The next run after 3×3 is 4x4. This is where the validation strategy has to change. Exact diagonalization is no longer the comfortable general reference, so the route needs tensor baselines.
The 4×4 run in this series uses:
- lattice:
4x4 - sites:
16 - qubits:
32 - holes:
2 - expected sector:
N_up = N_down = 7 - circuit family:
number_preserving_t_only - step:
1 dt = 0.1- hardware backend:
ibm_marrakesh
This is still a shallow early-time run. It is not the final interacting cuprate problem. Its purpose is to test whether the 2D stack still works when the lattice is large enough that exact ED is no longer the default reference.
Circuit depth
The 4×4 dry-run gave:
| route | logical depth | transpiled IBM depth | two-qubit gates | two-qubit depth |
|---|---|---|---|---|
| 3×3 IBM step 1 reference | 8 | 167 | 159 | 52 |
| 4×4 IBM step 1 | 8 | 199 | 341 | 65 |
| 4×4 Fire Opal abstract input | 8 | not hardware-compiled | 64 logical | 5 logical |
The two-qubit gate count roughly doubles relative to 3×3, but the two-qubit depth grows much less because the circuit has parallel structure.
That is encouraging, but not enough. On current hardware, two-qubit depth and sector survival remain the limiting issues.
MPS circuit baseline
For 4×4, this route adds a quimb MPS circuit baseline for the same shallow circuit family. This is important: it is an apples-to-apples circuit baseline for the hardware output. It is not a full Hamiltonian TDVP solution for all cuprate physics, but it is the right reference for the shallow circuit that was actually run.
The MPS convergence ladder was:
| max bond | observed max bond | elapsed s | occupation RMSE vs previous | total charge | total spin-z | mean doublon |
|---|---|---|---|---|---|---|
| 32 | 32 | 13.497 | None | 14.000000 | 0.000000 | 0.020073 |
| 64 | 64 | 49.455 | 0.000202737 | 14.000000 | 0.000000 | 0.020170 |
The chi32 -> chi64 occupation RMSE is about 2.0e-4. For this short shallow step, the MPS run is numerically stable enough to serve as the local reference.
Local baselines against MPS chi64:
| source | charge RMSE | spin-z RMSE | doublon RMSE |
|---|---|---|---|
| TDHF | 0.000660 | 0.002011 | 0.000995 |
| Gaussian/free | 0.000955 | 0.000917 | 0.000088 |
At this very early step, both TDHF and Gaussian/free are close to the shallow circuit baseline on the diagonal observables. That should not be overread. It means this run is still an early-time circuit diagnostic, not a late-time strong-correlation benchmark.
IBM hardware result
The 4×4 IBM job was:
- job id:
d93v9a6vtlqs73ftm6mg - backend:
ibm_marrakesh - status:
DONE - shots:
1024 - main transpiled depth:
199 - two-qubit count:
341 - two-qubit depth:
65
Hardware summary:
| profile | total charge | total spin-z | mean abs spin-z | mean doublon |
|---|---|---|---|---|
| raw | 13.766602 | -0.340820 | 0.650818 | 0.072632 |
| readout-corrected | 13.881690 | -0.242608 | 0.685372 | 0.064137 |
| MPS chi64 | 14.000000 | 0.000000 | 0.833332 | 0.020170 |
RMSE against MPS chi64:
| source | profile | charge RMSE | spin-z RMSE | doublon RMSE |
|---|---|---|---|---|
| IBM | raw | 0.119908 | 0.218926 | 0.059685 |
| IBM | readout-corrected | 0.130838 | 0.190883 | 0.056986 |
| IBM | exact-sector | 0.097011 | 0.123450 | 0.044959 |
The exact-sector result is better than raw and readout-corrected, but it is still not close enough to treat as a clean physical result.
Sector survival
Particle-sector survival is the central diagnostic:
| profile | kept shots | total shots | kept fraction |
|---|---|---|---|
exact sector N_up = N_down = 7 |
133 | 1024 | 0.129883 |
| near sector L1 <= 1 | 462 | 1024 | 0.451172 |
| near sector L1 <= 2 | 762 | 1024 | 0.744141 |
Only about 13 percent of shots remain in the exact expected sector. Near-sector analysis shows that many errors are close to the right sector, but not all the way there.
That is a useful hardware diagnostic. It is also the reason this result should not be presented as a final 4×4 physics result.
Fire Opal result
The 4×4 abstract/native Fire Opal route was not the final usable path. The working route was a qelib retry for the same shallow number_preserving_t_only step. That gives a cleaner comparison because IBM, Fire Opal, TDHF, Gaussian/free, and MPS are all evaluated on the same diagonal observables.
RMSE against MPS chi64:
| source | profile | charge RMSE | spin-z RMSE | doublon RMSE |
|---|---|---|---|---|
| IBM | raw | 0.119908 | 0.218926 | 0.059685 |
| IBM | readout-corrected | 0.130838 | 0.190883 | 0.056986 |
| IBM | exact-sector | 0.097011 | 0.123450 | 0.044959 |
| Fire Opal | raw | 0.065163 | 0.104742 | 0.045752 |
| Fire Opal | readout-corrected | 0.068299 | 0.097232 | 0.041630 |
| Fire Opal | exact-sector | 0.026794 | 0.021459 | 0.013010 |
The Fire Opal result is clearly closer to the MPS baseline than direct IBM. The exact-sector Fire Opal row is especially strong, but it must still be read with the survival fraction in mind.
Sector survival:
| route | exact-sector kept shots | total shots | kept fraction |
|---|---|---|---|
| IBM | 133 | 1024 | 0.129883 |
| Fire Opal | 267 | 1053 | 0.253561 |
Fire Opal roughly doubles the exact-sector survival for this 4×4 case and reduces the RMSE substantially. That is an engineering improvement, not a superconductivity claim.
Conclusion
The 4×4 run is useful because it exposes the next bottleneck.
Direct IBM hardware can run the circuit, but noise and sector leakage dominate. The MPS baseline is stable enough for this short shallow step. Fire Opal is much cleaner than direct IBM, especially after exact-sector conditioning, but the result is still diagnostic because the survival fraction is only about 25 percent.
So the next improvement is not simply a larger lattice. The next improvement is a better route:
- shallower circuits;
- better fermion routing;
- stronger layout selection;
- better readout and sector mitigation;
- a larger MPS convergence ladder, for example
chi = 32, 64, 128; - separate 6×6 analysis only with a matching tensor or operator baseline.
Only after that does a larger 2D dry-run become physically meaningful.