Operator Loschmidt echo circuits on 49 to 70 qubits

Classically estimate an operator Loschmidt echo on 49- to 70-qubit circuits, where IBM hardware and classical simulations still disagree.

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	1	+# Operator Loschmidt echo circuits on 49 to 70 qubits
	2	+
	3	+An operator Loschmidt echo asks how much of an observable survives when a many-qubit circuit is run forward, nudged, and run back. On these circuits IBM quantum processors and the best classical simulations give clearly different values, and no classical estimate yet has error bars that settle which is right.
	4	+
	5	+## Task
	6	+
	7	+For a fixed circuit $U$ on $n$ qubits, given as an OpenQASM file, estimate classically
	8	+
	9	+$$f_\delta(O)=\frac{1}{2^n}\,\mathrm{Tr}\left(U O U^\dagger\, V_\delta^\dagger\, U O U^\dagger\, V_\delta\right),$$
	10	+
	11	+where $O$ is a product of Pauli $Z$ operators and $V_\delta=e^{-i\delta G}$ with $G=\sum_{u\in\mathcal P}X_u$ over a set $\mathcal P$ of perturbed qubits; without the perturbation the echo is exactly 1. The circuit $U=(U_{b-\alpha}^\dagger)^L\,U_b^L$ runs $L$ Floquet layers of an Ising-type evolution on a heavy-hex patch, then $L$ inverse layers in which the transverse-field angles of some qubits are shifted, so the second half undoes the first only in part.
	12	+
	13	+Instance ids are `operator_loschmidt_echo_` plus qubits × two-qubit (CZ) gates. The tracker's active candidates are `49x1296` and `70x1872` ($L=6$, $\delta=0.15$, observable $Z_{52}Z_{59}Z_{72}$, indexed as in the circuit files' 156-qubit register) and `56x1488` ($L=6$, $\delta=0.30$, a 12-qubit $Z$ string, scattering potential $\alpha=0.15$). The shallower `49x648` ($L=3$) is a baseline benchmark.
	14	+
	15	+## Background
	16	+
	17	+The operator Loschmidt echo is an operator version of the Loschmidt echo, a standard probe of chaos and of sensitivity to perturbations. Algorithmiq and IBM turned it into a hardware experiment that needs only $n$ qubits, by averaging over random eigenstates of $O$ [1]. For small $\delta$ it reduces to an out-of-time-order correlator $C$, $f_\delta\approx 1-\tfrac{\delta^2}{2}C$, the kind of quantity behind Google's 2025 interference experiment [2], yet its signal does not shrink exponentially with system size [1]. Algorithmiq, IBM and the Flatiron Institute posted the first three circuits in November 2025 and the 56-qubit one in June 2026.
	18	+
	19	+## Data and evaluation
	20	+
	21	+- Circuits and parameters are in the tracker folder, which also has 56-qubit circuits for $\alpha$ from 0 to 0.25.
	22	+- The tracker records a value with optional lower and upper bounds, runtime and hardware, and compares results only within one instance.
	23	+- The tracker does not rank submissions by a score. The metric for this entry is our choice, not the tracker's: the width of a defensible error bar on $f_\delta(O)$, smaller is better, backed by evidence of convergence. There is no baseline score: no active instance has an agreed classical value, and no accepted echo entry on the tracker reports an error bar.
	24	+
	25	+## What is known
	26	+
	27	+Accepted tracker entries, from November and December 2025:
	28	+
	29	+- `49x1296`: hardware with global rescaling 0.662 and 0.657 (ibm_boston, two layouts) and 0.649 (ibm_pittsburgh); belief-propagation tensor network (BP-TN, bond dimension 512) 0.8816, or 0.9426 rescaled; single-path Monte Carlo 0.619.
	30	+- `70x1872`: ibm_boston 0.673; BP-TN 0.8738, or 0.9525 rescaled; single-path Monte Carlo 0.614.
	31	+- `49x648`: ibm_boston 0.824; BP-TN 0.8217; single-path Monte Carlo 0.808. An exact contraction under review gives 0.8225 and finds the exact cost grows about $2^{25}$-fold per unit of $L$.
	32	+
	33	+Global rescaling divides the echo by the measured echo at $\delta=0$, ideally 1; its authors treat it as a heuristic [1]. The single-path Monte Carlo ignores interference between Pauli paths, as its submitters note. For the 56-qubit circuit, BP-TN up to bond dimension 980 and a Pauli-propagation Monte Carlo each match the hardware only in opposite limits of scattering strength, while five mitigated hardware signals on different devices and noise settings agree within 0.03 [1]. Submissions under review for this instance ($\alpha=0.15$) report 0.232 to 0.259 from three hardware settings (ibm_boston and ibm_pittsburgh), 0.459 from BP-TN and $0.187\pm0.006$ (statistical error only) from Pauli-propagation Monte Carlo.
	34	+
	35	+## What would count as progress
	36	+
	37	+- A classical estimate for `49x1296` or `70x1872` with a convergence study and an error bar small enough to confirm or exclude the hardware values of 0.65 to 0.67; anyone can rerun the code on the QASM file.
	38	+- The same for `56x1488`, ideally across the $\alpha$ scan, where the published methods disagree.
	39	+- An exact or rigorously bounded value for any active instance, which would settle it.
	40	+
	41	+## References
	42	+
	43	+1. S. V. Barron et al., "Observable Estimation in the Absence of Classical Verification", preprint (2026). [arXiv:2607.25998](https://arxiv.org/abs/2607.25998)
	44	+2. Google Quantum AI and Collaborators, "Observation of constructive interference at the edge of quantum ergodicity", Nature 646, 825–830 (2025). [doi:10.1038/s41586-025-09526-6](https://doi.org/10.1038/s41586-025-09526-6)
	45	+
	46	+## Source
	47	+
	48	+Imported from [Quantum Advantage Tracker operator_loschmidt_echo](https://github.com/quantum-advantage-tracker/quantum-advantage-tracker.github.io/tree/main/data/observable-estimations/circuit-models/operator_loschmidt_echo), curated by Quantum Advantage Tracker reviewers (IBM, Flatiron Institute, BlueQubit, Algorithmiq and others). Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.