# Quantum convolutional neural network

A quantum convolutional neural network (QCNN) is a variational quantum machine learning model that applies parameterized two-qubit unitaries in a convolutional pattern, interspersed with measurement-based pooling, to extract features from quantum states or encoded classical data. It was introduced by Iris Cong, Soonwon Choi, and Mikhail D. Lukin in [Nature Physics](https://www.edgepedia.org/nature-physics) in 2019, and uses only \( O(\log(N)) \) variational parameters for an input of N qubits, which keeps training and implementation tractable on near-term devices.<sup>[1](https://doi.org/10.1038/s41567-019-0648-8)</sup> The QCNN has an analytical trainability guarantee: its cost-function landscape does not exhibit barren plateaus (flat regions in a model's training landscape where learning stalls).<sup>[2](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041011)</sup> Documented applications range from recognizing symmetry-protected topological phases of quantum matter to designing quantum error correction codes and classifying classical images.<sup>[1](https://doi.org/10.1038/s41567-019-0648-8)</sup>

| Key fact | Detail |
|---|---|
| Introducing paper | Cong, Choi, and Lukin, Nature Physics 15, 1273–1278 (2019)<sup>[1](https://doi.org/10.1038/s41567-019-0648-8)</sup> |
| Parameter scaling | \( O(\log(N)) \) variational parameters for N input qubits<sup>[1](https://doi.org/10.1038/s41567-019-0648-8)</sup> |
| Trainability | Gradient variance vanishes no faster than polynomially with system size, so no barren plateaus under random initialization<sup>[2](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041011)</sup> |
| Hardware demonstration | 7-qubit superconducting processor identifying symmetry-protected topological phases (2022)<sup>[3](https://www.nature.com/articles/s41467-022-31679-5)</sup> |
| Classical benchmarks | About 99% accuracy on MNIST and about 94% on Fashion MNIST with 12 to 51 free parameters<sup>[4](https://ar5iv.labs.arxiv.org/html/2108.00661)</sup> |
| Noise sensitivity | An 8-qubit QCNN at 96% clean test accuracy fell to 25% when five noise channels were applied at probabilities of 0.1, 0.5, and 1.0<sup>[5](https://www.nature.com/articles/s41598-025-17769-6)</sup> |
| Software | Qiskit Machine Learning, PennyLane, and TensorFlow Quantum all provide QCNN implementations<sup>[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html)</sup><sup> • </sup><sup>[7](https://www.tensorflow.org/quantum/tutorials/qcnn)</sup><sup> • </sup><sup>[8](https://pubmed.ncbi.nlm.nih.gov/41926404/)</sup> |

## How it works

The circuit's input is an unknown quantum state \( \rho_{\text{in}} \). A convolution layer applies a single quasi-local unitary \( U_{i} \) in a translationally invariant manner for finite depth. In the pooling step, a fraction of the qubits are measured, and their outcomes determine unitary rotations \( V_{j} \) applied to nearby qubits.<sup>[9](https://arxiv.org/pdf/1810.03787)</sup> The combination of entangling gates between neighboring qubits and measurement-conditioned single-qubit gates reduces the number of qubits while retaining the characteristic features of the input state vector.<sup>[3](https://www.nature.com/articles/s41467-022-31679-5)</sup>

Nonlinearity, the ingredient that gives classical convolutional networks much of their expressive power, arises here from reducing the number of degrees of freedom rather than from a nonlinear activation function. Convolution and pooling layers are repeated until the system is small; a fully connected layer then applies a unitary \( F \) on the remaining qubits, and the output is read by measuring a fixed number of output qubits.<sup>[9](https://arxiv.org/pdf/1810.03787)</sup> After \( d \) repetitions of the convolution-pooling procedure, the fully connected unitary maps the feature of interest onto a single output qubit.<sup>[3](https://www.nature.com/articles/s41467-022-31679-5)</sup>

The architecture carries a trainability guarantee. Under assumptions of independent 2-designs and a linear cost function, the variance of the cost function [partial derivatives](https://www.edgepedia.org/partial-derivative) is at most polynomially vanishing with system size, so the landscape does not exhibit a barren plateau and the model is trainable under random initialization. The proof uses a graph-based method for analyzing expectation values over Haar-distributed unitaries, verified numerically.<sup>[2](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041011)</sup>

## How it is done

A practitioner follows four stages: encoding, circuit construction, training, and measurement.

Encoding. For classical data, qubit encoding rescales each data point \( x_{i} \) to lie between 0 and \( \pi \) and prepares \( \ket{\phi(x_{i})} = \cos(x_{i}/2)\ket{0} + \sin(x_{i}/2)\ket{1} \), mapping N features to N qubits with constant circuit depth.<sup>[4](https://ar5iv.labs.arxiv.org/html/2108.00661)</sup> Hybrid direct and hybrid angle encoding schemes trade circuit depth against width: depth is reduced to \( O(2^{m}) \) for \( m < N \) while the qubit count becomes \( O(m \cdot N/2^{m}) \).<sup>[4](https://ar5iv.labs.arxiv.org/html/2108.00661)</sup> In Qiskit, pixelated images are encoded with a feature map such as ZFeatureMap or ZZFeatureMap from the circuit library.<sup>[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html)</sup>

Circuit construction. Alternating convolutional and pooling layers are applied until one qubit remains, and classification is performed by measuring that qubit.<sup>[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html)</sup> The circuit structure, such as the number of convolution and pooling layers, is a fixed hyperparameter; the unitaries themselves are learned.<sup>[9](https://arxiv.org/pdf/1810.03787)</sup> The pooling layer reduces qubits by disregarding certain qubits after operations, rather than by classical downsampling.<sup>[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html)</sup>

Training. The parameters of the parametrized circuits are adjusted to reduce the QCNN's loss function.<sup>[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html)</sup> Implementations in PennyLane use fully parameterized, shallow-depth circuits suited to NISQ devices,<sup>[4](https://ar5iv.labs.arxiv.org/html/2108.00661)</sup> and TensorFlow Quantum provides a simplified, translationally invariant 1D quantum convolution following the Cong and Lukin paper.<sup>[7](https://www.tensorflow.org/quantum/tutorials/qcnn)</sup>

## Origin

The QCNN was introduced by Iris Cong, Soonwon Choi, and Mikhail D. Lukin in Nature Physics volume 15, pages 1273–1278, published 26 August 2019.<sup>[1](https://doi.org/10.1038/s41567-019-0648-8)</sup> The design builds on two earlier lines of work. A QCNN circuit with multiple pooling layers can be viewed as a combination of MERA, a variational ansatz for many-body wavefunctions, and nested quantum error correction, a mechanism for detecting and correcting local errors without collapsing the wavefunction. For any state with a MERA representation, a QCNN exists that recognizes it with deterministic measurement outcomes, namely the inverse of the MERA circuit.<sup>[9](https://arxiv.org/pdf/1810.03787)</sup> The specific structure used in the superconducting demonstration was inspired by the MERA representation of the topological cluster state.<sup>[3](https://www.nature.com/articles/s41467-022-31679-5)</sup> A closely related architecture, the Hierarchical Quantum Classifier, uses a similar tree-like structure without translational invariance.<sup>[2](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041011)</sup><sup> • </sup><sup>[4](https://ar5iv.labs.arxiv.org/html/2108.00661)</sup>

## Variants

Several named modifications of the basic architecture exist.

**Branching QCNN.** The branching QCNN (bQCNN) is a generalization with substantially higher expressibility whose key feature is leveraging midcircuit measurements; it was motivated by classification tasks on nontrivial phases of quantum matter.<sup>[10](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.013117)</sup>

**Hybrid quantum-classical CNN.** The hybrid QCCNN implements the feature map of a classical convolutional layer with a parametric quantum circuit whose output is a correlational measurement, so the number of qubits required depends only on the feature-map window size, often from 3×3 to 9×9, within reach of current hardware. Each filter takes a 2×2 window, translates it into a separable 4-qubit state, and evolves it with Ry single-qubit gates and CNOT gates on nearest-neighbor pairs. This contrasts with the pure QCNN, which uses as many qubits as the input size.<sup>[11](https://ar5iv.labs.arxiv.org/html/1911.02998)</sup>

**Hierarchical structures.** Tree-like QCNN architectures consist of \( O(\log(n)) \) layers for \( n \) input qubits, permitting shallow circuit depth and avoiding the barren plateau problem.<sup>[4](https://ar5iv.labs.arxiv.org/html/2108.00661)</sup>

**Symmetry-preserving circuits.** A 2025 architecture based on Hamming weight preserving quantum circuits introduces convolutional layers and measurement-based pooling layers that preserve the symmetries of the quantum states while realizing nonlinearity using gates that are not subspace preserving, with reported polynomial running-time advantages over classical deep learning and an open-source GPU-oriented simulation library.<sup>[12](https://beta.iopscience.iop.org/article/10.1088/2058-9565/adbf43)</sup>

## Applications

The introducing paper demonstrated two uses: recognizing quantum states associated with a one-dimensional symmetry-protected topological phase, with performance surpassing existing approaches, and devising a quantum error correction scheme optimized for an unknown error model. The error correction connection works because any QCNN circuit, and its inverse, can be viewed as a decoding (or encoding) quantum channel between physical input qubits and a logical output qubit, so recovery fidelity can be maximized to design new codes.<sup>[1](https://doi.org/10.1038/s41567-019-0648-8)</sup><sup> • </sup><sup>[9](https://arxiv.org/pdf/1810.03787)</sup>

On hardware, a 2022 [Nature Communications](https://www.edgepedia.org/nature-communications) study realized a QCNN on a 7-qubit superconducting quantum processor to identify symmetry-protected topological phases of a spin model characterized by a non-zero string order parameter. Despite finite-fidelity gates, the QCNN recognized the topological phase with higher fidelity than direct measurements of the string order parameter for the prepared states.<sup>[3](https://www.nature.com/articles/s41467-022-31679-5)</sup>

For classical data, fully parameterized QCNN models using only two-qubit interactions were benchmarked on MNIST and Fashion MNIST with 12 to 51 free parameters, achieving about 99% accuracy on MNIST and about 94% on Fashion MNIST, and performing noticeably better than CNN models under similar training conditions.<sup>[4](https://ar5iv.labs.arxiv.org/html/2108.00661)</sup>

## Limitations and alternatives

The clearest failure mode demonstrated so far is noise. An 8-qubit QCNN trained without quantum noise reached 96% test accuracy, but after introducing five fundamental noise channels at probabilities of 0.1, 0.5, and 1.0, test accuracy dropped to 25% regardless of noise type or probability. The authors conclude that large-scale quantum models like 8-qubit QCNNs are not yet practical on current NISQ hardware, because with more qubits and deeper circuits error rates accumulate faster.<sup>[5](https://www.nature.com/articles/s41598-025-17769-6)</sup>

The comparison with classical CNNs is structural rather than performance-based: the QCNN does not perform spatial convolution. Its "convolution" and "pooling" occur via qubit entanglement and measurement reduction; they do not perform the mathematical spatial convolution of classical CNNs, although the parameter sharing of the circuit can impose translational invariance.<sup>[5](https://www.nature.com/articles/s41598-025-17769-6)</sup> The name reflects the layered, coarse-graining pattern of the classical architecture, not a shared operation.<sup>[3](https://www.nature.com/articles/s41467-022-31679-5)</sup>

## References

1. [Iris Cong, Soonwon Choi, Mikhail D. Lukin (2019). Quantum convolutional neural networks. Nature Physics.](https://doi.org/10.1038/s41567-019-0648-8)
2. [Absence of Barren Plateaus in Quantum Convolutional Neural Networks (Phys. Rev. X 11, 041011)](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041011)
3. [Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases](https://www.nature.com/articles/s41467-022-31679-5)
4. [Quantum convolutional neural network for classical data classification (arXiv:2108.00661)](https://ar5iv.labs.arxiv.org/html/2108.00661)
5. [A comparative analysis and noise robustness evaluation in quantum neural networks](https://www.nature.com/articles/s41598-025-17769-6)
6. [The Quantum Convolution Neural Network - Qiskit Machine Learning](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html)
7. [Quantum Convolutional Neural Network | TensorFlow Quantum](https://www.tensorflow.org/quantum/tutorials/qcnn)
8. [Quantum Convolutional Neural Networks: A Survey on Architectures, Applications, and Future Directions](https://pubmed.ncbi.nlm.nih.gov/41926404/)
9. [Quantum Convolutional Neural Networks (arXiv:1810.03787 preprint of the introducing paper; ar5iv rendering excerpts merged)](https://arxiv.org/pdf/1810.03787)
10. [Branching quantum convolutional neural networks (Phys. Rev. Research 4, 013117)](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.013117)
11. [Hybrid Quantum-Classical Convolutional Neural Networks (arXiv:1911.02998)](https://ar5iv.labs.arxiv.org/html/1911.02998)
12. [Subspace preserving quantum convolutional neural network architectures](https://beta.iopscience.iop.org/article/10.1088/2058-9565/adbf43)

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*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*

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