The first real run in this 2D route is a 3x3 lattice. That is small, but it is not arbitrary. A 3×3 lattice is small enough for exact fixed-sector diagonalization, and large enough to exercise the two-dimensional code path: site indexing, bonds, particle sectors, charge/spin observables, time evolution, and hardware measurement decoding.
The role of 3×3 is validation, not scale.
Configuration
The current 3×3 validation run uses:
- lattice:
3x3 - sites:
9 - qubits:
18 - holes:
1 - particle sector:
N_up = N_down = 4 - exact fixed-sector ED dimension:
15876 - full Hubbard Hamiltonian nonzeros:
368046 - hardware circuit family:
number_preserving_t_only
The one-hole sector is useful because it is not trivial but still exactly tractable. It also lets the pipeline test particle-sector survival explicitly: after measurement, each bitstring can be checked against the expected number of spin-up and spin-down particles.
Why exact fixed-sector ED comes first
Exact diagonalization is expensive, but for 3×3 it is still possible. That is why it should come before hardware interpretation.
The fixed-sector exact calculation gives a clean reference for the same observables measured on hardware:
- site charge;
- site spin-z;
- site double occupancy;
- nearest-neighbor bond charge correlations;
- nearest-neighbor bond spin-z correlations.
It also gives a time sweep. That is important because a circuit can look fine at step 1 while failing quickly at later time. In this route, the exact full Hubbard sweep through step 4 keeps total charge and total spin-z conserved and shows early spin melting and doublon formation.
Exact full ED time sweep
The full exact fixed-sector time sweep gives:
| step | total charge | total spin-z | mean abs spin-z | mean doublon |
|---|---|---|---|---|
| 0 | 8.000000 | 0.000000 | 0.888889 | 0.000000 |
| 1 | 8.000000 | 0.000000 | 0.855116 | 0.016223 |
| 2 | 8.000000 | 0.000000 | 0.778109 | 0.049209 |
| 3 | 8.000000 | -0.000000 | 0.707763 | 0.069466 |
| 4 | 8.000000 | -0.000000 | 0.674849 | 0.064354 |
This is the physical reference for the small run. Total charge remains 8. Total spin-z remains zero. The mean absolute spin-z decreases, which is a simple signature of early spin melting. Double occupancy rises and then bends, showing that the interacting dynamics are not just a static bitstring pattern.
TDHF and Gaussian/free baselines
The validation ladder also includes TDHF and Gaussian/free evolution.
TDHF is a mean-field method. It can be very good at early times for diagonal observables, but it is not exact. It can deviate as interaction effects and correlations build up.
The Gaussian/free baseline removes the interaction structure. It is useful because it asks: what would happen if the system were basically noninteracting? When the Gaussian/free result diverges quickly from exact full ED, that is a signal that the interacting part of the model is already visible in the chosen observables.
Against exact full ED, the early-time RMSEs are:
| step | source | charge RMSE | spin-z RMSE | doublon RMSE |
|---|---|---|---|---|
| 1 | exact t-only | 0.000519 | 0.000896 | 0.000079 |
| 2 | exact t-only | 0.001871 | 0.003245 | 0.000982 |
| 3 | exact t-only | 0.013111 | 0.006454 | 0.003472 |
| 4 | exact t-only | 0.031148 | 0.009996 | 0.007165 |
| 1 | TDHF | 0.000004 | 0.000029 | 0.000040 |
| 2 | TDHF | 0.000196 | 0.001375 | 0.001822 |
| 3 | TDHF | 0.001202 | 0.008724 | 0.011952 |
| 4 | TDHF | 0.003642 | 0.019529 | 0.032194 |
| 1 | Gaussian/free | 0.000378 | 0.001924 | 0.000889 |
| 2 | Gaussian/free | 0.005167 | 0.026965 | 0.011859 |
| 3 | Gaussian/free | 0.019970 | 0.109488 | 0.044513 |
| 4 | Gaussian/free | 0.042469 | 0.254813 | 0.093432 |
The pattern is useful. TDHF is excellent at the earliest step, but doublon starts to drift later. The Gaussian/free baseline separates faster, especially in spin-z and double occupancy.
That gives the hardware comparison a real reference frame. A hardware result should not merely look smooth. It should land near the right reference on the same observables.
What the 3×3 run proves and does not prove
The 3×3 run proves that the pipeline can define and compare a small 2D Hubbard problem across exact ED, TDHF, Gaussian/free, IBM hardware, and Fire Opal.
It does not prove cuprate physics at scale. It does not prove superconducting order. It does not prove quantum advantage.
That limitation is exactly why it is useful. It is a controlled place to find the errors before scaling up.