantum algorithms can calculate relativistic scattering probabilities in φ4 theory at weak and strong coupling using real-time dynamics.The proposed algorithm is polynomial in the number of particles, their energy, and the desired precision.
- Quantum Algorithms for Quantum Field Theories: Strong-coupling quantum algorithms provide an exponential acceleration over the best known classical algorithms.
- Quantum Algorithms for Quantum Field Theories: VQE combines quantum-state preparation and measurement with classical optimization in an iterative hybrid quantum-classical loop.
- Quantum Algorithms for Quantum Field Theories: Photonic VQE calculations obtained binding energies for the nuclei 3H, 3He, and 4He and determined an effective interaction potential from the Schwinger model.The latter demonstrates that effective field theory interactions can be implemented and determined from first principles using quantum simulations.
- Digital quantum simulation with trapped ions: The first digital high-energy gauge-theory simulation realized the 1+1-dimensional Schwinger model on a trapped-ion quantum computer.The experiment used N = 4 lattice sites, four qubits, and a gate sequence with more than 200 operations.
- Digital quantum simulation with trapped ions: Trapped-ion simulations reproduced particle-pair creation and measured entanglement corresponding to the original model's gauge fields and fermions.The encoded model's entanglement was shown to correspond to entanglement in the original model.
- Digital quantum simulation with superconducting circuits: Superconducting circuits were proposed for digital simulation of a non-Abelian dynamical SU(2) gauge theory on a triangular lattice.
- Rydberg and optical quantum simulators: Rydberg platforms implement coherent and dissipative spin dynamics using auxiliary atoms, while optical proposals encode U(1) gauge bosons in Rydberg-atom hyperfine levels.Auxiliary atoms mediate effective n-body interactions and can be optically pumped to generate dissipative dynamics; optical schemes digitally impose dynamics and the Gauss law.
6 Analog Quantum simulations
Analog quantum simulations use cold atoms and related platforms to implement Abelian and non-Abelian gauge fields, synthetic dimensions, and relativistic lattice models. The reviewed proposals demonstrate tunable gauge potentials, constrained dynamics, and routes toward observing gauge-theory phenomena and exotic phases.
- Relativistic lattice fermions: Cold-atom superlattices can simulate relativistic lattice fermions in 3 + 1 dimensions, including naive and Wilson fermions with tunable mass inversion.In the relevant regime, Maxwell electrodynamics is replaced by axion electrodynamics.
- Non-Abelian gauge potentials: Lattice shaking converts sublattice-dependent spin rotations into tunable non-Abelian SU(2) gauge fields with experimentally accessible Wilson loops.The synthetic gauge field arises from time-averaging driven tunnelling in a spin-dependent square lattice.
- Synthetic dimensions: Synthetic dimensions realize ladder geometries whose complex hopping phases generate background gauge potentials and steady-state chiral edge currents.Experiments observed counterpropagating motion on the outer legs and cyclotron-like skipping orbits along the edge.
- Synthetic dimensions: Measuring chiral currents across synthetic flux reveals an edge-current reversal above φ = π, reminiscent of a Chern-number sign change in the Hofstadter spectrum.The flux-dependent measurement used an electronic-state implementation in 173Yb.
- Abelian gauge fields: Cold-atom proposals implement Abelian gauge theories through quantum links, spin-gauge Hamiltonians, and independently tunable ring-exchange interactions.These designs target string breaking, convergence to the Kogut-Susskind cQED Hamiltonian, and exotic phases including a three-dimensional U(1) Coulomb phase.
- Constrained dynamics: Rydberg interactions can impose Gauss-law constraints and reproduce quantum-ice dynamics, including resonating valence-bond solid order and an imperfect Coulomb phase.The approach also supports engineered U(1)- and Z(2)-invariant spin exchanges through the Rydberg manifold.
7 Conclusions
Lattice gauge theories motivate and structure interdisciplinary quantum-technology research, while reviewed work combines classical tensor-network methods, proposed quantum simulators, and trapped-ion demonstrations. These efforts provide benchmarks for early quantum simulators and connect lattice-gauge-theory studies to longer-term goals in quantum computing and other sciences.
- The reviewed effort is a collaborative multidisciplinary project spanning quantum hardware and software for open problems from materials science to astrophysics.
- Its results serve as benchmarks for first-generation quantum simulators and are linked to the study and design of materials with topological error-correcting capabilities.
- The review presents tensor-network methods, quantum-simulation proposals, and a digital ion-trap variational optimization of the Schwinger-model ground state.The proposals include simulations of more complex theories, while tensor-network methods address one- and two-dimensional systems.
- Lattice gauge theories provide both motivation and a framework for interdisciplinary advancement of quantum technologies.
- Reliable simulation of intractable QCD aspects remains a long-term goal, while broader lattice gauge theories also have applications in condensed matter and quantum information science.
9 Authors contributions
All authors contributed to preparing the manuscript and approved its final version.
- All authors were involved in preparing the manuscript and approved the final manuscript.