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.
We propose an efficient variational method for
Z2 lattice gauge theory based on the matrix product ansatz. The method is applied to ladder and square lattices. The Gauss law needs to be imposed on quantum states to guarantee gauge invariance when one studies gauge theory in hamiltonian formalism. On the ladder lattice, we identify gauge invariant low-lying states by evaluating expectation values of the Gauss law operator after numerical diagonalization of the gauge hamiltonian. On the square lattice, the second order phase transition is well reproduced.
Glueball spectroscopy and real-time production with quantum computing require three distinct ingredients: a correlated gauge vacuum, a controlled construction of pure-gauge excitations, and a dynamical detector. We develop a physics-informed quantum-algorithm toolbox for these tasks in a
(2+1)-dimensional
Z2 lattice gauge theory. We use the term \emph{glueball-like} for localized closed-flux excitations on the confining side of this Abelian model, without identifying them with the non-Abelian glueballs of QCD. A loop-gas circuit and Hamiltonian variational ansatz prepare the gauge vacuum, Wilson-loop quantum subspace expansion constructs and characterizes low-lying excitations, and eigenvector continuation imports parameter-dependent dressing without a rapidly enlarged explicit loop basis. A Bethe--Salpeter-type transition amplitude quantifies the spatial broadening of the lightest state. For dynamics, a vacuum-dressed contractible-loop counter measures excess production of localized glueball-like structures. Although demonstrated in an Abelian model, the toolbox separates preparation, construction, compression, characterization, and dynamical detection in a form that is naturally extensible to non-Abelian lattice gauge theories.
In this work, exact solutions are obtained for a class of generalized gauge-invariant
n-chain Ising models (
n=1,2,3,4) with arbitrary multi-spin interactions that are invariant under the local
Z2 gauge group. On a strip lattice of finite length
L and width
n with periodic or free boundary conditions, an explicit expression for the partition function is derived using the transfer-matrix method. Two successive transformations are developed: elimination of gauge redundancy and reduction of the original model to an effective
n-chain Ising model with all possible interactions between neighboring vertical layers. On the basis of the spectral decomposition of the
2n×2n transfer matrix, general formulas are obtained for gauge-invariant correlation functions and Wilson loops of arbitrary width. For
n≤3, explicit expressions are derived in terms of eigenvalues and eigenvectors. A detailed analysis of the behavior of the Wilson loop is performed, which allows us to identify regimes exhibiting area-law (confinement-like) and perimeter-law (deconfinement-like) dependence. For specific Hamiltonians, the string tension is computed and the corresponding phase diagrams are constructed. The results generalize and substantially extend the classical works on the gauge-invariant Ising model.