This project does not start with qubits. It starts with a physical model: the one-dimensional Fermi-Hubbard model. The model describes fermions that can hop through a lattice and interact locally when two opposite spins occupy the same site.
You can picture the 1D system as a chain of sites
1 – 2 – 3 – 4 – … – 60
Each site has two fermionic modes: spin-up and spin-down. A site can be empty, contain a spin-up fermion, contain a spin-down fermion, or contain both. When both spin states are present on the same site, the interaction term contributes.
The Hamiltonian is compact
\[H=-t_h\sum_{i,\sigma}\left(c^\dagger_{i,\sigma}c_{i+1,\sigma}+c^\dagger_{i+1,\sigma}c_{i,\sigma}\right)+U\sum_i n_{i,\uparrow}n_{i,\downarrow}-\mu\sum_{i,\sigma}n_{i,\sigma}\]The first term is hopping. A fermion moves from site i to site i+1. The parameter t_h sets the hopping strength. The second term is the onsite interaction. It matters when spin-up and spin-down occupy the same site. The parameter U determines whether this double occupation is energetically costly or favorable.
In the Q-CTRL Fermi-Hubbard paper and in the larger local comparison behind this series, the attractive regime is central: U/t_h = -2. This is a regime where pairing and double occupancy become interesting observables.
Why is this hard?
Fermions are not ordinary bits. Fermionic operators anticommute. When two occupied fermionic modes are exchanged, a minus sign appears. This is not a bookkeeping luxury. It is a physical property of fermions.
For a quantum computer this means that we cannot simply place the sites on qubits and use ordinary swaps. The mapping must preserve the fermionic signs. That is why Jordan-Wigner ordering, fSWAP gates, and the snake layout become important later in the series.
What do we measure?
For a material-like view, we do not try to inspect the full quantum state. That state is far too large. Instead, we measure local observables.
\[n_\uparrow(i)=\langle n_{i,\uparrow}\rangle,\quad n_\downarrow(i)=\langle n_{i,\downarrow}\rangle\\ \mathrm{charge}(i)=n_\uparrow(i)+n_\downarrow(i)\\ \mathrm{spin}(i)=n_\uparrow(i)-n_\downarrow(i)\\ D(i)=\langle n_{i,\uparrow}n_{i,\downarrow}\rangle\]The charge tells us how many particles are locally present. The spin tells us whether there is locally more spin-up than spin-down. Double occupancy is directly sensitive to the interaction U, because it measures how often both spin states occupy the same site.
A chain with 60 sites has 120 spin-orbitals: 60 sites times two spin states. On a digital quantum computer, that naturally becomes 120 qubits. This is the scale where the Q-CTRL/IBM Fermi-Hubbard demonstration becomes interesting.
Why 1D?
A 1D chain is simpler than a real 2D or 3D material. But that is precisely why it is a useful testbed. Tensor-network methods such as MPS and TDVP are strong in one dimension. If quantum hardware becomes competitive here, it is not because classical methods are weak. It is because the specific task, layout, and observables form a serious benchmark.
There is also a real experimental analogy. Cold-atom groups can build 1D Fermi-Hubbard chains using optical lattices and quantum gas microscopes. They measure site-resolved charge and spin as a function of time. Our digital heatmaps are not the same experiment, but they use the same language: site position, time, charge, and spin.
The question in this series is therefore not: "Are quantum computers faster everywhere?" The sharper question is:
How quickly can we obtain a useful observable for this concrete Fermi-Hubbard task, and how much classical validation or tuning is needed to trust it?
That is a better near-term question than a broad quantum-advantage slogan.
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
- Time-resolved 1D Hubbard cold-atom experiment: https://arxiv.org/abs/1905.13638