Instance-wise accuracy guarantees for trained QCNNs

Establish an explicit instance-wise approximation-error guarantee for low-weight Pauli propagation applied to trained quantum convolutional neural networks, beyond the existing average-case guarantee for random circuit ensembles.

Background

The paper studies classical simulation of quantum convolutional neural networks (QCNNs) using low-weight Pauli propagation. The method truncates the Heisenberg-picture observable by discarding Pauli strings whose weight exceeds a threshold, yielding an efficient approximate simulation under locally scrambling random-circuit assumptions.

The cited results provide an average-case mean-squared truncation-error bound for random circuit ensembles, and numerical studies suggest that the approximation remains accurate during training on selected benchmark problems. However, the paper identifies a gap between these ensemble-level results and a rigorous guarantee for individual trained QCNN circuits. Establishing such an instance-wise bound would certify the accuracy of the classical surrogate for specific trained networks rather than only on average over random circuit instances.

References

Second, the selection of the dominant Pauli-string subset $\Omega$ in Sec.~\ref{sec approximated be} is currently heuristic, based on the observed $n$-independence of the dominant pattern; a rigorous a-priori bound on $|\Omega|$ versus the target error would strengthen the result.

— Quantum algorithms for the exponentiation of Toeplitz matrices and applications in partial differential equations  (2609.31372 - Gutiérrez et al., 25 Sep 2026) in Section Discussion and conclusions; see also Section Approximated block encoding

However, beyond the average-case bound for random circuit ensembles, no explicit instance-wise guarantee has been provided yet for the trained circuits.

— The cost of simulating classically tractable quantum circuits and dynamics  (2609.11847 - Chang et al., 10 Sep 2026) in Section 3.6, subsection “Quantum convolutional neural networks”