Predicting optimal parameters with random forest for quantum key distribution

Abstract

For practical quantum key distribution (QKD) in finite-data case, full optimized parameters can greatly improve its key rate. To gain such optimal parameters, traditional search algorithms are performed quite frequently despite the high time and hardware overhead, which may be a severe challenge for real-time QKD systems and large-scale QKD networks. In this paper, instead of searching optimal parameters, we employ random forest to directly predict those parameters. Firstly, we illustrate the feasibility of this method with 3-intensity measurement-device-independent QKD (MDI-QKD). Later, we rebuild a versatile model applicable to MDI and BB84 protocol simultaneously. Both numerical simulations demonstrate our method enjoys a low time and hardware overhead compared with traditional search method and achieves over 99% of the optimal secure key rate as well, which is very promising in future QKD applications.

Appendix: derivation about the function f

Assume the training set is ${{\varvec{D}}}=\{(\textit{X}_i, \textit{Y}_i)\}_{i=1}^n$, where $\textit{X}_i$ denotes input features with responses $\textit{Y}_i$. In the input space of training set, the binary decision tree is generated by recursively dividing each region into two subregions and determining the output value of each one.

• (1) Select the optimal segmentation variable j and the segmentation point s to solve:

  $$\begin{aligned} \mathop {\min }\limits _{j,s} \left[ {\mathop {\min }\limits _{{c_1}} \sum _{{x_i} \in {R_1}\left( {j,s} \right) } {{{\left( {{y_i} - {c_1}} \right) }^2}} + \mathop {\min }\limits _{{c_2}} \sum _{{x_i} \in {R_2}\left( {j,s} \right) } {{{\left( {{y_i} - {c_2}} \right) }^2}} } \right] . \end{aligned}$$

  (4)

  By traversing the variable j, the segmentation point s is scanned for the fixed segmentation variable j, and (j, s) is finally obtained that makes Eq. (4) reach the minimum.

• (2) Devide the region with selected (j, s) and determine the corresponding output value:

  $$\begin{aligned}&{R_1}\left( {j,s} \right) = \left\{ {x|{x^{\left( j \right) }} \le s} \right\} , {R_2}\left( {j,s} \right) = \left\{ {x|{x^{\left( j \right) }} > s} \right\} ,\nonumber \\&{{\hat{c}}_m} = \frac{1}{{{N_m}}}\sum _{{x_i} \in {R_m} \left( {j,s} \right) } {y_i} , x \in {R_m},m = 1,2. \end{aligned}$$

  (5)

• (3) Continue to call steps (1,2) on the two subregions until the stop condition is met.

• (4) Divide the input space into M subregions ($R_1, R_2,\ldots , R_M$) to generate a decision tree:

  $$\begin{aligned} f\left( x \right) = \sum _{m = 1}^M {{{\hat{c}}_m} I\left( {x \in {R_m}} \right) }. \end{aligned}$$

  (6)