{ "id": "epa4kj56zj", "slug": "quantum-convolutional-neural-network", "title": "Quantum convolutional neural network", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.ml_computation" }, { "id": "technology.computing.ai.neural_deep", "label": "Neural networks and deep learning", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.neural_deep" }, { "id": "technology.computing.ai.neural_deep.alpha-i-to-z", "label": "Titles I to Z", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.neural_deep.alpha-i-to-z" } ], "geo": [ { "id": "geo.nongeo.t2001.technology.computing.ai.neural_deep", "label": "Non-geographic · 2001 to 2020: Neural networks and deep learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.neural_deep", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t2001", "label": "Non-geographic · 2001 to 2020", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001" }, { "id": "geo.nongeo.t2001.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology" }, { "id": "geo.nongeo.t2001.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing" }, { "id": "geo.nongeo.t2001.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai" }, { "id": "geo.nongeo.t2001.technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.ml_computation" }, { "id": "geo.nongeo.t2001.technology.computing.ai.neural_deep", "label": "Neural networks and deep learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.neural_deep" } ] } ], "excerpt": "A quantum convolutional neural network (QCNN) is a variational quantum machine learning model introduced by Cong, Choi, and Lukin in Nature Physics in 2019, using O(log N) parameters.", "snippet": "A quantum convolutional neural network (QCNN) is a variational quantum machine learning model introduced by Cong, Choi, and Lukin in Nature Physics in 2019, using O(log N) parameters.", "node": "technology.computing.ai.neural_deep.alpha-i-to-z", "markdown": "# Quantum convolutional neural network\n\nA 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.[1](https://doi.org/10.1038/s41567-019-0648-8) 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).[2](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041011) Documented applications range from recognizing symmetry-protected topological phases of quantum matter to designing quantum error correction codes and classifying classical images.[1](https://doi.org/10.1038/s41567-019-0648-8)\n\n| Key fact | Detail |\n|---|---|\n| Introducing paper | Cong, Choi, and Lukin, Nature Physics 15, 1273–1278 (2019)[1](https://doi.org/10.1038/s41567-019-0648-8) |\n| Parameter scaling | \\( O(\\log(N)) \\) variational parameters for N input qubits[1](https://doi.org/10.1038/s41567-019-0648-8) |\n| Trainability | Gradient variance vanishes no faster than polynomially with system size, so no barren plateaus under random initialization[2](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041011) |\n| Hardware demonstration | 7-qubit superconducting processor identifying symmetry-protected topological phases (2022)[3](https://www.nature.com/articles/s41467-022-31679-5) |\n| Classical benchmarks | About 99% accuracy on MNIST and about 94% on Fashion MNIST with 12 to 51 free parameters[4](https://ar5iv.labs.arxiv.org/html/2108.00661) |\n| 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[5](https://www.nature.com/articles/s41598-025-17769-6) |\n| Software | Qiskit Machine Learning, PennyLane, and TensorFlow Quantum all provide QCNN implementations[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html) • [7](https://www.tensorflow.org/quantum/tutorials/qcnn) • [8](https://pubmed.ncbi.nlm.nih.gov/41926404/) |\n\n## How it works\n\nThe 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.[9](https://arxiv.org/pdf/1810.03787) 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.[3](https://www.nature.com/articles/s41467-022-31679-5)\n\nNonlinearity, 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.[9](https://arxiv.org/pdf/1810.03787) After \\( d \\) repetitions of the convolution-pooling procedure, the fully connected unitary maps the feature of interest onto a single output qubit.[3](https://www.nature.com/articles/s41467-022-31679-5)\n\nThe 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.[2](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041011)\n\n## How it is done\n\nA practitioner follows four stages: encoding, circuit construction, training, and measurement.\n\nEncoding. 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.[4](https://ar5iv.labs.arxiv.org/html/2108.00661) 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}) \\).[4](https://ar5iv.labs.arxiv.org/html/2108.00661) In Qiskit, pixelated images are encoded with a feature map such as ZFeatureMap or ZZFeatureMap from the circuit library.[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html)\n\nCircuit construction. Alternating convolutional and pooling layers are applied until one qubit remains, and classification is performed by measuring that qubit.[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html) The circuit structure, such as the number of convolution and pooling layers, is a fixed hyperparameter; the unitaries themselves are learned.[9](https://arxiv.org/pdf/1810.03787) The pooling layer reduces qubits by disregarding certain qubits after operations, rather than by classical downsampling.[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html)\n\nTraining. The parameters of the parametrized circuits are adjusted to reduce the QCNN's loss function.[6](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html) Implementations in PennyLane use fully parameterized, shallow-depth circuits suited to NISQ devices,[4](https://ar5iv.labs.arxiv.org/html/2108.00661) and TensorFlow Quantum provides a simplified, translationally invariant 1D quantum convolution following the Cong and Lukin paper.[7](https://www.tensorflow.org/quantum/tutorials/qcnn)\n\n## Origin\n\nThe QCNN was introduced by Iris Cong, Soonwon Choi, and Mikhail D. Lukin in Nature Physics volume 15, pages 1273–1278, published 26 August 2019.[1](https://doi.org/10.1038/s41567-019-0648-8) 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.[9](https://arxiv.org/pdf/1810.03787) The specific structure used in the superconducting demonstration was inspired by the MERA representation of the topological cluster state.[3](https://www.nature.com/articles/s41467-022-31679-5) A closely related architecture, the Hierarchical Quantum Classifier, uses a similar tree-like structure without translational invariance.[2](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.041011) • [4](https://ar5iv.labs.arxiv.org/html/2108.00661)\n\n## Variants\n\nSeveral named modifications of the basic architecture exist.\n\n**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.[10](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.013117)\n\n**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.[11](https://ar5iv.labs.arxiv.org/html/1911.02998)\n\n**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.[4](https://ar5iv.labs.arxiv.org/html/2108.00661)\n\n**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.[12](https://beta.iopscience.iop.org/article/10.1088/2058-9565/adbf43)\n\n## Applications\n\nThe 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.[1](https://doi.org/10.1038/s41567-019-0648-8) • [9](https://arxiv.org/pdf/1810.03787)\n\nOn 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.[3](https://www.nature.com/articles/s41467-022-31679-5)\n\nFor 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.[4](https://ar5iv.labs.arxiv.org/html/2108.00661)\n\n## Limitations and alternatives\n\nThe 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.[5](https://www.nature.com/articles/s41598-025-17769-6)\n\nThe 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.[5](https://www.nature.com/articles/s41598-025-17769-6) The name reflects the layered, coarse-graining pattern of the classical architecture, not a shared operation.[3](https://www.nature.com/articles/s41467-022-31679-5)\n\n## References\n\n1. [Iris Cong, Soonwon Choi, Mikhail D. Lukin (2019). Quantum convolutional neural networks. Nature Physics.](https://doi.org/10.1038/s41567-019-0648-8)\n2. [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)\n3. [Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases](https://www.nature.com/articles/s41467-022-31679-5)\n4. [Quantum convolutional neural network for classical data classification (arXiv:2108.00661)](https://ar5iv.labs.arxiv.org/html/2108.00661)\n5. [A comparative analysis and noise robustness evaluation in quantum neural networks](https://www.nature.com/articles/s41598-025-17769-6)\n6. [The Quantum Convolution Neural Network - Qiskit Machine Learning](https://qiskit-community.github.io/qiskit-machine-learning/tutorials/11_quantum_convolutional_neural_networks.html)\n7. [Quantum Convolutional Neural Network | TensorFlow Quantum](https://www.tensorflow.org/quantum/tutorials/qcnn)\n8. [Quantum Convolutional Neural Networks: A Survey on Architectures, Applications, and Future Directions](https://pubmed.ncbi.nlm.nih.gov/41926404/)\n9. [Quantum Convolutional Neural Networks (arXiv:1810.03787 preprint of the introducing paper; ar5iv rendering excerpts merged)](https://arxiv.org/pdf/1810.03787)\n10. [Branching quantum convolutional neural networks (Phys. Rev. Research 4, 013117)](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.4.013117)\n11. [Hybrid Quantum-Classical Convolutional Neural Networks (arXiv:1911.02998)](https://ar5iv.labs.arxiv.org/html/1911.02998)\n12. [Subspace preserving quantum convolutional neural network architectures](https://beta.iopscience.iop.org/article/10.1088/2058-9565/adbf43)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Machine learning and neural computation › Neural networks and deep learning › Titles I to Z*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/quantum-convolutional-neural-network", "markdown_url": "https://www.edgepedia.org/quantum-convolutional-neural-network.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. 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Edgepedia Community License 1.0.", "speakable": "A quantum convolutional neural network is a variational quantum machine learning model introduced by Cong, Choi, and Lukin in Nature Physics in 2019, using O parameters." }