Structured Unitary Tensor Network Representations for Circuit-Efficient Quantum Data Encoding

TL;DR

TNQE addresses the data-encoding bottleneck in quantum machine learning by leveraging structured tensor networks to produce circuit-efficient quantum encodings. It decouples data representation (TN decomposition) from circuit realization (core-to-circuit conversion), introducing three instantiations—TNQE-full, TNQE-core, and TNQE-unitary—to cover sequential and parallel encoding with a unitary-aware optimization. It achieves depths as low as $0.04\times$ the depth of amplitude encoding and scales to high-resolution inputs up to $256\times256$, with validation on real IBM hardware. The results show preserved semantic information and practical feasibility, highlighting TNQE as a promising path for end-to-end quantum data encoding in quantum machine learning.

Abstract

Encoding classical data into quantum states is a central bottleneck in quantum machine learning: many widely used encodings are circuit-inefficient, requiring deep circuits and substantial quantum resources, which limits scalability on quantum hardware. In this work, we propose TNQE, a circuit-efficient quantum data encoding framework built on structured unitary tensor network (TN) representations. TNQE first represents each classical input via a TN decomposition and then compiles the resulting tensor cores into an encoding circuit through two complementary core-to-circuit strategies. To make this compilation trainable while respecting the unitary nature of quantum operations, we introduce a unitary-aware constraint that parameterizes TN cores as learnable block unitaries, enabling them to be directly optimized and directly encoded as quantum operators. The proposed TNQE framework enables explicit control over circuit depth and qubit resources, allowing the construction of shallow, resource-efficient circuits. Across a range of benchmarks, TNQE achieves encoding circuits as shallow as $0.04\times$ the depth of amplitude encoding, while naturally scaling to high-resolution images ($256 \times 256$) and demonstrating practical feasibility on real quantum hardware.

Figures (10)

• Figure 1: Different-resolution visual comparisons in simulation are conducted using $32 \times 32$, $256 \times 256$, and $256 \times 256 \times 3$ images. The ordering of methods is consistent across resolutions, and the circuit structure is reported as (qubit, depth). Amplitude encoding enables accurate encoding in simulation at the cost of deep circuits, TNQEs provide different trade-offs between qubit and depth.
• Figure 2: Illustration of the proposed TNQEs. First, a) classical data is represented using quantized tensor train decomposition. Then, b) full tensor network encoding and c) core-wise encoding are developed to convert the core tensors into quantum circuits.
• Figure 3: Illustration of TNQE-unitary. Each core is parameterized by a learnable block unitary acting on the bond and physical qubits, with $N_\ell$ layers of single-qubit rotations and fixed entanglers.
• Figure 4: Comparison of quantum circuit for (a) amplitude encoding (partially shown due to its depth), (b) TNQE-full encoding and TNQE-unitary encoding, and (c) TNQE-core encoding.
• Figure 5: Scalability of different quantum data encoding methods. (a) The number of qubits, (b) circuit depth, and (c) the number of quantum operators as image size increases from $4 \times 4$ to $512 \times 512$. (d) Classification accuracy (%) on approximated MNIST images using CNN and MLP classifiers.