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Published January 2022 | Accepted Version
Journal Article Open

Early fault-tolerant simulations of the Hubbard model

Abstract

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 × L lattice with length L = 2^k. On lattices with length L ≠ 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× 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× to 9× (depending on the parameter regime) over the best previous estimates for 8 × 8 and 16 × 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.

Additional Information

© 2021 IOP Publishing Ltd. Received 4 August 2021; Revised 15 October 2021; Accepted 19 October 2021; Published 18 November 2021. We thank Fernando Brandão for proposing a study of the Hubbard model and useful early discussions. We thank Yuan Su for discussions on commutator bounds and Sam McArdle for detailed feedback on the manuscript.

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Created:
August 22, 2023
Modified:
October 23, 2023