  Quantum algorithms for simulating electronic ground states are slower than
popular classical mean-field algorithms such as Hartree-Fock and density
functional theory, but offer higher accuracy. Accordingly, quantum computers
have been predominantly regarded as competitors to only the most accurate and
costly classical methods for treating electron correlation. However, here we
tighten bounds showing that certain first quantized quantum algorithms enable
exact time evolution of electronic systems with exponentially less space and
polynomially fewer operations in basis set size than conventional real-time
time-dependent Hartree-Fock and density functional theory. Although the need to
sample observables in the quantum algorithm reduces the speedup, we show that
one can estimate all elements of the $k$-particle reduced density matrix with a
number of samples scaling only polylogarithmically in basis set size. We also
introduce a more efficient quantum algorithm for first quantized mean-field
state preparation that is likely cheaper than the cost of time evolution. We
conclude that quantum speedup is most pronounced for finite temperature
simulations and suggest several practically important electron dynamics
problems with potential quantum advantage.
