Quantum Convolutional Neural Networks are (Effectively) Classically Simulable

TL;DR

The paper argues that Quantum Convolutional Neural Networks (QCNNs) derive their apparent power from operating in a low-bodyness subspace and from being tested on locally-easy datasets, making their action classically simulable when augmented with Pauli shadows. It provides both a theoretical framework and extensive numerical demonstrations showing that random QCNN initializations and the use of shadow tomography enable efficient classical simulations across quantum- and classical-data tasks, including large-scale quantum datasets up to 1024 qubits. The results suggest that current QCNN benchmarks do not imply true quantum advantage and that non-trivial, hard datasets are essential to justify QCNNs, while offering a dequantization perspective and practical routes for quantum-measurement-enhanced learning. The authors also discuss implications for extending these ideas to other quantum neural architectures and advocate re-evaluating benchmarking tasks to avoid trivial datasets.

Abstract

Quantum Convolutional Neural Networks (QCNNs) are widely regarded as a promising model for Quantum Machine Learning (QML). In this work we tie 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. We further show that the QCNN's action on this subspace can be efficiently classically simulated by a classical algorithm equipped with Pauli shadows on the dataset. Indeed, we present a shadow-based simulation of QCNNs on up-to $1024$ qubits for phases of matter classification. 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 non-trivial 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.

Figures (9)

• Figure 1: Schematic representation of our main results. We conceptualize the success of QCNN as a consequence of two facts: (1) When randomly initialized, they operate on a polynomially-sized subspace of low-bodyness observables, (2) They are benchmarked on locally-easy datasets that are classifiable via the information encoded in low-bodyness measurements. The combination of these two facts allows us to show that there exist efficient classical algorithms that can simulate the action of the QCNN in this small subspace, provided that we are given access to Pauli classical shadows on the input data.
• Figure 2: QCNN architecture. (a) A QCNN is composed of alternating convolutional and pooling layers. In the convolutional layers, information is usually being processed by parametrized quantum gates. In the pooling layers, the dimension of the QCNN feature space is reduced by tracing out or measuring qubits. By design, QCNNs have a depth that only scales logarithmically with the number of qubits $n$, and the measurements at its output are local. (b) Examples of unitary QCNNs, where the convolutional layers are composed of two-local gates acting on nearest neighbors, and a pooling layer where half of the qubits are being traced out.
• Figure 3: Bond-Alternating XXX model. a) Phase diagram for the Hamiltonian in Eq. \ref{['eq:alt_xxx_model']}. b) Test classification accuracy for the simulated QCNN acting only on the low-bodyness operator subspace. We show the accuracy as a function of the number of training points and Pauli classical shadows on each state of the dataset.
• Figure 4: Haldane Chain. a) Phase diagram for the Hamiltonian in Eq. \ref{['eq:haldane_model']}. b) Test classification accuracy for the simulated QCNN acting only on the low-bodyness operator subspace. We show the accuracy as a function of the number of training points and Pauli classical shadows on each state of the dataset.
• Figure 5: ANNNI model. a) Phase diagram for the Hamiltonian in Eq. \ref{['eq:ANNNI_model']}. The phases are: (I) ferromagnetic, (II) paramagnetic, (III) floating, (IV) antiphase. b) Predicted phase diagram when training the simulated QCNN acting only on the low-bodyness operator subspace. The model is trained on with $200$ and $20$ states. The crosses mark the training points, while the circle the test ones. Blue circles means correct phase prediction, while a red color indicates that an incorrect phase was assigned.

Theorems & Definitions (1)

• Definition 1: Locally-easy dataset