The quantum hardware was much faster than our capped classical simulator. How does sample quality change as the MPS bond dimension χ grows?
Measured classical runtimes
We used Qiskit Aer’s matrix-product-state (MPS) simulator on the same logical 61-qubit circuit. Each classical run produced 1,000 samples. Maximum bond dimension χ limits how much entanglement the approximation retains.
| χ | Local time for 1,000 shots | Per-bit RMS difference from IBM |
|---|---|---|
| 8 | 1.45 s | 0.195 |
| 16 | 4.92 s | 0.140 |
| 32 | 34.31 s | 0.119 |
| 64 | 173.81 s | 0.109 |
| 128 | 1,248.95 s | 0.090 |
With 1,000 samples, the 95% noise-only threshold for this RMS difference is about 0.018 if the distribution truly matches the measured IBM distribution. χ=128 is still about five times higher. Its strings average 31.371 ones versus IBM’s 30.440. The distribution of the total number of ones does not move smoothly toward a single limit; that one summary cannot test the full output.
What does the runtime ratio say?
The quantum job yielded a million samples in 19 seconds of QPU time. The local χ=128 MPS run took 1,249 seconds for a thousand samples. This is a large measured runtime lead over this particular classical implementation, even before accounting for shot count. The clocks differ: QPU usage versus local program runtime. Moreover χ=128 has not produced a quality-matched classical output.
Extrapolation and conclusion
A fit to measured χ=32–128 runtimes gives approximately time ∝ χ2.59: around two hours at χ=256 and twelve hours at χ=512 for 1,000 samples if the same scaling persists. Yet the error in the zero/one distribution does not consistently halve with each doubling. We cannot identify a reliable χ at which MPS matches the ideal-circuit fidelity.
Our result supports a sharp but limited statement: Nighthawk sampled this circuit far faster than our tested capped-MPS runs, whose simple marginals have not converged. The paper’s broader claim rests on its separate fidelity experiments and specified classical tensor-contraction model. A new quality-matched MPS or tensor benchmark remains a useful next test.
Sources: the paper, code and original data.
Project overview and article source
Edukaizen · 25 september 2026 / 25 September 2026