The two-dimensional Hubbard model is one of the standard minimal models for strongly correlated electrons. It is simple enough to write down, but hard enough that it remains a serious benchmark for classical and quantum methods.

For the cuprate route, the basic idea is that the CuO2 planes in cuprate materials are effectively two-dimensional, and that strong local repulsion, antiferromagnetism, hole doping, and pairing-like correlations all matter.

The one-band Hubbard model is not a full material model. Real cuprates have multiple orbitals, phonons, disorder, material-dependent parameters, finite temperature effects, and other complications. But as a first computational target, the 2D Hubbard model is the right place to build the simulation stack.

The model

The route in this repository uses the square-lattice one-band Hubbard model:

\[
\begin{aligned}
H ={}&
-t \sum_{\langle i,j\rangle,\sigma}
\left(c^{\dagger}_{i\sigma} c_{j\sigma} + \mathrm{h.c.}\right) \\
&-t’ \sum_{\langle\langle i,j\rangle\rangle,\sigma}
\left(c^{\dagger}_{i\sigma} c_{j\sigma} + \mathrm{h.c.}\right) \\
&+U \sum_i n_{i\uparrow} n_{i\downarrow}
-\mu \sum_{i,\sigma} n_{i\sigma}.
\end{aligned}
\]

Here:

• t is nearest-neighbor hopping;
• t' is next-nearest-neighbor hopping;
• U is the onsite repulsion;
• mu is the chemical potential.

For the cuprate-oriented route in this project, the typical parameters are:

• U/t = 8
• t'/t = -0.25
• open boundary conditions
• hole-doped Neel or stripe-like initial states

The first hardware circuits do not implement every term equally. Some runs use a shallow number_preserving_t_only circuit family to keep depth under control. That is a deliberate engineering choice, not a claim that the full interacting model has been solved on hardware.

Why the observables are modest

The first layer of observables is diagonal in the measurement basis:

\[
\begin{aligned}
\mathrm{charge}_i &= n_{i\uparrow} + n_{i\downarrow}, \\
\mathrm{spin}^z_i &= n_{i\uparrow} – n_{i\downarrow}, \\
\mathrm{double\ occupancy}_i &= n_{i\uparrow} n_{i\downarrow}.
\end{aligned}
\]

We also track nearest-neighbor charge and spin-z bond correlations.

This choice is conservative. These observables are directly accessible from computational-basis measurements and are useful for debugging physical structure:

• total charge conservation;
• total spin-z conservation;
• particle-sector survival;
• local spin melting;
• doublon formation;
• bond-level charge and spin correlations.

They do not yet measure off-diagonal d-wave pairing. That would require extra basis rotations, correlator circuits, or more tomographic information. So the present series should be read as a validation ladder for 2D Hubbard dynamics, not as a superconductivity measurement.

Why Google/Willow matters as context

The relevant external comparison is recent 2D Fermi-Hubbard work on Google's Willow processor. Alam et al. report programmable digital simulation of 2D Fermi-Hubbard dynamics using 72 superconducting qubits, with lattice sizes up to 6x6. Their validation uses exact calculations where possible and tensor or operator-propagation methods once exact calculation becomes too expensive. The associated Zenodo dataset includes raw shots and expectation values per physical system.

That matters because it sets the right scale. A serious 2D Hubbard route needs more than a submitted circuit:

• optimized circuit structure;
• hardware-aware layout;
• mitigation or error suppression;
• exact validation where possible;
• tensor/operator baselines where exact validation is no longer possible;
• observable definitions that match across all routes.

That is also the standard this repository is trying to grow toward. The present IBM/Fire Opal route started smaller, with 3x3 and 4x4, because those runs make the validation problem visible before the 72-qubit 6x6 diagnostic step.

Why not jump straight to 6×6?

It is tempting to ask why not submit a 6x6 circuit immediately. The reason is that a larger circuit without a reliable comparison stack is not automatically more informative.

The 3×3 run can be compared to exact ED. The 4×4 run can be compared to a tensor baseline for the same shallow circuit family. Those comparisons reveal whether the hardware output is mostly physics, mostly noise, mostly sector leakage, or some combination of all three.

Only after those smaller runs are understood does a larger dry-run become useful. In this series, the current 6x6 step is therefore reported as a diagnostic comparison against an MPS tensor baseline, not as a standalone physics claim.

Sources

• Alam et al., "Programmable digital quantum simulation of 2D Fermi-Hubbard dynamics using 72 superconducting qubits", arXiv:2510.26845, https://arxiv.org/abs/2510.26845
• Dataset for Alam et al., Zenodo 17881822, https://zenodo.org/records/17881822
• Hartnett et al., "Fast, accurate, high-resolution simulation of large-scale Fermi-Hubbard models on a digital quantum processor", arXiv:2605.04025, https://arxiv.org/abs/2605.04025