Pre-training Tensor-Train Networks Facilitates Machine Learning with Variational Quantum Circuits

TL;DR

The paper tackles the exponential cost of amplitude encoding for classical data in quantum machine learning. It introduces the Pre-TT-Encoder, a two-stage pipeline that pre-trains a tensor-train decomposition to compress data before amplitude encoding, reducing state-preparation complexity from $\mathcal{O}(2^{U})$ to $O(U r^{2})$ and providing fidelity guarantees that link TT-rank to approximation error. The authors derive a fidelity bound and demonstrate that higher TT-ranks yield closer approximations, with practical trade-offs between efficiency and accuracy. Empirically, the method improves encoding efficiency and downstream variational quantum circuit classification performance on MNIST and quantum-dot spectra while reducing runtime, suggesting TT-based preprocessing as a scalable interface between classical data and near-term quantum hardware.

Abstract

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.

Figures (1)

• Figure 1: The architecture of a variational quantum circuit. The circuit consists of three main components: (i) data encoding, where classical data are mapped to quantum states $\vert x_{1} \rangle$, $\vert x_{2} \rangle$, ..., $\vert x_{U} \rangle$; (ii) parametric quantum gates, including rotation gates $R_{X}(\alpha_u)$, $R_{Y}(\beta_{u})$, $R_{Z}(\gamma_{u})$, which are trained to optimize the model; and (iii) measurement, where observables $\langle \sigma_{z}^{(1)}\rangle$, $\langle \sigma_{z}^{(2)}\rangle$, ..., $\langle \sigma_{z}^{(U)} \rangle$ are evaluated to produce classical outputs for learning tasks.