  We provide an explicit recursive divide and conquer approach for simulating
quantum dynamics and derive a discrete first quantized non-relativistic QED
Hamiltonian based on the many-particle Pauli Fierz Hamiltonian. We apply this
recursive divide and conquer algorithm to this Hamiltonian and compare it to a
concrete simulation algorithm that uses qubitization. Our divide and conquer
algorithm, using lowest order Trotterization, scales for fixed grid spacing as
$\widetilde{O}(\Lambda N^2\eta^2 t^2 /\epsilon)$ for grid size $N$, $\eta$
particles, simulation time $t$, field cutoff $\Lambda$ and error $\epsilon$.
Our qubitization algorithm scales as $\widetilde{O}(N(\eta+N)(\eta +\Lambda^2)
t\log(1/\epsilon)) $. This shows that even a na\"ive partitioning and low-order
splitting formula can yield, through our divide and conquer formalism, superior
scaling to qubitization for large $\Lambda$. We compare the relative costs of
these two algorithms on systems that are relevant for applications such as the
spontaneous emission of photons, and the photoionization of electrons. We
observe that for different parameter regimes, one method can be favored over
the other. Finally, we give new algorithmic and circuit level techniques for
gate optimization including a new way of implementing a group of
multi-controlled-X gates that can be used for better analysis of circuit cost.
