Quantum neural networks (QNNs) use parameterized quantum circuits with data-dependent inputs and generate outputs through the evaluation of expectation values. Calculating these expectation values necessitates repeated circuit evaluations, thus introducing fundamental finite-sampling noise even on error-free quantum computers. We reduce this noise by introducing the variance regularization, a technique for reducing the variance of the expectation value during the quantum model training. This technique requires no additional circuit evaluations if the QNN is properly constructed. Our empirical findings demonstrate the reduced variance speeds up the training and lowers the output noise as well as decreases the number of necessary evaluations of gradient circuits. This regularization method is benchmarked on the regression of multiple functions. We show that in our examples, it lowers the variance by an order of magnitude on average and leads to a significantly reduced noise level of the QNN. We finally demonstrate QNN training on a real quantum device and evaluate the impact of error mitigation. Here, the optimization is feasible only due to the reduced number of necessary shots in the gradient evaluation resulting from the reduced variance.

翻译：量子神经网络利用参数化量子电路，通过依赖数据的输入并基于期望值的评估生成输出。计算这些期望值需要重复进行电路评估，因此即使在无误差的量子计算机上也会引入固有的有限采样噪声。我们通过引入方差正则化来降低这种噪声，这是一种在量子模型训练期间减少期望值方差的技术。若量子神经网络构建得当，该技术无需额外进行电路评估。我们的实证结果表明，方差减小能够加速训练过程、降低输出噪声，并减少梯度电路的必要评估次数。这种正则化方法在多个函数的回归任务上进行了基准测试。研究表明，在我们的示例中，它使方差平均降低一个数量级，并显著降低了量子神经网络的噪声水平。最后，我们展示了在真实量子设备上进行的量子神经网络训练，并评估了误差缓解的影响。在此过程中，由于方差减小导致的梯度评估所需量子采样次数降低，才使得优化得以实现。