A larger quantum circuit is not automatically a better simulation. First, we must establish which model the circuit implements. That distinction proved important in our project: a shallow, particle-number-preserving XY approach is not necessarily the same calculation as the full two-dimensional fermion model.
Hopping, repulsion and spin
The Hubbard model describes particles moving between neighbouring sites. The hopping parameter t sets the energy scale. Two particles with opposite spins can occupy the same site, but in our repulsive model this costs an energy U. Double occupancy is therefore an interesting observable: it responds to interactions and to the dynamics.
Our target also includes diagonal hopping t’=-0.25, alongside t=1 and U=8. Thus U=8 means U/t=8. It does not mean eight interactions or eight seconds. The evolution time T=0.4 is also a model quantity, expressed in units of hbar/t. It is not directly equal to the duration of the quantum job.
\[H=-t\sum_{\langle i,j\rangle,\sigma}(c^\dagger_{i\sigma}c_{j\sigma}+\mathrm{h.c.})-t’\sum_{\langle\!\langle i,j\rangle\!\rangle,\sigma}(c^\dagger_{i\sigma}c_{j\sigma}+\mathrm{h.c.})+U\sum_i n_{i\uparrow}n_{i\downarrow}.
\]
Here c-dagger creates a fermion and c annihilates one. The first two sums move particles; the final term penalises double occupancy. We use open boundaries, not lattice edges joined periodically to one another.
The minus signs that must not be forgotten
Fermions obey anticommutation relations. Exchanging two fermions can introduce a minus sign. A qubit does not automatically have that property. A fermion-to-qubit mapping must therefore preserve the algebra explicitly.
Jordan-Wigner does this using parity strings of Z operators. These strings can become long for modes that are far apart in the chosen ordering. In two dimensions, this creates tension between a local physical lattice and the qubit ordering used to implement it.
The early bare-XY/t-only approaches were useful technical pilots, but did not contain every nonlocal fermion sign and model term. The later full approach retains parity, t’, and U. Both remain in the archive, with distinct labels and references.
What is conserved, and what must genuinely be predicted
We start on 6×6 with 32 particles: sixteen up and sixteen down. The model preserves both numbers. A result close to N=32 is a useful check, but we already know the value 32 before running the simulation.
The nontrivial questions are where the charge moves, how much local spin structure remains, and where double occupancy develops. The known total charge cannot replace that unknown local dynamics.
Our model is inspired by correlated-electron physics. This does not mean we have simulated a real cuprate material or superconductivity. This series concerns a concrete, programmable model and the quality of its calculation.
Sources: theory/HANDBOOK.md, cuprate_2d_fermionic_mapping.py, cuprate_2d_fermionic_dynamics.py and the independent small-lattice checks in validate_cuprate_2d_fermionic_dynamics.py in the project repository.