Quantum Convolutional Neural Networks are Effectively Classically Simulable

Quantum convolutional neural networks (QCNNs) are widely regarded as a promising model for quantum machine learning (QML). In this work, we analyze the most widely used variants of these models (i.e., tracing out- and measurement-based QCNNs), and we relate their heuristic success to two facts. First, that when randomly initialized, they can only operate on the information encoded in low-bodyness measurements of their input states. And second, that they are commonly benchmarked on “locally easy” datasets whose states are precisely classifiable by the information encoded in these low-bodyness observables subspace. From these insights, we argue that the QCNN’s action on this subspace should be efficiently classically simulable. Indeed, we construct and train a purely classical QCNN surrogate—based on low-bodyness Pauli propagation, tensor networks, and classical shadows—that matches or outperforms standard QCNNs on all benchmark datasets and on up-to 1024 qubits, thereby empirically realizing our simulability claims. Our results can then be understood as highlighting a deeper symptom of QML: Models could only be showing heuristic success because they are benchmarked on simple problems, for which their action can be classically simulated. This insight points to the fact that nontrivial datasets are a truly necessary ingredient for moving forward with QML. To finish, we discuss how our results can be extrapolated to classically simulate other architectures.