The 1D Fermi-Hubbard chain as a test case
A cold-atom lab builds a Hubbard chain with ultracold lithium atoms, lasers, and a quantum gas microscope. A digital quantum computer encodes the same chain in qubits, translates the Hamiltonian into gates, and reads out bitstrings after many repetitions. In both cases, researchers prepare an initial state, induce a local disturbance, let the system evolve, and measure how spin and charge spread.
Have these become two implementations of the same experiment?
Short answer: for a precisely defined question about the 1D Fermi-Hubbard model, a digital quantum processor can take over an increasing number of functions of a cold-atom experiment. It does not thereby replace the complete physics laboratory. It becomes a different kind of quantum lab: more programmable, but also dependent on fermion-to-qubit mapping, Trotter approximation, calibration, and error suppression.
The same Hamiltonian, different matter
The comparison starts with the model:
\[H=-t_h\sum_{i,\sigma}\left(c^\dagger_{i,\sigma}c_{i+1,\sigma}+c^\dagger_{i+1,\sigma}c_{i,\sigma}\right)+U\sum_i n_{i,\uparrow}n_{i,\downarrow}.\]The first term lets fermions hop between neighboring lattice sites. The second assigns an energy to double occupation. For U > 0, the interaction is repulsive; for U < 0, it is attractive.
In an optical-lattice experiment, the lattice sites are physically formed by laser light and the fermions are ultracold atoms. The Fermi-Hubbard dynamics there emerge largely and directly from the physical apparatus.
On a gate-based quantum computer, the fermions are not physically present. Each spin orbital is encoded in a qubit. A Jordan-Wigner transformation preserves the fermionic anticommutation rules, and a sequence of compiled gates approximates the time evolution. The quantum processor therefore does not simulate how its own qubits behave spontaneously; it forces the qubits to follow the mathematical dynamics of the selected model.
If the scientific question concerns only the Hubbard model, the difference in physical carrier need not be an obstacle. If the question concerns lithium atoms, finite temperature, optical confinement, or the operation of a quantum gas microscope, the digital processor cannot replace the cold-atom lab.
The cold-atom experiment: removing one atom
In 2020, Vijayan and colleagues published the clearest laboratory observation of time-dependent spin-charge deconfinement in a fermionic Hubbard chain. Using lithium-6 atoms, they created repulsive chains with U/t approximately between 8 and 20. The chains contained about thirteen atoms on average, with a nearly half-filled central region of about nine sites and short-range antiferromagnetic spin correlations.
With a narrow laser pulse, they removed one atom from the central site. The atom was removed from the intended site with a probability of about 78 percent. They then let the system evolve for an adjustable time and recorded a snapshot using spin- and density-resolved quantum gas microscopy. The experiment was repeated thousands of times for each evolution time.
Removing one fermion in one dimension creates two collective excitations:
- holon: carries charge but no spin;
- spinon: carries spin but no charge.
At U/t = 15, the researchers measured a propagation speed of 3.08 +/- 0.09 sites per millisecond for the charge excitation and 0.58 +/- 0.04 sites per millisecond for the spin excitation. The ratio was 5.31 +/- 0.43. The charge excitation appeared in the hole density. The spin excitation was tracked through spin correlations in so-called squeezed space, in which holes and doublons are removed from the chain during the analysis.
The paper goes beyond observing two different velocities. Using spin-hole-spin correlators and local magnetization fluctuations, it shows that the spinon and holon truly separate in space. The experiment also contains genuine laboratory effects, including finite temperature, harmonic confinement, imperfect atom removal, and a finite chain.
Hilker and colleagues had already revealed the static correlation side in 2017. In doped 1D Hubbard chains, they found hidden antiferromagnetic order using nonlocal string correlators. Holes dilute the visible spin pattern, but after correcting for their positions, the underlying order reappears. This paper explains why charge defects and spin order cannot be described by a single local density alone.
In 2019, Salomon and colleagues made a related mechanism visible: holes, doublons, and excess spins act as delocalized domain walls in the antiferromagnetic order of doped Hubbard chains. Hilker and Salomon thus provide the equilibrium and correlation background for the time-dependent deconfinement that Vijayan later tracked directly.
Google 2020: the first digital bridge
In 2020, Google AI Quantum simulated an 8-site Fermi-Hubbard chain on 16 superconducting qubits. The researchers prepared localized charge and spin distributions, suddenly removed the confining potential, and switched on the interaction. They then measured
\[\rho_i^{\mathrm{c}}(t)=\langle n_{i,\uparrow}(t)+n_{i,\downarrow}(t)\rangle\]and
\[\rho_i^{\mathrm{s}}(t)=\langle n_{i,\uparrow}(t)-n_{i,\downarrow}(t)\rangle.\]At stronger interaction, the charge density spread faster than the spin density. The deepest circuits contained more than six hundred two-qubit gates. To make such circuits usable, Google combined rapid Floquet calibration with postselection on the correct particle number, averaging over different qubit assignments, and rescaling of damped expectation values.
This was an important digital demonstration, but not a digital copy of Vijayan’s experiment. The initial state and quench differed. Google released a preformed density peak from a trap; Vijayan locally removed one atom from a nearly half-filled antiferromagnetic chain. Google focused mainly on one-point densities and their spreading; Vijayan additionally used conditional and multipoint correlators to establish the spatial deconfinement of the spinon and holon.
Google 2020 showed that a gate-based processor can approach the same phenomenon in the same model language. It did not yet replace the same laboratory experiment.
Q-CTRL and IBM 2026: almost the same digital experiment
The Q-CTRL/IBM preprint by Hartnett and colleagues, revised to version 2 in July 2026, makes the comparison much more direct. The researchers started with a 31-site Neel chain, introduced a vacancy in the middle, and simulated repulsive interactions from U/t_h = 0 through 14. They used 62 qubits, evolution times up to t = 9/t_h, and as many as 90 Trotter steps.
They measured a charge tracer:
\[C_i^{\mathrm{c}}(t)=\langle n_{i,\uparrow}(t)+n_{i,\downarrow}(t)\rangle-\langle n_{i,\uparrow}(0)+n_{i,\downarrow}(0)\rangle\]and a connected spin tracer:
\[C_i^{\mathrm{s}}(t)=4\left[\langle S_i^z(t)S_{i_0}^z(t)\rangle-\langle S_i^z(t)\rangle\langle S_{i_0}^z(t)\rangle\right],\]where i_0 is the location of the initial vacancy. Separate charge and spin wavefronts and their velocities were extracted from the two space-time heatmaps. The results were compared with TDVP calculations and, in appropriate limits, with free-fermion results and Bethe-ansatz predictions.
This protocol is structurally close to the cold-atom experiment:
| Experimental function | Vijayan 2020, cold atoms | Hartnett 2026, digital processor |
|---|---|---|
| Initial state | Nearly half-filled repulsive chain with short-range antiferromagnetic correlations | Pure Neel Fock state |
| Local quench | Remove one lithium atom with a laser | Omit one central occupation from the bitstring |
| Charge signal | Spatial hole density | Change in local total density |
| Spin signal | Squeezed-space spin correlations and multipoint correlators | Connected two-point spin tracer |
| Dynamical result | Holon and spinon have different velocities and separate spatially | Charge and spin wavefronts have different velocities |
| Main error source | Temperature, confinement, quench efficiency, and imaging | Gate errors, decoherence, Trotter error, readout, and finite shot count |
The observables are therefore still not identical. The spin measurement differs in particular. Vijayan tracks nearest-neighbor correlations in squeezed space and uses additional multipoint correlators. Hartnett uses a connected spin correlator relative to the original defect site. Both measurements nevertheless isolate a spatially and temporally varying spin signal alongside a separate charge signal.
Important boundary: the spin-charge figures in the Q-CTRL preprint come from the repulsive 31-site, 62-qubit experiment. The widely discussed largest run in the same paper uses 60 sites and 120 qubits, but focuses on U/t_h = -2 and primarily studies local occupation dynamics. That 120-qubit run is therefore not, by itself, the digital repetition of the repulsive spinon-holon experiment.
When does this truly count as replacement?
The word replacement is defensible only if we separate it into different levels.
| Level | Assessment | Reason |
|---|---|---|
| Run the same mathematical model | Yes | Both platforms realize the time evolution of a 1D Fermi-Hubbard Hamiltonian. |
| Answer the same physics question | Largely | Both can measure whether spin and charge propagate at different velocities after a local disturbance. |
| Run exactly the same protocol | Not yet | The initial state, quench, temperature, boundaries, and spin correlators differ. |
| Replace the complete cold-atom lab | No | A digital processor investigates the model, not all physics and engineering of the atomic realization. |
| Establish new physics without independent validation | Not yet convincingly | Where classical methods and quantum data diverge, it is not automatically known which result is correct. |
The last row is essential. For the 120-qubit run, Q-CTRL reports agreement with TDVP up to approximately t = 5.2/t_h at a bond dimension as high as chi = 4096. At t = 6/t_h, the discrepancy rises to about four percent, and the paper calls the accuracy of the digital simulation indeterminate. This is not a failure, but it shows that the disappearance of a classical reference does not automatically validate the quantum result.
A replacement digital lab therefore needs a validation ladder:
| Step | Check |
|---|---|
| 1 | Recover the exact limit at U = 0. |
| 2 | Compare short times with exact diagonalization or a converged tensor-network calculation. |
| 3 | Compare strong coupling with analytical results, such as Bethe ansatz where applicable. |
| 4 | Reproduce the same trend across different qubit selections, calibration times, or hardware platforms. |
| 5 | Compare an overlapping quench and the same dimensionless observables with the cold-atom experiment. |
Only then is it responsible to explore a harder parameter regime that is no longer classically manageable.
What does this mean for the EduKaizen project?
The current EduKaizen series and repository already demonstrate the digital-lab workflow: a 1D Fermi-Hubbard chain is encoded in qubits, evolved on IBM hardware, and compared with TDVP through local occupation, charge, spin, and doublon observables. The stored 60-qubit run uses 30 sites, eight Trotter steps, t = 1.6/t_h, and the attractive setting U/t_h = -2. The repository correctly makes no claim of practical quantum advantage.
The result need not have exactly the same velocity in sites per millisecond. The cold-atom clock is set by the tunneling rate in hertz; the digital simulation generally uses the dimensionless time t_h t / hbar, or t_h t when hbar = 1. The correct comparison therefore uses dimensionless velocities and the same ratio U/t_h, not raw laboratory seconds versus processor runtime.
Conclusion
A quantum computer cannot replace a quantum lab in general. It can functionally replace a specific Hubbard experiment when five elements match: the Hamiltonian, initial state, quench, observables, and validation procedure.
Hilker 2017 provides the correlation background. Vijayan 2020 shows in a cold-atom lab how a locally removed fermion separates into a faster holon and a slower spinon. Google 2020 shows the first gate-based separation of charge and spin density. Hartnett 2026 brings the digital method closest to the same local quench and the same wavefront question.
The most accurate picture is therefore not that the quantum computer makes the physical lab obsolete. They are two complementary quantum laboratories. The cold-atom lab provides a direct material realization with temperature and experimental imperfections. The digital processor provides a programmable realization in which parameters, initial states, and measurement functions can be changed systematically. Precisely when both platforms give the same answer in their overlapping regime, we gain confidence to take the digital simulator farther into regions where classical calculations or analog platforms become more difficult.
Sources and project links
- T. A. Hilker et al., “Revealing Hidden Antiferromagnetic Correlations in Doped Hubbard Chains via String Correlators”, Science 357, 484-487 (2017). DOI
- J. Vijayan et al., “Time-Resolved Observation of Spin-Charge Deconfinement in Fermionic Hubbard Chains”, Science 367, 186-189 (2020). DOI
- G. Salomon et al., “Direct Observation of Incommensurate Magnetism in Hubbard Chains”, Nature 565, 56-60 (2019). DOI
- Google AI Quantum and collaborators, “Observation of Separated Dynamics of Charge and Spin in the Fermi-Hubbard Model”, arXiv:2010.07965 (2020). arXiv
- G. S. Hartnett et al., “Fast, Accurate, High-Resolution Simulation of Large-Scale Fermi-Hubbard Models on a Digital Quantum Processor”, arXiv:2605.04025v2 (2026), preprint. arXiv
- EduKaizen, “Fermi-Hubbard on a Quantum Computer, Part 1: the 1D Hubbard Model”. EduKaizen article
- EduKaizen project repository, “fermi-hubbard-60q-tdvp”. GitHub repository