  We describe methods to construct digital quantum simulation algorithms for
quantum spin systems on a regular lattice with local interactions. In addition
to tools such as the Trotter-Suzuki expansion and graph coloring, we also
discuss the efficiency gained by parallel execution of an extensive number of
commuting terms. We provide resource estimates and quantum circuit elements for
the most important cases and classes of spin systems. As resource estimates we
indicate the total number of gates $N$ and simulation time $T$, expressed in
terms of the number $n$ of spin 1/2 lattice sites (qubits), target accuracy
$\epsilon$, and simulated time $t$. We provide circuit constructions that
realize the simulation time $T^{(1)}\propto nt^2/\epsilon$ and $T^{(2q)}\propto
t^{1+\eta}n^\eta/\epsilon^\eta$ for arbitrarily small $\eta=1/2q$ for the
first-order and higher-order Trotter expansions. We also discuss the potential
impact of scaled gates, which have not been fully explored yet.
