  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 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
loop-string-hadron 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 quantumly simulating other complex theories of
relevance to nature.
