This project follows a concrete 2D Fermi-Hubbard simulation route: from the earlier 1D IBM/TDVP validation series, through a cuprate-oriented square-lattice Hubbard model, to exact 3×3 validation, IBM and Fire Opal diagnostics, and tensor-network baselines up to the current 6×6 diagnostic step.
The claim boundary is practical and cautious. The current 2D results are diagnostic validation data. They are not a claim of d-wave superconductivity, not material evidence for cuprates, and not a quantum-advantage claim.
Two 2D routes: XY approximation and fermionic simulation
Within this cuprate-inspired 2D Fermi-Hubbard project, we distinguish two simulation routes. Both simulate time evolution. Dynamic describes evolution in time; it does not, by itself, mean that fermionic statistics are implemented.
- 2D XY approximation (t, t′, U). This route includes nearest-neighbour hopping t, diagonal hopping t′ and onsite interaction U. Its bare XY hopping omits the nonlocal Jordan-Wigner parity strings needed for the corresponding fermionic hopping in this 2D encoding.
- 2D fermionic simulation (t, t′, U). This route includes the same model terms together with the fermionic exchange signs, using explicit Jordan-Wigner strings or a fermionic swap network with physical hopping and interaction gates. Correct fermionic signs do not remove finite-time-step or classical-reference errors.
The historical t-only route is different
The earlier number_preserving_t_only route is not the current XY route described above: it also omits diagonal hopping. It was retired from new runs on 22 October 2026. Its saved results, original circuit identities and retrieval remain available. See the retirement note.
Classical references and hardware measurements
MPS and other tensor-network outputs are classical calculations, not hardware measurements. The recent 6×6 XY-versus-fermionic comparison is a finite-bond-dimension, finite-time-step classical estimate, not a new hardware result or a certified hardware-error bound. Neither bond-dimension nor time-step convergence is established for that 6×6 comparison.
Earlier hardware results remain attributed to the circuits that actually ran; they are not retrospectively relabelled as measurements of the newer fermionic route. Fez and Nighthawk identify processors, not particle statistics. The Nighthawk series documents its separate full-fermion experiments.
New report: orbital rotation versus fermionic swap network
29 October 2026. Exact checks on 2×3 and 3×3 and an offline 6×6 compilation comparison. Both routes retain t, diagonal t′ and U with fermionic signs. The tested orbital implementation has comparable small-system splitting error, but uses about 14.2% more native CZ gates at two steps. No new hardware execution.
Article series
- Part 1: from 1D to 2D
- Part 2: 2D Hubbard, cuprates, and Google/Willow context
- Part 3: our 3×3 implementation
- Part 4: time evolution and hardware diagnostics
- Part 5: 4×4 hardware and tensor baselines
- Part 6: 6×6 tensor baseline, IBM diagnostics, and Fire Opal Fez