本期导读

1. Why：为什么引入$\alpha$？
2. What：$\alpha$的计算依据是什么？
3. How：如何计算$\alpha$？

参数$\alpha$的解读

Why：为什么引入$\alpha$？

【背景】在学习HHL算法时，研读了一些关于HHL算法的实验实现。下面给出这些算法的线路截图及参考来源：

• [1] Y. Cao, A. Daskin, S. Frankel, and S. Kais,Quantum circuits for solving linear systems of equations. Molecular Physics 110, 1675 (2012).

• [2] J. Pan, Y. Cao, X. Yao, Z. Li, C. Ju, H. Chen, X. Peng, S. Kais, and J. Du,Experimental realization of quantum algorithm for solving linear systems of equations. Phys. Rev. A, 89, 022313 (2014).

• [3] X. Cai, C. Weedbrook, Z. Su, M. Chen, M. Gu, M. Zhu, L. Li, N. Liu, C. Lu, and J. Pan, Experimental quantum computing to solve systems of linear equations. Phys. Rev. Lett. 110, 230501 (2013).

$R_y(\theta)={\cos(\theta/2)\qquad-\sin(\theta/2)\choose \sin(\theta/2)\qquad\cos(\theta/2)}$

[4] Iris Cong and Luming Duan，Quantum discriminant analysis for dimensionality reduction and classification. New J. Phys. 18 073011, 2016.

What：$\alpha$的计算依据是什么？

$\theta=0.\theta_1\cdots\theta_d=\sum\nolimits_{j=1}^d\theta_j\cdot2^{-j},(\theta_j\in\{0,1\})$

$R_y(2\alpha\theta)=e^{-i\alpha\theta Y}=\prod\limits_{j=1}^{d}e^{-i\alpha\theta Y/{2j}}=\prod\limits_{j=1}^{d}R_y^{\theta_j}(2^{1-j}\alpha)\\=R_y^{\theta_1}(\alpha)R_y^{\theta_2}(\alpha)\cdots R_y^{\theta_d}(\alpha/{2^{d-1}})$

$P(\alpha)=\frac{N_{\alpha}}{N_1}=\frac{\sum\limits_{k=1}^{r}\sigma_k^2\sin^2(y_k\alpha)}{\sum\limits_{k=1}^{r}\sigma_k^2}$

$F(\alpha)=\langle\psi_{\hat{S}}\mid\psi_S\rangle=\frac{1}{aqrt{N_2 N_{\alpha}}}\sum\nolimits_{k=1}^{r}\sigma_k^2y_k\sin(y_k\sigma)\\=\frac{\sum\limits_{k=1}^{r}\sigma_k^2y_k\sin(y_k\alpha)}{\sqrt{\sum\nolimits_{k=1}{r}\sigma_k^2y_k^2\times\sum\nolimits_{k=1}^r\sigma_k^2\sin^2(y_k\alpha)}}$

How：如何计算$\alpha$？

$G(\alpha)=\sqrt{P(\alpha)}\times F(\alpha)=\sqrt{\frac{N_{\alpha}}{N_1}}\times\frac{1}{\sqrt{N_2N_{\alpha}}}\sum\nolimits_{k=1}^{r}\sigma_k^2y_k\sin(y_k\alpha)\\=\frac{\sum\nolimits_{k=1}{r}\sigma_k^2y_k\sin(y_k\sigma)}{\sqrt{\sum\nolimits_{k=1}{r}\sigma_k^2\times\sum\nolimits_{k=1}{r}\sigma_k^2y_k^2}}$

G的引入使得在目标函数的分母中消去了变量$\alpha$，而使求解问题变得简单。因而最终的求解问题变成了：

$\arg_{\alpha}\max G(\alpha)$

$G^{`}(\alpha)=\frac{\sum\nolimits_{k=1}^{r}\sigma_k^2y_k\cos(y_k\sigma)}{\sqrt{\sum\nolimits_{k=1}^{r}\sigma_k^2\times\sum\nolimits_{k=1}{r}\sigma_k^2y_k^2}}=0$

小结

• [5] Bojia Duan, Jiabin Yuan, Ying Liu and Dan Li. Efficient quantum circuit for singular value thresholding. arXiv:1711.08896. (2017)