The 1D Fermi-Hubbard chain is a good testbed. But the bigger physics ambition is in 2D.
Why? Because the 2D Hubbard model is much closer to the questions around strongly correlated materials, including the classic motivation from cuprates and high-temperature superconductivity. A 1D chain is instructive. A 2D lattice is physically more ambitious.
The 2D Hamiltonian looks almost the same
\[H=-t_h\sum_{\langle i,j\rangle,\sigma}\left(c^\dagger_{i,\sigma}c_{j,\sigma}+h.c.\right)+U\sum_i n_{i,\uparrow}n_{i,\downarrow}\]The difference is in the neighbors. In 1D, a site usually has a left and right neighbor. In 2D, a site has neighbors in two directions. That sounds like a small change, but computationally it changes almost everything.
Why 2D is harder classically
MPS and TDVP are strong in 1D because the entanglement along a line remains relatively manageable. In 2D, we often have to unroll a lattice into a 1D path. Then some physical neighbors become far apart in the MPS ordering. The entanglement structure becomes much harder.
As a result, the required bond dimension grows much faster. What is still possible in 1D with a large but manageable chi can become extremely expensive in 2D.
There are 2D tensor-network methods, such as PEPS, but they also have heavy costs and difficult convergence questions. 2D is not a small extension of 1D. It is a different class of problem.
Why 2D is also harder on quantum hardware
For quantum hardware, 2D is not automatically easy. The mapping becomes more complex. fSWAP routing becomes even more important. The hardware connectivity has to support more interaction directions. A snake layout may still help, but now it has to embed a 2D structure into a limited hardware graph.
The core question becomes
Can we map 2D Hubbard locality so that the hardware is not overwhelmed by routing?
This is exactly where co-design becomes important again. We do not want to build an abstract 2D circuit first and hope that the compiler fixes it. We want the mapping, layout, and hardware to be designed together.
A realistic route
The path toward 2D does not have to start with a large square lattice. A sensible route is:
- small plaquettes, for example 2 by 2;
- ladders, for example 2 by L;
- cylinders with limited width;
- larger 2D patches;
- only then truly material-like 2D regimes.
Every step should be validated exactly or classically where that is still possible. The lesson from 1D remains: larger hardware plots are interesting, but small exact checks are still needed for strong physics claims.
What do we take from 1D?
The 1D series teaches five lessons that remain true for 2D
- the physical observable must be clear in advance;
- the fermion-to-qubit mapping is not a detail;
- fSWAP routing is physics, not just compiler work;
- quantum timing must be compared with classical convergence workflows;
- heatmaps and local observables are often more understandable than global statevector claims.
The role of quantum hardware
2D Hubbard is exactly the kind of problem where quantum hardware may become really interesting. Not because classical methods disappear, but because the classical costs and validation questions grow much faster than in 1D.
The right ambition is not
we did 1D, so 2D is solved.
The right ambition is
1D is the testbed where we learn to measure, map, mitigate, and compare. 2D is where the real physical pressure begins.
That makes 2D the natural next step for this project line.
Sources and project links
- Q-CTRL Fermi-Hubbard paper: https://arxiv.org/abs/2605.04025
- Project repository: https://github.com/BramDo/fermi-hubbard-60q-tdvp
- Classical response paper: https://arxiv.org/abs/2606.04771