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arXiv:2504.09250

Adiabatic Encoding of Pre-trained MPS Classifiers into Quantum Circuits

Keisuke Murota

Published Apr 12, 2025• 0 citations
arXivHTMLPDF

Abstract

Although Quantum Neural Networks (QNNs) offer powerful methods for classification tasks, the training of QNNs faces two major training obstacles: barren plateaus and local minima. A promising solution is to first train a tensor-network (TN) model classically and then embed it into a QNN.\ However, embedding TN-classifiers into quantum circuits generally requires postselection whose success probability may decay exponentially with the system size. We propose an \emph{adiabatic encoding} framework that encodes pre-trained MPS-classifiers into quantum MPS (qMPS) circuits with postselection, and gradually removes the postselection while retaining performance. We prove that training qMPS-classifiers from scratch on a certain artificial dataset is exponentially hard due to barren plateaus, but our adiabatic encoding circumvents this issue. Additional numerical experiments on binary MNIST also confirm its robustness.

DOI
10.48550/arxiv.2504.09250

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Although Quantum Neural Networks (QNNs) offer powerful methods for classification tasks, the training of QNNs faces two major training obstacles: barren plateaus and local minima. A promising solution is to first train a tensor-network (TN) model classically and then embed it into a QNN.\ However, embedding TN-classifiers into quantum circuits generally requires postselection whose success probability may decay exponentially with the system size. We propose an \emph{adiabatic encoding} framework that encodes pre-trained MPS-classifiers into quantum MPS (qMPS) circuits with postselection, and gradually removes the postselection while retaining performance. We prove that training qMPS-classifiers from scratch on a certain artificial dataset is exponentially hard due to barren plateaus, but our adiabatic encoding circumvents this issue. Additional numerical experiments on binary MNIST also confirm its robustness.
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 FTNclassifiers into 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.
Abstract There have been numerous quantum neural networks reported, but they struggle to match traditional neural networks in accuracy. Given the huge improvement of the neural network models’ accuracy by two-dimensional tensor network (TN) states in classical tensor network machine learning (TNML), it is promising to explore whether its application in quantum machine learning can extend the performance boundary of the models. Here, we transform two-dimensional TNs into quantum circuits for supervised learning. Specifically, we encode two-dimensional TNs into quantum circuits through rigorous mathematical proofs for constructing model ansätze, including string-bond states, entangled-plaquette states and isometric TN states. In addition, we propose adaptive data encoding methods and combine with TNs. We construct a tensor-network-inspired quantum circuit (TNQC) supervised learning framework for transferring TNML from classical to quantum, and build several novel two-dimensional TN-inspired quantum classifiers based on this framework. Finally, we propose a parallel quantum machine learning method for multi-class classification to construct 2D TNQC-based multi-class classifiers. Classical simulation results on the MNIST benchmark dataset show that our proposed models achieve the state-of-the-art accuracy performance, significantly outperforming other quantum classifiers on both binary and multi-class classification tasks, and beat simple convolutional classifiers on a fair track with identical inputs. The noise resilience of the models makes them successfully run and work in a real quantum computer.
One key step in performing quantum machine learning (QML) on noisy intermediate-scale quantum (NISQ) devices is the dimension reduction of the input data prior to their encoding. Traditional principle component analysis (PCA) and neural networks have been used to perform this task; however, the classical and quantum layers are usually trained separately. A framework that allows for a better integration of the two key components is thus highly desirable. Here we introduce a hybrid model combining the quantum-inspired tensor networks (TN) and the variational quantum circuits (VQC) to perform supervised learning tasks, which allows for an end-to-end training. We show that a matrix product state based TN with low bond dimensions performs better than PCA as a feature extractor to compress data for the input of VQCs in the binary classification of MNIST dataset. The architecture is highly adaptable and can easily incorporate extra quantum resource when available.
Training deep quantum neural networks (QNNs) for image classification is notoriously difficult due to vanishing gradients (barren plateaus) and limited nonlinearity in purely unitary circuits. We propose a novel gradient-free surrogate-driven framework combined with mid-circuit measurement and reset of ancillary qubits to induce effective nonunitarity. Our approach uses a classical neural surrogate to predict measurement outcomes from circuit parameters to avoid direct gradients. Theoretical results prove that bypassing quantum gradients mitigates plateau issues. Experiments on MNIST, CIFAR-10, and CIFAR-100 with 15-qubit, 6-layer circuits using four resettable ancillas demonstrate superior accuracy compared to direct-gradient QNNs and classical baselines. Our method also serves as a potential for a generalized training framework applicable to various QNN architectures beyond image classification.
Data encoding remains a fundamental bottleneck in quantum machine learning, where amplitude encoding of high-dimensional classical vectors into quantum states incurs exponential cost. In this work, we propose a pre-trained tensor-train (TT) encoding network (Pre-TT-Encoder) that significantly reduces the computational complexity of amplitude encoding while preserving essential data structure. The Pre-TT-Encoder exploits low-rank TT decompositions learned from classical data, enabling polynomial-time state preparation in the number of qubits and TT-ranks. We provide a theoretical analysis of the encoding complexity and establish fidelity bounds that quantify the trade-off between TT-rank and approximation error. Empirical evaluations on classical (MNIST) and quantum-native (semiconductor quantum dot) datasets demonstrate that our approach achieves substantial gains in encoding efficiency over direct amplitude encoding and PCA-based dimensionality reduction, while maintaining competitive performance in downstream variational quantum circuit classification tasks. The proposed method highlights the role of tensor networks as scalable intermediaries between classical data and quantum processors.
Quantum Neural Networks (QNNs), or the so-called variational quantum circuits, are important quantum applications both because of their similar promises as classical neural networks and because of the feasibility of their implementation on near-term intermediate-size noisy quantum machines (NISQ). However, the training task of QNNs is challenging and much less understood. We conduct a quantitative investigation on the landscape of loss functions of QNNs and identify a class of simple yet extremely hard QNN instances for training. Specifically, we show for typical under-parameterized QNNs, there exists a dataset that induces a loss function with the number of spurious local minima depending exponentially on the number of parameters. Moreover, we show the optimality of our construction by providing an almost matching upper bound on such dependence. While local minima in classical neural networks are due to non-linear activations, in quantum neural networks local minima appear as a result of the quantum interference phenomenon. Finally, we empirically confirm that our constructions can indeed be hard instances in practice with typical gradient-based optimizers, which demonstrates the practical value of our findings.