Fault-tolerant cost to beat classical methods on the 2D Hubbard model
In plain words
The two-dimensional Hubbard model, a simple model of electrons hopping on a grid and repelling each other, is a likely first case where a quantum computer could outdo classical methods. How many error-corrected qubits and operations that would take is not pinned down.
Precise statement
For the $2\mathrm{D}$ Fermi-Hubbard model at $U/t = 8$ and hole doping near $1/8$ on L x L lattices, determine the minimum logical qubit count and Toffoli count for phase estimation to give the energy per site to accuracy $10^{-3} t$ at a size L beyond reliable classical extrapolation, including state-preparation cost. Published estimates (e.g. Kivlichan et al., Quantum 2020) give roughly $10^{6}-10^{8}$ Toffoli gates for lattices of about $10 \times 10 \text{ to } 20 \times 20$ sites, without full overlap costs (figures approximate).
What would settle it
An end-to-end resource estimate including state preparation, paired with the best classical error bars at the same L.
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