Abstract We study the Hamiltonian lattice Yang-Mills theory based on spin networks that provide a useful basis to represent the physical states satisfying the Gauss law constraints. We focus on SU(2) Yang-Mills theory in (2 + 1) dimensions. Following the string-net model, we introduce a regularization of the Kogut-Susskind Hamiltonian of lattice Yang-Mills theory based on the q deformation, which respects the (discretized) SU(2) gauge symmetry as quantum group, i.e., SU(2)k, and enables implementation of the lattice Yang-Mills theory both in classical and quantum algorithms by referring to those of the string-net model. Using the regularized Hamiltonian, we study quantum scars in a nonabelian gauge theory. Quantum scars are nonthermal energy eigenstates arising in the constrained quantum many-body systems. We find that quantum scars from zero modes, which have been found in abelian gauge theories arise even in a nonabelian gauge theory. We also show the spectrum of a single-plaquette model for SU(2)k and SU(3)k with naive cutoff and that based on the q-deformation to discuss cutoff dependence of the formulation.
Abstract We study the Hamiltonian lattice Yang-Mills theory based on spin networks that provide a useful basis to represent the physical states satisfying the Gauss law constraints. We focus on SU(2) Yang-Mills theory in (2 + 1) dimensions. Following the string-net model, we introduce a regularization of the Kogut-Susskind Hamiltonian of lattice Yang-Mills theory based on the q deformation, which respects the (discretized) SU(2) gauge symmetry as quantum group, i.e., SU(2)k, and enables implementation of the lattice Yang-Mills theory both in classical and quantum algorithms by referring to those of the string-net model. Using the regularized Hamiltonian, we study quantum scars in a nonabelian gauge theory. Quantum scars are nonthermal energy eigenstates arising in the constrained quantum many-body systems. We find that quantum scars from zero modes, which have been found in abelian gauge theories arise even in a nonabelian gauge theory. We also show the spectrum of a single-plaquette model for SU(2)k and SU(3)k with naive cutoff and that based on the q-deformation to discuss cutoff dependence of the formulation.
Abstract We study SU(3) Yang-Mills theory in (2 + 1) dimensions based on networks of Wilson lines. With the help of the q deformation, networks respect the (discretized) SU(3) gauge symmetry as a quantum group, i.e., SU(3)k, and may enable implementations of SU(3) Yang-Mills theory in quantum and classical algorithms by referring to those of the stringnet model. As a demonstration, we perform a mean-field computation of the groundstate of SU(3)k Yang-Mills theory, which is in good agreement with the conventional Monte Carlo simulation by taking sufficiently large k. The variational ansatz of the mean-field computation can be represented by the tensor networks called infinite projected entangled pair states. The success of the mean-field computation indicates that the essential features of Yang-Mills theory are well described by tensor networks, so that they may be useful in numerical simulations of Yang-Mills theory.
Treating the infinite-dimensional Hilbert space of non-Abelian gauge theories is an outstanding challenge for classical and quantum simulations. Here, we employ q-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. In contrast to other formulations, this approach simultaneously provides a controlled regularization of the infinite-dimensional local Hilbert space while preserving essential symmetry-related properties. This enables the development of both quantum as well as quantum-inspired classical spin-network algorithms for q-deformed gauge theories. To be explicit, we focus on SU(2)_{k} gauge theories with k∈N that are controlled by the deformation parameter q=e^{2πi/(k+2)}, a root of unity, and converge to the standard SU(2) Kogut-Susskind model as k→∞. In particular, we demonstrate that this formulation is well suited for efficient tensor network representations by variational ground-state simulations in 2D, providing first evidence that the continuum limit can be reached with k=O(10). Finally, we develop a scalable quantum algorithm for Trotterized real-time evolution by analytically diagonalizing the SU(2)_{k} plaquette interactions. Our work gives a new perspective for the application of tensor network methods to high-energy physics and paves the way for quantum simulations of non-Abelian gauge theories far from equilibrium where no other methods are currently available.
Hamiltonian formulation of lattice gauge theories (LGTs) is the most natural framework for the purpose of quantum simulation, an area of research that is growing with advances in quantum-computing algorithms and hardware. It, therefore, remains an important task to identify the most accurate, while computationally economic, Hamiltonian formulation(s) in such theories, considering the necessary truncation imposed on the Hilbert space of gauge bosons with any finite computing resources. This paper is a first step toward addressing this question in the case of non-Abelian LGTs. Such theories require the imposition of non-Abelian Gauss's laws on the Hilbert space, introducing additional computational complexity. Focusing on the case of SU(2) LGT in 1+1 dimensions coupled to one flavor of fermionic matter, a number of different formulations of the original Kogut-Susskind framework are analyzed with regard to the dependence of the dimension of the physical Hilbert space on boundary conditions, system size, and the cutoff on the excitations of gauge bosons. The impact of such dependencies on the accuracy of the spectrum and dynamics obtained from a Hamiltonian computation is examined, and the (classical) computational-resource requirements given these considerations are studied. Besides the well-known angular-momentum formulation of the theory, the cases of purely fermionic and purely bosonic formulations (with open boundary conditions) and the loop-string-hadron formulation are analyzed, along with a brief discussion of a quantum-link-model formulation of the same theory. Clear advantages are found in working with the loop-string-hadron framework which implements non-Abelian Gauss's laws a priori using a complete set of gauge-invariant operators. It is argued that the substantially reduced cost of simulation within the loop-string-hadron formulation compared with the angular-momentum basis has implications for more efficient and robust quantum simulations of the same theory. Future studies will extend this investigation to the analysis of resource requirements for quantum simulations of non-Abelian LGT, with the goal of shedding light on the most efficient Hamiltonian formulation of gauge theories of relevance in nature.
The development of quantum technology and the advent of quantum computation have renewed interest in Hamiltonian formulations of lattice gauge theories. In contrast to Euclidean Monte Carlo calculations, Hamiltonian simulation requires a direct representation of the physical Hilbert space, an efficient treatment of enforcing Gauss' law as constraints, and a formulation of real-time dynamics in terms of degrees of freedom that can be encoded on quantum or quantum-inspired classical platforms. The starting point corresponds to finding a suitable basis for Hamiltonian lattice gauge theories. The existing literature focuses on several complementary approaches: electric basis, magnetic basis, gauge-fixed, quantum-link, qubit-regularised, finite-group, and orbifold formulations. This article primarily focuses on the prepotential and loop--string--hadron formulations, where non-Abelian Gauss laws are solved locally and the dynamics is written directly in terms of gauge-invariant loop, string and hadron excitations. The discussion covers the SU(2) and SU(3) constructions in one spatial dimension, the role of residual Abelian Gauss laws, and the extension to higher dimensions through point splitting. We conclude by highlighting its direct applications: eliminating redundant gauge degrees of freedom makes exact diagonalisation and tensor network calculations feasible for studying real-time dynamics, provides insights into thermalisation properties, enables mapping of physical degrees of freedom onto analog and digital quantum hardware, and enables quantum simulation of dynamics.
We consider the spectrum of a U(1) quantum link model where gauge fields are realized as S=1/2 quantum spins and demonstrate a new mechanism for generating quantum many-body scars (high-energy eigenstates that violate the eigenstate thermalization hypothesis) in a constrained Hilbert space. Many-body dynamics with local constraints has attracted much attention due to the recent discovery of non-ergodic behavior in quantum simulators based on Rydberg atoms. Lattice gauge theories provide natural examples of constrained systems since physical states must be gauge-invariant. In our case, the Hamiltonian H=Okin+λOpot, where Opot (Okin) is diagonal (off-diagonal) in the electric flux basis, contains exact mid-spectrum zero modes at λ=0 whose number grows exponentially with system size. This massive degeneracy is lifted at any non-zero λ but some special linear combinations that simultaneously diagonalize Okin and Opot survive as quantum many-body scars, suggesting an order-by-disorder mechanism in the Hilbert space. We give evidence for such scars and show their dynamical consequences on ladders with up to 56 spins, which may be tested using available proposals of quantum simulators.
We examine the Kogut-Susskind formulation of lattice gauge theories under the light of fermionic and bosonic degrees of freedom that provide a description useful to the development of quantum simulators of gauge-invariant models. We consider both discrete and continuous gauge groups and adopt a realistic multicomponent Fock space for the definition of matter degrees of freedom. In particular, we express the Hamiltonian of the gauge theory and the Gauss law in terms of Fock operators. The gauge fields are described in two different bases based on either group elements or group representations. This formulation allows for a natural scheme to achieve a consistent truncation of the Hilbert space for continuous groups, and provides helpful tools to study the connections of gauge theories with topological quantum double and string-net models for discrete groups. Several examples, including the case of the discrete D3 gauge group, are presented.