Abstract
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, which further 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 D coupled to 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's size, and the cutoff on the excitations of gauge bosons. The impact of such dependencies on the accuracy of the spectrum and dynamics 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 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. Although small lattices are studied in the numerical analysis of this work, and only the simplest algorithms are considered, a range of conclusions will be applicable to larger systems and potentially to higher dimensions.
Highlights
Gauge-field theories are at the core of our modern understanding of nature, from the descriptions of quantum Hall effect and superconductivity in condensed-matter physics [1,2] to the mechanisms underlying the interactions of subatomic particles at the most fundamental level within the Standard Model of particle physics [3]
Focusing on the case of SU(2) lattice gauge theories (LGTs) in 1 þ 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 dimension of the Hilbert space of the fermionic theory is larger than the dimension of the physical Hilbert space of the KS formulation in the angular momentum basis for cutoff values that allow the full physical Hilbert space to be constructed with open boundary conditions (OBCs) (i.e., Λ ≥ N þ 2ε0)
Summary
Gauge-field theories are at the core of our modern understanding of nature, from the descriptions of quantum Hall effect and superconductivity in condensed-matter physics [1,2] to the mechanisms underlying the interactions of subatomic particles at the most fundamental level within the Standard Model of particle physics [3]. As a first step in the computation, a mechanism to construct the physical Hilbert space and its corresponding Hamiltonian must be implemented Different approaches to this construction from the basis of various formulations studied in this work. Instead the focus is on the representation of the gauge theory itself in terms of the chosen basis states for fermions and bosons, and the gauge group will be kept exact despite the imposed truncation on the high excitations One exception to this trend is the discussion of the quantum link model (QLM) [98,99] of the SU(2) LGT that, given its popularity in the context of quantum simulation, is discussed in some length in Appendix C.2. VI summarizes the main points of the study more crisply, along with presenting an outlook of this work
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