Many quantum algorithms make use of oracles which evaluate classical
functions on a superposition of inputs. In order to facilitate implementation,
testing, and resource estimation of such algorithms, we present quantum
circuits for evaluating functions that are often encountered in the quantum
algorithm literature. This includes Gaussians, hyperbolic tangent, sine/cosine,
inverse square root, arcsine, and exponentials. We use insights from classical
high-performance computing in order to optimize our circuits and implement a
quantum software stack module which allows to automatically generate circuits
for evaluating piecewise smooth functions in the computational basis. Our
circuits enable more detailed cost analyses of various quantum algorithms,
allowing to identify concrete applications of future quantum computing devices.
Furthermore, our resource estimates may guide future research aiming to reduce
the costs or even the need for arithmetic in the computational basis
altogether.

该论文介绍了为了更好地实现、测试和资源评估量子算法，使用量子电路来评估量子算法文献中经常遇到的函数，以及借鉴目前高性能计算领域的经验优化这些电路，并实现了一个量子软件堆栈模块来自动生成用于计算基础函数的电路，以更详尽的成本分析启发未来量子计算设备的具体应用和降低计算基础算术成本的未来研究。

量子电路算法优化

Optimizing Quantum Circuits for Arithmetic

In this paper, we revise two commonly used saturated functions, the logistic
sigmoid and the hyperbolic tangent (tanh).
We point out that, besides the well-known non-zero centered property, slope
of the activation function near the origin is another possible reason making
training deep networks with the logistic function difficult to train. We
demonstrate that, with proper rescaling, the logistic sigmoid achieves
comparable results with tanh.
Then following the same argument, we improve tahn by penalizing in the
negative part. We show that "penalized tanh" is comparable and even outperforms
the state-of-the-art non-saturated functions including ReLU and leaky ReLU on
deep convolution neural networks.
Our results contradict to the conclusion of previous works that the
saturation property causes the slow convergence. It suggests further
investigation is necessary to better understand activation functions in deep
architectures.

本文在深度神经网络中研究常用的饱和函数：logistic sigmoid 和双曲正切 (tanh)，发现使用 logistic sigmoid 函数训练困难的原因不仅在于其非零中心属性，还在于其在原点附近的斜率过大。通过适当的重新调整，logistic sigmoid 和 tanh 函数的性能相当。接着，通过在负部分加罚项可以改进 tanh 函数，形成了 “带惩罚的 tanh” 函数，其性能甚至优于 ReLu 和 Leaky ReLU 等饱和函数。本文的结果与之前的研究结论产生冲突，表明有必要进一步研究深度架构中的激活函数。