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 et al 2019 Quantum 4 296] found that the lowest resource costs were achieved by split-operator Trotterization combined with the fast-fermionic Fourier transform (FFFT) on an L x L lattice with length L = 2(k). On lattices with length L not equal 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 16x 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 x to 9x (depending on the parameter regime) over the best previous estimates for 8 x 8 and 16 x 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.