Though parameter shift rules have drastically improved gradient estimation methods for several types of quantum circuits, leading to improved performance in downstream tasks, so far they have not been transferable to linear optics with single photons. In this work, we derive an analytical formula for the gradients in these circuits with respect to phaseshifters via a generalized parameter shift rule, where the number of parameter shifts depends linearly on the total number of photons. Experimentally, this enables access to derivatives in photonic systems without the need for finite difference approximations. Building on this, we propose two strategies through which one can reduce the number of shifts in the expression, and hence reduce the overall sample complexity. Numerically, we show that this generalized parameter-shift rule can converge to the minimum of a cost function with fewer parameter update steps than alternative techniques. We anticipate that this method will open up new avenues to solving optimization problems with photonic systems, as well as provide new techniques for the experimental characterization and control of linear optical systems.

翻译：尽管参数偏移规则已显著改进了多种量子电路的梯度估计方法，从而提升了下游任务的性能，但迄今为止这些规则尚未能适用于单光子线性光学系统。本研究通过广义参数偏移规则推导出此类电路中关于相移器的梯度解析公式，其中参数偏移次数与总光子数呈线性关系。实验上，这使得光子系统无需借助有限差分近似即可获得导数。在此基础上，我们提出两种可减少表达式中偏移次数的策略，从而降低总体采样复杂度。数值实验表明，相较于其他技术，该广义参数偏移规则能够以更少的参数更新步骤收敛至代价函数的最小值。我们预期该方法将为解决光子系统优化问题开辟新途径，并为线性光学系统的实验表征与控制提供新技术。