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Abstract

The optimal selection of parameters in practical quantum key distribution can greatly improve the key generation rate and maximum transmission distance of the system. Owing to the high cost of global search algorithm, local search algorithm is widely used. However, there are two shortcomings in local search algorithm. One is that the solution obtained is not always the global optimal solution, and the other is that the effectiveness of the algorithm is greatly dependent on the choice of initial value. This paper uses the Monte Carlo method to prove whether the key generation rate function is convex, and also simulates and analyzes the projection of the key generation rate function on each dimension of the parameter, in contrast to the approach in previous article. In order to eliminate the effect of the initial value, this paper proposes the particle swarm local search optimization algorithm which integrates particle swarm optimization algorithm and local search algorithm. The first step is to use the particle swarm optimization to find a valid parameter which leads to nonzero key generation rate, and the second step is to adopt the parameter as the initial value of local search algorithm to derive the global optimal solution. Then, the two algorithms are used to conduct simulation and their results are compared. The results show that the key generation rate function is non-convex because it does not satisfy the definition of a convex function, however, since the key generation rate function has only one non-zero stagnation point, the LSA algorithm can still obtain the global optimal solution with an appropriate initial value. When the transmission distance is relatively long, the local search algorithm is invalid because it is difficult to obtain an effective initial value by random value method. The particle swarm optimization algorithm can overcome this shortcoming and improve the maximum transmission distance of the system at the cost of slightly increasing the complexity of the algorithm.

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Authors and contacts

Corresponding author:Zhou Yuan-Yuan,EPJZYY@aliyun.com

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Cited By

DownLoad: Full-Size Img PowerPoint

目标函数$ {f_{{\text{LSA}}}}\left( {{x_1}, \cdots , {x_n}} \right) $
根据经验初始化搜索位置$ {p_0} = \left( {x_1^0, \cdots , x_n^0} \right) $, 初始化迭代次数 t= 0; while (迭代判决条件) do 　fork= 1∶n 　　对$ {f_{{\text{LSA}}}}\left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^t, x_{k + 1}^t, \cdots , x_n^t} \right) $在$ x_k^t $维度上　　进行线性回溯搜索　　if ${f_{ {\text{LSA} } } }\left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^{t + 1}, x_{k + 1}^t, \cdots , x_n^t} \right) > $ 　　 $ {f_{ {\text{LSA} } } }\left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^t, x_{k + 1}^t, \cdots , x_n^t} \right)$ 　　　更新$ x_k^t $为$ x_k^{t + 1} $ 　　　$ {p_{t + 1}} = \left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^{t + 1}, x_{k + 1}^t, \cdots , x_n^t} \right) $ 　　else 　　　$ {p_{t + 1}} = \left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^t, x_{k + 1}^t, \cdots , x_n^t} \right) $ 　　end 　end 　更新迭代次数t=t+ 1 　更新搜索位置为$ {p_{t + 1}} $ end 输出最终结果$ \left( {x_1^{t + 1}, \cdots , x_n^{t + 1}} \right) $和$ {f_{{\text{LSA}}}}\left( {x_1^{t + 1}, \cdots , x_n^{t + 1}} \right) $

目标函数$ {f_{{\text{LSA}}}}\left( {{x_1}, \cdots , {x_n}} \right) $

根据经验初始化搜索位置$ {p_0} = \left( {x_1^0, \cdots , x_n^0} \right) $, 初始化迭代次数
t= 0;
while (迭代判决条件) do
　fork= 1∶n
　　对$ {f_{{\text{LSA}}}}\left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^t, x_{k + 1}^t, \cdots , x_n^t} \right) $在$ x_k^t $维度上
　　进行线性回溯搜索
　　if ${f_{ {\text{LSA} } } }\left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^{t + 1}, x_{k + 1}^t, \cdots , x_n^t} \right) > $
　　 $ {f_{ {\text{LSA} } } }\left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^t, x_{k + 1}^t, \cdots , x_n^t} \right)$
　　　更新$ x_k^t $为$ x_k^{t + 1} $
　　　$ {p_{t + 1}} = \left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^{t + 1}, x_{k + 1}^t, \cdots , x_n^t} \right) $
　　else
　　　$ {p_{t + 1}} = \left( {x_1^{t + 1}, \cdots , x_{k - 1}^{t + 1}, x_k^t, x_{k + 1}^t, \cdots , x_n^t} \right) $
　　end
　end
　更新迭代次数t=t+ 1
　更新搜索位置为$ {p_{t + 1}} $
end
输出最终结果$ \left( {x_1^{t + 1}, \cdots , x_n^{t + 1}} \right) $和$ {f_{{\text{LSA}}}}\left( {x_1^{t + 1}, \cdots , x_n^{t + 1}} \right) $

DownLoad: CSV

目标函数$ f\left( {\boldsymbol{x}} \right) $
根据经验初始化粒子1的位置$ {{\boldsymbol{x}}_1} $ 初始化剩余$ n - 1 $个粒子的位置$ {{\boldsymbol{x}}_i} $和所有粒子的速度$ {{\boldsymbol{\nu }}_i} $ 寻找全局最优解$ {{\boldsymbol{g}}_{\text{a}}} $, $ {{\boldsymbol{g}}_{\text{a}}} = \min \left\{ {f\left( {{{\boldsymbol{x}}_1}} \right), \cdots , f\left( {{{\boldsymbol{x}}_n}} \right)} \right\} $ while(迭代判决条件) do 　for 所有的粒子 do 　　更新速度$ {\boldsymbol{\nu }}_i^{t + 1} $ 　　更新位置$ {\boldsymbol{x}}_i^{t + 1} $ 　　计算目标函数在新位置的值　　更新当前粒子的历史最优位置$ {\boldsymbol{x}}_i^* $ 　end 更新全局最优解$ {{\boldsymbol{g}}_{\text{a}}} $ 　end 输出最终结果$ {\boldsymbol{x}}_i^* $和$ {{\boldsymbol{g}}_{\text{a}}} $ 以$ {{\boldsymbol{g}}_{\text{a}}} $为初始点, 采用LSA算法求局部最优解

目标函数$ f\left( {\boldsymbol{x}} \right) $

根据经验初始化粒子1的位置$ {{\boldsymbol{x}}_1} $
初始化剩余$ n - 1 $个粒子的位置$ {{\boldsymbol{x}}_i} $和所有粒子的速度$ {{\boldsymbol{\nu }}_i} $
寻找全局最优解$ {{\boldsymbol{g}}_{\text{a}}} $, $ {{\boldsymbol{g}}_{\text{a}}} = \min \left\{ {f\left( {{{\boldsymbol{x}}_1}} \right), \cdots , f\left( {{{\boldsymbol{x}}_n}} \right)} \right\} $
while(迭代判决条件) do
　for 所有的粒子 do
　　更新速度$ {\boldsymbol{\nu }}_i^{t + 1} $
　　更新位置$ {\boldsymbol{x}}_i^{t + 1} $
　　计算目标函数在新位置的值
　　更新当前粒子的历史最优位置$ {\boldsymbol{x}}_i^* $
　end
更新全局最优解$ {{\boldsymbol{g}}_{\text{a}}} $
　end
输出最终结果$ {\boldsymbol{x}}_i^* $和$ {{\boldsymbol{g}}_{\text{a}}} $
以$ {{\boldsymbol{g}}_{\text{a}}} $为初始点, 采用LSA算法求局部最优解

DownLoad: CSV

${\eta _{\text{d}}}$/%	${e_{\text{d}}}$/%	${Y_0}$	${f_{\text{e}}}$	$\chi $	$N$	$\alpha $
14.5	1.5	$6.02 \times {10^{ - 6}}$	1.16	${10^{ - 7}}$	${10^{12}}$	0.21

DownLoad: CSV

参数	$ {{\boldsymbol{x}}_1} $	$ {{\boldsymbol{x}}_2} $	$ {{\boldsymbol{x}}_3} $	$ {{\boldsymbol{x}}_4} $
$ \mu $	0.40	0.60	0.50	0.075
$ \nu $	0.037	0.045	0.029	$5.9 \times {10^{ - 3} }$
$ \omega $	$7.9 \times {10^{ - 4} }$	$2.6 \times {10^{ - 3} }$	$2.0 \times {10^{ - 3} }$	$2.57 \times {10^{ - 4} }$
$ {P_\mu } $	0.15	0.48	0.54	0.020
$ {P_\nu } $	0.18	0.13	0.26	0.48
$ {P_{{\text{X}}\|\mu }} $	0.8	0.14	0.76	0.14
$ {P_{{\text{X}}\|\nu }} $	0.92	0.56	0.89	0.64
$ {P_{{\text{X}}\|\omega }} $	0.25	0.99	0.54	0.55
$ \theta $	0.47		0.64
$ F\left( {\theta , {\boldsymbol{x}}, {\boldsymbol{y}}} \right) $	$ - 5.50 \times {10^{ - 6}} $		$ 3.84 \times {10^{ - 6}} $

DownLoad: CSV

算法	迭代次数	时间/s	密钥生成率
LSA	858	0.12	$ 4.13 \times {10^{ - 6}} $
PSO	40000	4.2	$ 2.97 \times {10^{ - 6}} $
PSLSA	40559	4.29	$ 4.13 \times {10^{ - 6}} $

DownLoad: CSV

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Vol. 72, Issue 12 - 2023-06-20

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