  Quantum simulations of chemistry in first quantization offer important
advantages over approaches in second quantization including faster convergence
to the continuum limit and the opportunity for practical simulations outside
the Born-Oppenheimer approximation. However, as all prior work on quantum
simulation in first quantization has been limited to asymptotic analysis, it
has been impossible to compare the resources required for these approaches to
those for more commonly studied algorithms in second quantization. Here, we
analyze and optimize the resources required to implement two first quantized
quantum algorithms for chemistry from Babbush et al that realize block
encodings for the qubitization and interaction picture frameworks of Low et al.
The two algorithms we study enable simulation with gate complexities
$\tilde{\cal O}(\eta^{8/3}N^{1/3}t+\eta^{4/3}N^{2/3}t)$ and $\tilde{\cal
O}(\eta^{8/3} N^{1/3} t)$ where $\eta$ is the number of electrons, $N$ is the
number of plane wave basis functions, and $t$ is the duration of time-evolution
($t$ is inverse to target precision when the goal is to estimate energies). In
addition to providing the first explicit circuits and constant factors for any
first quantized simulation and introducing improvements which reduce circuit
complexity by about a thousandfold over naive implementations for modest sized
systems, we also describe new algorithms that asymptotically achieve the same
scaling in a real space representation. We assess the resources required to
simulate various molecules and materials and conclude that the qubitized
algorithm will often be more practical than the interaction picture algorithm.
We demonstrate that our qubitized algorithm often requires much less surface
code spacetime volume for simulating millions of plane waves than the best
second quantized algorithms require for simulating hundreds of Gaussian
orbitals.
