  The question of how to efficiently formulate Hamiltonian gauge theories is
experiencing renewed interest due to advances in building quantum simulation
platforms. We introduce a reformulation of an SU(2) Hamiltonian lattice gauge
theory---a loop-string-hadron (LSH) formulation---that describes dynamics
directly in terms of its loop, string, and hadron degrees of freedom, while
alleviating several disadvantages of quantumly simulating the Kogut-Susskind
formulation. This LSH formulation transcends the local loop formulation of
$d+1$-dimensional lattice gauge theories by incorporating staggered quarks,
furnishing the algebra of gauge-singlet operators, and being used to
reconstruct dynamics between states that have Gauss's law built in to them. LSH
operators are then factored into products of "normalized" ladder operators and
diagonal matrices, priming them for classical or quantum information
processing. Self-contained expressions of the Hamiltonian are given up to
$d=3$. The LSH formalism makes little use of structures specific to SU(2) and
its conceptual clarity makes it an attractive approach to apply to other
non-Abelian groups like SU(3).
