In this work we present a novel strategy to evaluate multi-variable integrals with quantum circuits. The procedure first encodes the integration variables into a parametric circuit. The obtained circuit is then derived with respect to the integration variables using the parameter shift rule technique. The observable representing the derivative is then used as the predictor of the target integrand function following a quantum machine learning approach. The integral is then estimated using the fundamental theorem of integral calculus by evaluating the original circuit. Embedding data according to a reuploading strategy, multi-dimensional variables can be easily encoded into the circuit’s gates and then individually taken as targets while deriving the circuit. These techniques can be exploited to partially integrate a function or to quickly compute parametric integrands within the training hyperspace.

中文翻译：

---

具有变分量子电路的多变量积分


在这项工作中，我们提出了一种利用量子电路评估多变量积分的新颖策略。该过程首先将积分变量编码到参数电路中。然后使用参数移位规则技术根据积分变量导出所获得的电路。然后，按照量子机器学习方法，表示导数的可观测量被用作目标被积函数的预测器。然后通过评估原始电路，使用积分微积分的基本定理来估计积分。根据重新上传策略嵌入数据，可以轻松地将多维变量编码到电路的门中，然后在推导电路时单独将其作为目标。这些技术可用于部分积分函数或快速计算训练超空间内的参数被积函数。