Enhancing Variational Quantum Eigensolvers for SU(2) Lattice Gauge Theory via Systematic State Preparation
Klaus Liegener, Dominik Mattern, Alexander Korobov, Lisa Krüger, Manuel Geiger, Malay Singh, Longxiang Huang, Christian Schneider, Federico Roy, Stefan Filipp
quant-phhep-latSubmitted Mar 4, 2026• Announced Mar 5, 2026• 0 citations
Computing the vacuum and energy spectrum in non-Abelian, interacting lattice gauge theories remains an open challenge, in part because approximating the continuum limit requires large lattices and huge Hilbert spaces. To address this difficulty with near-term quantum computing devices, we adapt the variational quantum eigensolver to non-Abelian gauge theories. We outline scaling advantages when using a spin-network basis to simulate the gauge-invariant Hilbert space and develop a systematic state preparation ansatz that creates gauge-invariant excitations while alleviating the barren plateau problem. We illustrate our method in the context of SU(2) Yang-Mills theory by testing it on a minimal toy model consisting of a single vertex in 3+1 dimensions. In this toy model, simulations allow us to investigate the impact of noise expected in current quantum devices.
Computing the vacuum and energy spectrum in non-Abelian, interacting lattice gauge theories remains an open challenge, in part because approximating the continuum limit requires large lattices and huge Hilbert spaces. To address this difficulty with near-term quantum computing devices, we adapt the variational quantum eigensolver to non-Abelian gauge theories. We outline scaling advantages when using a spin-network basis to simulate the gauge-invariant Hilbert space and develop a systematic state preparation ansatz that creates gauge-invariant excitations while alleviating the barren plateau problem. We illustrate our method in the context of SU(2) Yang-Mills theory by testing it on a minimal toy model consisting of a single vertex in 3+1 dimensions. In this toy model, simulations allow us to investigate the impact of noise expected in current quantum devices.
In this work, we implement a variational quantum eigensolver (VQE) suitable for investigating ground states and static string breaking in a Z 2 lattice gauge theory (LGT). We consider a two-leg ladder lattice with Kogut-Susskind staggered fermions and verify the results of the VQE simulations using tensor network methods. We find that for varying Hamiltonian parameter regimes and in the presence of external charges, the VQE is able to arrive at the gauge-invariant ground state without explicitly enforcing gauge invariance through penalty terms. Additionally, experiments showing string breaking are performed on IBM’s quantum platform. Thus, VQEs are seen to be a promising tool for Z 2 LGTs, and could serve as a stepping stone toward studies of other gauge groups. We find that the scaling of gradients with the number of qubits is favorable for avoiding barren plateaus. Furthermore, strategies that avoid barren plateaus arise naturally as features of LGTs, such as choosing the initialization by setting the Gauss law sector and restricting the Hilbert space to the gauge-invariant subspace.
Quantum simulation of lattice gauge theories (LGTs) provides a powerful framework for understanding nonperturbative phenomena. However, to realize the hope of answering challenging physics questions on near-term devices it is crucial to develop resource-efficient formulations, in particular for non-Abelian LGTs and beyond (1+1) dimensions. In this work, we address this issue by focusing on the difficulty of simulating fermionic degrees of freedom and on mitigating the Hilbert space redundancy, i.e., the presence of exponentially many nonphysical states that do not obey gauge invariance. First, we show a procedure that removes the matter and improves the hardware-resource efficiency. We demonstrate it for the simplest non-Abelian group addressable with this procedure, D 8 , in the cases of both one and two spatial dimensions. Then, with the objective of running a variational quantum simulation on real quantum hardware, we map the D 8 LGT onto qudit systems with local interactions. We propose a variational scheme for the qudit system with a local Hamiltonian, which can be implemented on a universal qudit quantum device. Our results show the effectiveness of the matter-removing procedure, solving the redundancy problem, and reducing the amount of quantum resources. This approach can serve as a way of simulating LGTs in high spatial dimensions, with non-Abelian gauge groups, and including dynamical fermions.
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.
Studying QCD and other gauge theories on quantum hardware requires the preparation of physically interesting states. The variational quantum eigensolver provides a way of performing vacuum state preparation on quantum hardware. In this work, variational quantum eigensolver is applied to pure SU(3) lattice Yang-Mills on a single plaquette and one dimensional plaquette chains. Bayesian optimization and gradient descent were investigated for performing the classical optimization. Ansatz states for plaquette chains are constructed in a scalable manner from smaller systems using domain decomposition and a stitching procedure analogous to the density matrix renormalization group. Small examples are performed on IBM's superconducting Manila processor.
Non-Abelian gauge theories provide the most accurate description of fundamental interactions, showing remarkable agreement with experimental data in cosmology and particle physics. Highly precise predictions can be made using standard techniques, both in the continuum and in the lattice frameworks. However, classical methods have limitations, particularly when attempting to extrapolate the continuum limit from the study of lattice gauge theories. Complementing classical computations or combining them with quantum computational methods, to improve the predictions toward the continuum limit with current quantum resources, is a formidable open challenge. In this paper, we propose a resource-efficient method to compute the running of the coupling in non-Abelian gauge theories beyond one spatial dimension. We first represent the Hamiltonian on periodic lattices in terms of loop variables and conjugate loop electric fields, exploiting the Gauss law to retain the gauge-independent ones. Then, we identify a local basis for small and large loops variationally to minimize the truncation error while computing the running of the coupling on small tori. Our method enables computations at arbitrary values of the bare coupling and lattice spacing with current quantum computers, simulators, and tensor-network calculations, in regimes otherwise inaccessible.
Tensor-network methods enable probing dynamics of strongly interacting quantum many-body systems, including gauge theories, via Hamiltonian simulation, hence bypassing sign problems. They also have the potential to inform efficient quantum-simulation algorithms of the same theories. We develop and benchmark a matrix-product-state ansatz for the SU(2) lattice gauge theory using the loop-string-hadron formulation. This formulation has been demonstrated to be advantageous in Hamiltonian simulation of non-Abelian gauge theories. It is applicable to both SU(2) and SU(3) gauge groups, to periodic and open boundary conditions, and to 1+1 and higher dimensions. In this work, we report on progress in computing static and dynamical observables in a SU(2) gauge theory in (1+1)D, pushing the boundary of existing studies.