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A Tutorial on Quantum Convolutional Neural Networks (QCNN)

Seunghyeok Oh, Jaeho Choi, Joongheon Kim

arXiv:2009.09423v1quant-ph

TL;DR

Classical processing of many-body quantum data is difficult because conversion to classical data grows exponentially with system size. The paper presents two QCNN approaches—applying CNN structure in quantum systems and adding quantum convolution to CNNs—and reports MNIST learning results similar to CNN.

  • Problem

    Classical handling of many-body quantum data is difficult because conversion to classical data grows exponentially with system size.

    Source: §I · p. 1

  • Method

    The paper presents QCNNs that apply CNN structure to quantum systems and quantum convolution layers that can be added to CNN-based image classification.

    Source: §I · p. 1, §II · p. 1, §III · p. 3

  • Results

    In MNIST simulations, QCNN showed good learning results compared with a fully connected layer and similar learning results to CNN.

    Source: §IV · p. 3

  • Takeaways & Limitations

    The presented QCNN approaches may serve quantum physics and chemistry classification and provide a learning method alongside existing CNNs.

    Source: §V · p. 3

    The simulations were performed at microscopic scale; applying QCNN to more complex data remains future work.

    Source: §V · p. 4

Abstract

from arXiv · show

Convolutional Neural Network (CNN) is a popular model in computer vision and has the advantage of making good use of the correlation information of data. However, CNN is challenging to learn efficiently if the given dimension of data or model becomes too large. Quantum Convolutional Neural Network (QCNN) provides a new solution to a problem to solve with CNN using a quantum computing environment, or a direction to improve the performance of an existing learning model. The first study to be introduced proposes a model to effectively solve the classification problem in quantum physics and chemistry by applying the structure of CNN to the quantum computing environment. The research also proposes the model that can be calculated with O(log(n)) depth using Multi-scale Entanglement Renormalization Ansatz (MERA). The second study introduces a method to improve the model's performance by adding a layer using quantum computing to the CNN learning model used in the existing computer vision. This model can also be used in small quantum computers, and a hybrid learning model can be designed by adding a quantum convolution layer to the CNN model or replacing it with a convolution layer. This paper also verifies whether the QCNN model is capable of efficient learning compared to CNN through training using the MNIST dataset through the TensorFlow Quantum platform.

I. INTRODUCTION

The introduction motivates quantum computing as an alternative for problems that are difficult to process classically, including many-body quantum data whose classical representation grows exponentially. It presents QCNN research as two approaches: applying CNN structure in quantum systems and adding quantum computing to CNN-based learning.

  • CNN preserves correlations among neighboring image pixels through convolution and pooling, unlike fully connected layers that do not retain those correlations.Convolution forms hidden data from nearby pixels; pooling reduces feature-map size and learning resources.Source: §I · p. 1
  • Many-body quantum data grows exponentially when converted to classical data as system size increases, making effective machine-learning solutions difficult.The paper frames quantum computing as an alternative when data and models are no longer efficiently processed in existing environments.Source: §I · p. 1
  • The paper introduces two QCNN approaches: applying CNN structure to quantum systems and adding quantum computing to problems previously handled by CNNs.The introduction situates these approaches among quantum machine-learning research, including VQE, and QAOA [4],.Source: §I · p. 1

II. QCNN USING THE STRUCTURE OF CNN

This section describes QCNNs that apply CNN's convolution and pooling structure to quantum systems, then use a fully connected circuit for classification. It also explains reverse MERA and a QEC-based correction approach.

  • The figure contrasts CNN and QCNN convolution and pooling, with measurement in pooling enabling additional QEC [9].Both models use convolution to find a new state and pooling to reduce system size.Source: Figure 1 · p. 2, §II · p. 2
  • Adjacent-qubit gates find hidden states, pooling reduces the quantum system, and a fully connected circuit classifies once the system is sufficiently small.The convolution and pooling steps repeat; pooling can observe a fraction of qubits or apply two-qubit gates such as CNOT.Source: §II · p. 2
  • MERA simulates many-body quantum states, while QCNN uses it in reverse to exponentially reduce the input system's size.MERA adds |0⟩ qubits as depth increases; its reverse direction suits QCNN's reduction of the system from given data.Source: §II · p. 2
  • For an input state MERA cannot generate, QCNN may lack a definitive classification; QEC adds degrees of freedom to address this problem.For a generated state, pooling should measure |0⟩; a |1⟩ result can trigger a gate on surrounding qubits to correct the result.Source: §II · p. 2

III. QCNN FOR IMAGE CLASSIFICATION

This section presents quantum convolution as a filter operation on image data, designed to work with small quantum computers. It describes patch encoding, learnable quantum circuits, measurement, and filter-depth efficiency.

  • Quantum convolution processes image data a filter-sized portion at a time, making the layer constructible on small quantum computers.The quantum layer applies a filter to an input feature map using a quantum environment.Source: §III · p. 2
  • Encoding stores filter-sized pixel data in qubits, learnable circuits apply filters, and measurement decodes new classical data for the feature map.The process repeats to form the new feature map; encoding and measurement choices can change qubit requirements and learning efficiency.Source: §III · p. 3
  • With MERA, quantum filters can be designed with O(log(n)) depth, compared with O(n2) classical operations for an n2-sized filter.Source: §III · p. 3
  • The learnable quantum circuit can use different designs, and the chosen design can affect learning performance.The MNIST circuit example uses variable gates with learnable hyperparameters.Source: §III · p. 3, Figure 3 · p. 3

IV. LEARNING MNIST USING QCNN SIMULATION

The paper simulates QCNN image classification on MNIST using TensorFlow Quantum, with reduced image size, filter size, and per-epoch training sample count. QCNN outperforms the fully connected-only model and shows similar learning results to CNN.

IV. LEARNING MNIST USING QCNN SIMULATION

  • The simulation uses TensorFlow Quantum to test QCNN image classification on MNIST.Source: §IV · p. 3
  • The MNIST images are downscaled from 28×28 to 10×10, and the quantum convolution filter is limited to 2 × 2.Source: §IV · p. 3
  • Each epoch trains on 2,500 randomly selected images from the 60,000-image dataset.The QCNN produces an 8-channel feature map connected to fully connected layers with 64 and 10 hidden units.Source: §IV · p. 3
  • The fully connected-only model tests the effect of the quantum convolution layer, while the CNN replaces it with a convolution layer of the same channel length.Source: §IV · p. 3
  • QCNN outperforms the fully connected-only model and shows learning results similar to CNN.Fig. 4 compares the QCNN with both baselines.Source: §IV · p. 3, Figure 4 · p. 3

V. CONCLUDING REMARKS AND FUTURE WORK

The conclusion presents QCNN as a possible approach for physical and chemical classification and for augmenting existing CNN models. The reported simulation is microscopic; future work will explore more complex data, circuit designs, and encoding methods.

V. CONCLUDING REMARKS AND FUTURE WORK

  • QCNN is presented as a possible approach to physical and chemical classification and as a way to augment existing CNN models.Source: §V · p. 3
  • The experiment is microscopic in scale, and applying QCNN to more complex data is future work.Source: §V · p. 4
  • Future exploration includes alternative internal quantum-circuit designs and encoding methods for image processing.The paper suggests that encoding more information in one qubit may permit more efficient learning.Source: §V · p. 4

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