If we want to benchmark a quantum computer, we need a strong classical competitor. For 1D quantum chains, that competitor is usually a tensor-network method, especially MPS and TDVP.
This matters. A 1D Fermi-Hubbard chain is not the worst case for classical computers. It is exactly where tensor networks are strong. If quantum hardware becomes interesting here, that is a serious signal.
What is an MPS?
A matrix product state writes a large quantum state as a chain of smaller tensors. Instead of storing an amplitude for every possible bitstring, we factorize the state along the 1D chain.
\[|\psi\rangle=\sum_{s_1,\ldots,s_L}A^{[1]s_1}A^{[2]s_2}\cdots A^{[L]s_L}|s_1\ldots s_L\rangle\]The bond dimension chi controls how much correlation and entanglement the MPS can carry between two parts of the chain. A larger chi can be more accurate. It is also more expensive.
Why chi is not a detail
In an MPS calculation, we cannot simply pick a chi and declare victory. We have to check whether the observables stabilize as chi grows. If chi64, chi128, and chi256 still move visibly, the calculation is not fully converged.
In our local 120q/30 comparison, chi256 was a major improvement over lower bond dimensions, especially for spin and double occupancy. But the run still hit the maximum bond cap. That means chi256 is a better reference, but not a mathematical proof of full convergence.
The local chi256 run
For the 120-qubit / 60-site / 30-step instance, the local quimb MPS chi256 run took:
9033 seconds wall time.
That is about 2 hours, 30 minutes, and 33 seconds.
The global values were physically reasonable
- total charge around 60;
- total spin around 0.044;
- mean double occupancy around 0.2091.
But the main role of this run is as a classical reference. It lets us score the hardware and other classical shortcuts.
How do we score?
For local observables, one useful metric is RMSE
\[\mathrm{RMSE}=\sqrt{\frac{1}{N}\sum_i\left(x_i-y_i\right)^2}\]This compares the hardware observable with the classical reference site by site. We can do this for occupation, charge, spin, and double occupancy.
One important result was that raw hardware sometimes scored better than readout-corrected hardware for local RMSE against chi256, while readout correction improved some global quantities. This shows why we should not trust a single metric.
TDVP in the paper
The Q-CTRL paper compares with much larger TDVP bond dimensions, such as chi2048 and chi4096. That is much heavier than our local chi256 run. Therefore our local comparison is a poor man's overlap with the paper, not a full reproduction of the headline claim.
That is not a weakness if we say it clearly. It makes the local comparison useful: we can see the classical cost grow on an ordinary machine without pretending to reproduce the full paper.
The lesson
Tensor networks are strong, especially in 1D. But their reliability is not free. We have to test bond-dimension convergence. We have to check whether the observables stabilize. We have to count runtime and memory.
For the quantum comparison, this means
The classical time is not just the runtime of one chosen chi. The classical workflow also includes the question of whether that chi is trustworthy enough.
This brings us back to time-to-answer. The quantum run can quickly give a useful observable. The tensor-network route can be more accurate and more controlled, but it pays for convergence checks.
That is not a simple victory for either side. It is exactly the interesting boundary.
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
- Local chi256 summary:
results_recomputed/quimb_mps_q120_steps30_chi256_20260624/classical_tensor_120q30_chi256_result.md