Sample-complexity control for randomized measurement ensembles

Determine how to control the sample complexity of certified fidelity-susceptibility estimation for different randomized-measurement ensembles, including random single-qubit Pauli measurements and random n-qubit Clifford circuits, while accounting for the trade-off between observable locality and measurement cost.

Background

The paper’s certification protocol estimates response moments using randomized measurements of a prepared ground state. With random single-qubit Pauli measurements, the sample complexity depends on the locality of the observables used to estimate higher-order moments, and that locality typically grows with the Krylov moment order. Random n-qubit Clifford circuits reduce the dependence on locality but incur substantially greater circuit and, in the worst case, Hilbert-space costs.

The authors identify a trade-off between these measurement ensembles and explicitly leave its proper analysis unresolved. The open problem is therefore to derive rigorous, practically useful sample-complexity bounds or optimization criteria that determine how the choice of randomized-measurement ensemble affects certified fidelity-susceptibility estimation.

References

The next crucial question is how to control sample complexity, as it depends on the ensembles selected for randomized measurements. If the ensemble is random single-qubit Pauli measurements (as done in this work), the sample complexity depends on the locality of the observable $X_k{(t)}$, which typically increases as $k$ increases. If, instead, the ensemble consists of Random $n$-qubit Clifford circuits, then the dependence on locality drops. So, we have a trade-off, and proper analysis is left for future work.

— Certified Fidelity Susceptibility from Classical Shadows  (2608.24732 - Adhikari, 25 Aug 2026) in Section Conclusion