A good quantum-advantage discussion needs a strong classical competitor. In this series, tensor-network TDVP is the standard competitor, but there is another interesting route: Majorana propagation.
This route does not try to keep the full quantum state. Instead, it follows a selected observable in the Heisenberg picture. For a specific observable, that can be much cheaper than simulating the entire state.
That makes Majorana propagation dangerous for a simple quantum-advantage claim. If we only want a local observable, why should we follow the full wavefunction?
The local test
In a separate laptop project, a Majorana-propagation baseline was compared with the 120-qubit / 60-site / 30-step Fermi-Hubbard run.
The observable was mean double occupancy. That is narrower than the full charge/spin heatmap, but physically important.
The comparison
\[\mathrm{cutoff}=2:\ 20.96\,\mathrm{s},\quad |D-D_{\chi256}|=0.0382328677\\ \mathrm{cutoff}=4:\ 1153.51\,\mathrm{s},\quad |D-D_{\chi256}|=0.0004838001\]Cutoff 2 was very fast. The wall time was about 20.96 seconds. That is even faster than the local Fire Opal main+readout execution proxy of about 33 seconds. But the error against the chi256 MPS reference was clearly visible.
Cutoff 4 was much more accurate. The error against chi256 for mean double occupancy was about 0.00048. But the runtime rose to about 1153.51 seconds, or just over 19 minutes.
Why this is interesting
If we know after the fact that cutoff 4 is sufficient, Majorana propagation is a strong classical method. For this single observable, cutoff 4 was much closer to chi256 than the hardware routes.
But the benchmark question is hidden in the phrase “after the fact.”
Beforehand, we do not automatically know which cutoff is trustworthy. Cutoff 2 looked fantastic in runtime, but it was not accurate enough. Cutoff 4 worked much better, but took much longer. To know that, we needed a more expensive reference or a cutoff-convergence study.
This is exactly the time-to-answer point.
Not every classical shortcut is free
A classical shortcut with a truncation parameter has two costs
- the runtime of the chosen parameter;
- the validation needed to know that the parameter is good enough.
In papers and benchmarks, we often quote mainly the first time. For practical use, the second also matters. If the correct cutoff is known only after using a chi256 or TDVP reference, that search belongs to the workflow.
What does this mean for the quantum run?
The quantum computer does not win on every observable. For mean double occupancy, Majorana cutoff 4 was much closer to chi256 than the hardware. We should say that honestly.
But the quantum route had a different advantage: it quickly produced a full-instance hardware measurement without choosing a classical cutoff in advance. The output was not perfect, but the route was direct.
The conclusion is subtle
Majorana propagation weakens a simple quantum-advantage claim, but it makes the time-to-answer question sharper. The correct claim is not:
the quantum computer beats Majorana propagation.
The better claim is
For this instance, a quantum run can quickly produce a useful observable, while a classical Majorana shortcut needs a cutoff choice whose reliability is not known in advance.
That is the kind of nuance a serious benchmark needs.
Sources and project links
- Project repository: https://github.com/BramDo/fermi-hubbard-60q-tdvp
- Majorana benchmark in this repository:
docs/120q30_majorana_benchmark_summary.md - Q-CTRL Fermi-Hubbard paper: https://arxiv.org/abs/2605.04025