  Simulation of the Hubbard model is a leading candidate for the first useful
applications of a fault-tolerant quantum computer. A recent study of quantum
algorithms for early simulations of the Hubbard model [Kivlichan \textit{et
al.} Quantum 4 296 (2019)] found that the lowest resource costs were achieved
by split-operator Trotterization combined with the fast-fermionic Fourier
transform (FFFT) on an $L \times L$ lattice with length $L=2^k$. On lattices
with length $L \neq 2^k$, Givens rotations can be used instead of the FFFT but
lead to considerably higher resource costs. We present a new analytic approach
to bounding the simulation error due to Trotterization that provides much
tighter bounds for the split-operator FFFT method, leading to $16 \times$
improvement in error bounds. Furthermore, we introduce plaquette Trotterization
that works on any size lattice and apply our improved error bound analysis to
show competitive resource costs. We consider a phase estimation task and show
plaquette Trotterization reduces the number of non-Clifford gates by a factor
$5.5\times$ to $9 \times$ (depending on the parameter regime) over the best
previous estimates for $8 \times 8$ and $16 \times 16$ lattices and a much
larger factor for other lattice sizes not of the form $L=2^k$. In conclusion,
we find there is a potentially useful application for fault-tolerant quantum
computers using around one million Toffoli gates.
