An upstream implementation changes the classical comparison
Part 7 of this series introduced Majorana propagation as a classical competitor for a gate-based Fermi-Hubbard simulation. That first comparison used our own EduKaizen implementation. We can now repeat the calculation with Monoprop, Algorithmiq’s official high-performance implementation of Majorana and Pauli propagation.
This distinction matters. Two programs can both use the label “Majorana propagation” and still apply different coefficient thresholds, term caps, operator representations, and execution strategies. A runtime result from one implementation is not automatically a benchmark of the method as a whole.
The same observable
The earlier quick Monoprop tutorial run propagated one local occupation operator. That took about one second, but it was not comparable with the EduKaizen benchmark. EduKaizen computed the mean double occupancy across all 60 sites:
\[D=\frac{1}{60}\sum_{i=1}^{60}\left\langle n_{i,\uparrow}n_{i,\downarrow}\right\rangle.\]This observable contains 60 onsite quartic terms. We therefore built it as one Monoprop FermiOperator and propagated it through the same 60-site, 120-mode, 30-step circuit used by the existing comparison. The parameters were U=-2, dt=0.2, and final model time t=6. The run used Monoprop 0.8.0 under Python 3.12 in WSL2. It used the standard single-process Python wheel, not an MPI build.
Before running the full problem, the observable construction was checked on four sites without truncation. Propagating one combined mean operator agreed with averaging four separately propagated onsite operators to within 2.78e-17.
The result
| Method | Mean double occupancy | Recorded time | Absolute difference from MPS chi256 |
|---|---|---|---|
| Local Fire Opal, readout corrected | 0.2306772151 | 33.15 s execution proxy | 0.0215794919 |
| Monoprop 0.8.0, cutoff 4, lower_atol=1e-8 | 0.1976589194 | 234.03 s | 0.0114388037 |
| EduKaizen Majorana, cutoff 4 | 0.2086139231 | 1,153.51 s | 0.0004838001 |
| MPS chi256 | 0.2090977231 | 9,033 s | reference |
The official Monoprop run is about 4.93x faster than the EduKaizen cutoff-4 run and 38.60x faster than the recorded MPS chi256 run. It is also 7.06x slower than the local 4,096-shot Fire Opal execution proxy.
The time ranking is therefore clear for these recorded runs: local quantum execution first, then Monoprop, then EduKaizen Majorana propagation, and finally MPS chi256. The accuracy ranking against chi256 is different. The EduKaizen result is closest, Monoprop is next, and the local readout-corrected hardware value is furthest away.
Why the fastest Monoprop setting is not enough
At the tutorial threshold lower_atol=1e-4, Monoprop completed the same observable in only 2.40 s. Its value was 0.2502868623, however, an absolute difference of about 0.04119 from chi256. Tightening the threshold to 1e-8 expanded the propagated operator to 11,443,150 active terms and increased the compute time to 234 seconds.
The values at lower_atol=1e-6 and 1e-8 differ by only 3.86e-5. The coefficient threshold is therefore nearly stable over that interval. The remaining difference from chi256 is not removed by coefficient pruning alone; the length cutoff and implementation details still matter.
The earlier EduKaizen code also used cutoff 4, but additionally retained at most 5,000 terms. Official Monoprop retained every term above its coefficient threshold. The matching cutoff label does not define the same approximation, and the close EduKaizen/chi256 agreement is not independent proof that cutoff 4 has converged.
Which quantum time should be compared?
The local 33.15 s number is an execution estimate for one 4,096-shot main circuit plus readout circuits. The Q-CTRL study used a larger 20,000-shot budget. For the 60-site, 30-step experiment it reports 166 s of main-circuit QPU time and 265 s when readout-error-mitigation and decay-recovery characterization are included.
Against the paper’s shot budget, the 234-second Monoprop run lies between the two quantum timings: slower than main-circuit QPU execution, but faster than the characterization-inclusive QPU time. QPU usage also excludes queueing and the surrounding cloud workflow, while the classical numbers are local wall-clock measurements. They are informative engineering measurements, not perfectly interchangeable definitions of time-to-answer.
What this benchmark establishes
Official Monoprop is a much stronger classical baseline than the earlier one-second local-occupation demonstration. For the same 60-site mean-double-occupancy observable, it reaches an answer in minutes rather than the 19 minutes of the EduKaizen implementation or the 2.5 hours of MPS chi256.
This benchmark does establish a quantum runtime advantage on the local execution metric. The 33.15 s hardware proxy is 234.03/33.15 = 7.06x faster than the strict official Monoprop run. Quantum execution uses about 14.2 percent of the Monoprop time for this recorded comparison.
The scope of that advantage remains specific. The hardware value differs more from chi256 than the Monoprop value, the QPU figure is an execution proxy rather than full cloud time-to-answer, and chi256 itself hit its configured bond cap. These caveats prevent a general matched-accuracy, end-to-end quantum-advantage claim, but they do not remove the measured 7.06x quantum runtime advantage over official Monoprop.
The conclusion is therefore a time-accuracy frontier with a clear runtime winner: hardware is 7.06x faster than official Monoprop on the local execution metric, Monoprop is more accurate against the current MPS reference than the hardware result, the EduKaizen truncation happens to lie closest to that reference, and MPS remains the most expensive calculation in this comparison.
Sources and reproducibility
- Algorithmiq, official Monoprop repository and documentation.
- A. Miller et al., Simulation of Fermionic Circuits Using Majorana Propagation.
- Q-CTRL Fermi-Hubbard study, arXiv:2605.04025.
- EduKaizen project code and benchmark artifacts, fermi-hubbard-60q-tdvp.