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operator_loschmidt_echo_56x1488 Hybrid Monte Carlo Pauli Propagation #204
Description
Name
Hybrid Monte Carlo Pauli Propagation estimate of the OLE signal for operator_loschmidt_echo_56x1488 at sc=0.15
Circuit
operator_loschmidt_echo_56x1488
Observable value
0.1867
Error bound (low)
No response
Error bound (high)
No response
Method
Hybrid Monte Carlo Pauli Propagation
Method proof
The resulting signal with statistical error bars is:
$$
f_{\delta=0.3} = 0.187 \pm 0.006
$$
Hybrid Monte Carlo Pauli Propagation, Heisenberg picture, cache size 500M Pauli terms, minimum absolute coefficient 1e-7, 800 Monte Carlo samples. The OLE signal is reconstructed from the diagonal (P = P') approximation of the commutator moments up to 20th order in the perturbation strength δ.
See Algorithmiq's model description and theory for details and applications.
Please see the attached document for the method proof. This submission reports the Hybrid Monte Carlo Pauli Propagation counterpart at the scattering coefficient sc=0.15. The showcased value is the OLE signal at δ=0.3 (perturbation V_δ = e^{−iδG}, G = Σ_{q∈V_P} X_q), reconstructed from the diagonal approximation of the commutator moments up to 20th order in δ, averaged over 800 Monte Carlo samples.
one-page-summary-hybridMC-56x1488.pdf
Quantum runtime (seconds)
No response
Classical runtime (seconds)
2100000
Compute resources (quantum)
No response
Compute resources (classical)
Intel(R) Xeon(R) Platinum 8168 CPU @ 2.70GHz, 32-vCPU / ~334 GB RAM, 16 nodes in total.
Notes
Average propagation time ≈ 2700 s per Monte Carlo sample, average memory ≈ 94 GB per sample; total CPU time over the 800 samples ≈ 2.1×10⁶ s (samples run in parallel across the cluster).
Authors
No response
Institutions
Algorithmiq, EPFL
Activity
[-]Hybrid Monte Carlo Pauli Propagation estimate of the OLE signal for `operator_loschmidt_echo_56x1488` at `sc=0.15`[/-][+]q56_hex_rzhalf_ABmC_FL6_sc0.15_hmc_cache500M_classical[/+][-]q56_hex_rzhalf_ABmC_FL6_sc0.15_hmc_cache500M_classical[/-][+]operator_loschmidt_echo_56x1488 Hybrid Monte Carlo Pauli Propagation[/+]n-e-sherman commentedon Jul 24, 2026
Hi @FranceGrieco,
Overall the method makes sense, but I am a bit confused on how to gauge the accuracy of the result. I'm not super familiar with this method, but it looks like there are two main approximations: the MC sampling step, and the diagonal approximation. Could you discuss the error that is expected from these approximations? As in why these approximations are valid here, or if they're not, do you have an idea of how these approximations typically effect the result? I suspect in the limit of large samples the MC method might converge, but the diagonal approximation in particular I am unsure about. If this is looked at elsewhere, just including a reference and maybe a brief discussion would be fine.
Best,
Nick
FranceGrieco commentedon Oct 14, 2026
Hi @n-e-sherman,
I am sorry for the belated reply. Thank you for you feedback.
This method is fairly new, and we considered it a development of the Monte Carlo simulations done in Google OTOC paper, described in their supplementary material.
Given the numerical limitations of "vanilla" Pauli Propagation for these methods, many "tricks" and approximations were added to be able to run and obtain these results. As you noted, is possible for the method to converge given enough samples, but the remaining bias is hard to estimate.
In the paper that we published, Observable Estimation in the Absence of Classical Verification, Appendix C1, we describe in more details the formulas we give a more extensive derivation and discussion on why we think these results have significance.
The paper was not published at the time of the submission. I could include it as a reference in the one page summary
We are still investigating on the capabilities and potentials of this method.
Ciao,
Francesco