Paper ID: 2406.18316

Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks

Koki Chinzei, Shinichiro Yamano, Quoc Hoan Tran, Yasuhiro Endo, Hirotaka Oshima

Quantum neural networks (QNNs) require an efficient training algorithm to achieve practical quantum advantages. A promising approach is the use of gradient-based optimization algorithms, where gradients are estimated through quantum measurements. However, general QNNs lack an efficient gradient measurement algorithm, which poses a fundamental and practical challenge to realizing scalable QNNs. In this work, we rigorously prove a trade-off between gradient measurement efficiency, defined as the mean number of simultaneously measurable gradient components, and expressivity in a wide class of deep QNNs, elucidating the theoretical limits and possibilities of efficient gradient estimation. This trade-off implies that a more expressive QNN requires a higher measurement cost in gradient estimation, whereas we can increase gradient measurement efficiency by reducing the QNN expressivity to suit a given task. We further propose a general QNN ansatz called the stabilizer-logical product ansatz (SLPA), which can reach the upper limit of the trade-off inequality by leveraging the symmetric structure of the quantum circuit. In learning an unknown symmetric function, the SLPA drastically reduces the quantum resources required for training while maintaining accuracy and trainability compared to a well-designed symmetric circuit based on the parameter-shift method. Our results not only reveal a theoretical understanding of efficient training in QNNs but also provide a standard and broadly applicable efficient QNN design.

Submitted: Jun 26, 2024