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, it is generally difficult to efficiently measure
gradients in QNNs because the quantum state collapses upon measurement. In this
work, we prove a general trade-off between gradient measurement efficiency 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.

量子神经网络（QNNs）需要一种高效的训练算法以实现实际的量子优势。我们提出了一种通用的权衡关系，即高测量效率会导致较低的表达能力，而减少 QNN 的表达能力可以提高梯度测量效率，同时提出了一种称为稳定 - 逻辑乘积 ansatz 的 QNN 设计方案，可以实现权衡不等式的上限，在学习未知对称函数时显著减少了量子资源需求，保持了准确性和可训练性。