Trainable Quantum Neural Network for Multiclass Image Classification with the Power of Pre-trained Tree Tensor Networks

TL;DR

The paper tackles multiclass image classification by embedding pre-trained TTN-based classifiers into QNNs, addressing two main hurdles: large gate depths from high bond dimensions and the exponential overhead of mid-circuit postselection. It introduces FTN-classifiers that combine multiple small-bond TTNs and extends an adiabatic encoding framework to TTNs to produce postselection-free qFTN-classifiers, preserving performance. Numerical results on MNIST and CIFAR-10 show that FTN-classifiers can be trained and encoded into qFTN-classifiers with high training accuracy and competitive test performance, illustrating a robust quantum-classical hybrid approach. Overall, the work demonstrates a scalable synergy between TTN-based models and QNNs for quantum-enhanced multiclass image classification, with clear pathways for improving generalization and hardware implementation.

Abstract

Tree tensor networks (TTNs) offer powerful models for image classification. While these TTN image classifiers already show excellent performance on classical hardware, embedding them into quantum neural networks (QNNs) may further improve the performance by leveraging quantum resources. However, embedding TTN classifiers into QNNs for multiclass classification remains challenging. Key obstacles are the highorder gate operations required for large bond dimensions and the mid-circuit postselection with exponentially low success rates necessary for the exact embedding. In this work, to address these challenges, we propose forest tensor network (FTN)-classifiers, which aggregate multiple small-bond-dimension TTNs. This allows us to handle multiclass classification without requiring large gates in the embedded circuits. We then remove the overhead of mid-circuit postselection by extending the adiabatic encoding framework to our setting and smoothly encode the FTN-classifiers into a quantum forest tensor network (qFTN)- classifiers. Numerical experiments on MNIST and CIFAR-10 demonstrate that we can successfully train FTN-classifiers and encode them into qFTN-classifiers, while maintaining or even improving the performance of the pre-trained FTN-classifiers. These results suggest that synergy between TTN classification models and QNNs can provide a robust and scalable framework for multiclass quantum-enhanced image classification.

Figures (12)

• Figure 1: Overview of our proposed method and key contributions. In the context of the TN-QNN synergetic learning framework, the typical pipeline consists of three steps: (1) training a classical TN-based ML model for initialization of QNN, (2) embedding the pre-trained TN model into a quantum circuit, and (3) further training the QNN model directly on quantum circuits. Our contributions address the first two steps: (I) we propose FTN-classifiers, classical TTN-based learning models for multiclass image classification that are efficient and easy to embed into quantum circuits, and (II) we extend the adiabatic encoding framework to TTNs to remove the need for postselection. The last step should be conducted in future work. In this paper, we refer to the quantum circuits that have the same underlying structures as TTN as qTTN-circuit, and the corresponding classifier as qTTN-classifier. The same applies to the notation of qFTN-circuit and qFTN-classifier.
• Figure 2: Conceptual diagram of basic TTN-classifier.
• Figure 3: Schematic diagram of our proposed FTN-classifier. We use multiple TTN-classifiers to extract diverse features from a single image.
• Figure 4: TN diagram of isometry condition for three-leg tensor $V_{ijk}$. In TN diagram notation, the identity tensor is represented as a simple line.
• Figure 5: Illustration of the embedding procedure. $V$ is an isometry of dimension $N \times M$, and $U$ is a unitary of dimension $N \times N$. In order to exactly encode $V$ into $U$, we need to perform postselection on the first $k$ qubits after applying $U$. $\alpha, \beta, \gamma$ correspond to the $k$ qubit states.