Solving Gauss's Law on Digital Quantum Computers with Loop-String-Hadron Digitization

We show that using the loop-string-hadron (LSH) formulation of SU(2) lattice gauge theory (arXiv:1912.06133) as a basis for digital quantum computation easily solves an important problem of fundamental interest: implementing gauge invariance (or Gauss's law) exactly. We first discuss the structure of the LSH Hilbert space in $d$ spatial dimensions, its truncation, and its digitization with qubits. Error detection and mitigation in gauge theory simulations would benefit from physicality "oracles,'"so we decompose circuits that flag gauge invariant wavefunctions. We then analyze the logical qubit costs and entangling gate counts involved with the protocols. The LSH basis could save or cost more qubits than a Kogut-Susskind-type representation basis, depending on how the bases are digitized as well as the spatial dimension. The numerous other clear benefits encourage future studies into applying this framework.Comment: 10 pages, 9 figures. v3: Journal version. A few added remarks and plots regarding qubit cost

General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory

With a focus on universal quantum computing for quantum simulation, and through the example of lattice gauge theories, we introduce rather general quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple (bosonic and fermionic) quantum numbers with non-trivial functional coefficients. In particular, we analyze diagonalization of Hamiltonian terms using a singular-value decomposition technique, and discuss how the achieved diagonal unitaries in the digitized time-evolution operator can be implemented. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions, for which a complete quantum-resource analysis within different computational models is presented. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories. The example chosen further demonstrates the importance of adopting efficient theoretical formulations: it is shown that an explicitly gauge-invariant formulation using loop, string, and hadron (LSH) degrees of freedom simplifies the algorithms and lowers the cost compared with the standard formulations based on angular-momentum as well as the Schwinger-boson degrees of freedom. The LSH formulation further retains the non-Abelian gauge symmetry despite the inexactness of the digitized simulation, without the need for costly controlled operations. Such theoretical and algorithmic considerations are likely to be essential in quantum simulating other complex theories of relevance to nature.Comment: 59+17+7 pages, 16 figure

Loop-string-hadron formulation of an SU(3) gauge theory with dynamical quarks

Towards the goal of quantum computing for lattice quantum chromodynamics, we present a loop-string-hadron (LSH) framework in 1+1 dimensions for describing the dynamics of SU(3) gauge fields coupled to staggered fermions. This novel framework was previously developed for an SU(2) lattice gauge theory in $d\leq3$ spatial dimensions and its advantages for classical and quantum algorithms have thus far been demonstrated in $d=1$. The LSH approach uses gauge invariant degrees of freedoms such as loop segments, string ends, and on-site hadrons, it is free of all nonabelian gauge redundancy, and it is described by a Hamiltonian containing only local interactions. In this work, the SU(3) LSH framework is systematically derived from the reformulation of Hamiltonian lattice gauge theory in terms of irreducible Schwinger bosons, including the addition of staggered quarks. Furthermore, the superselection rules governing the LSH dynamics are identified directly from the form of the Hamiltonian. The SU(3) LSH Hamiltonian with open boundary conditions has been numerically confirmed to agree with the completely gauge-fixed Hamiltonian, which contains long-range interactions and does not generalize to either periodic boundary conditions or to $d>1$.Comment: 35 pages plus references, 5 figures. v2 includes typo corrections, trivial adjustments to text sectioning, and added reference

Compiling Quantum Gauge Theories for Quantum Computation

Thesis (Ph.D.)--University of Washington, 2020This dissertation gives a survey of formal issues of Hamiltonian lattice gauge theories in the context of simulation by quantum computers. The basic properties of gauge field theories and their lattice regularizations are first reviewed, especially as they pertain to the local constraint that arises in canonical quantization: Gauss’s law, the satisfaction of which is synonymous with gauge invariance and charge conservation. Digital quantum algorithms are developed for the basic task of checking Gauss’s law in U(1) and Z(N) Abelian gauge theories, as they are conventionally formulated. We then analyze U(1) gauge theories by reconstructing them in terms of dual variables that make Gauss’s law manifest. The task of quantum simulation is then studied for the non-Abelian gauge group SU(2). The first quantum simulation of SU(2) gauge bosons using existing IBM quantum hardware is presented, made possible by partially solving the Gauss law constraints in a small system. The quest for the “right” variables to use for quantum algorithms begun with U(1) is then taken up for SU(2). Building on the prepotential formulation of lattice gauge theories, a complete ‘loop-string-hadron’ (LSH) framework is developed for one fermion flavor interacting with SU(2) gauge bosons, in terms of strictly SU(2)-invariant variables. The LSH Hamiltonian is unpacked at a low level, making it transparent what would have to be implemented on a quantum computer. This LSH framework is then applied to provide the first quantum circuits for validating wave functions in SU(2) gauge theories and the associated resource requirements are discussed