Early fault-tolerant simulations of the Hubbard model

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.