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Boson sampling is a candidate for quantum protocols to truly realize the quantum computation advantage and to be used in advanced fields where complex computations are needed, such as quantum chemistry. However, this proposal is hard to achieve due to the existence of noise sources such as photon losses. In order to quantificationally analyze the influences of photon losses in optical networks, boson sampling is classically simulated based on the equivalent beam splitter mechanism, where the photon loss happening in optical units is equivalent to the photon transmission into the environmental paths through a virtual beam splitter. In our simulation, networks corresponding to random unitary matrices are made up, considering both the Reck structure and the Clements structure. The photon loss probability in an optical unit is well controlled by adjusting the parameters of the virtual beam splitter. Therefore, to simulate boson sampling with photon losses in optical networks is actually to simulate ideal boson sampling with more modes. It is found that when the photon loss probability is constant, boson sampling with Clements structures distinctly performs much better than that with Reck structures. Furthermore, the photon loss probability is also set to follow the normal distribution, which is thought to be closer to the situation in reality. It is found that when the mean value of photon loss probability is constant, for both network structures, errors of outputs become more obvious with the increase of standard deviation. It can be inferred that the increase of error rate can be explained by the network depth and the conclusion is suitable for larger-scale boson sampling. Finally, the number of output photons is taken into consideration, which is directly related to the classical computation complexity. It is found that with the photon loss probability, the ratio of output combinations without photon losses decreases sharply, implying that photon losses can obviously affect the quantum computation advantage of boson sampling. Our results indicate that photon losses can result in serious errors for boson sampling, even with a stable network structure such as that of Clements. This work is helpful for boson sampling experiments in reality and it is desired to develop a better protocol, for example, a well-designed network or excellent optical units, to well suppress photon losses.
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Keywords:
- boson sampling/
- optical network/
- photon loss/
- quantum computing
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算法1 任意幺正矩阵符合Reck结构的分解算法[20] 输入 任意幺正矩阵U 输出 符合Reck结构的矩阵列表M_list 1:M_list ← $\phi$ //初始化M_list 为空集 2:fork← 1 to (m– 1)do //m为U的维数 3:fort← 0 to (k– 1)do 4: ω← arctan(|um–t,k–t/um–t,k–t+1|) //计算分束器参数ω,u为U中元素, 下标分别表示u所在的行数和列数 5: temp ← (um–t,k–t/um–t,k–t+1)cotω //计算临时变量temp 6: φ← arctan(Imag(temp)/Real(temp)) //计算相移器参数φ, Imag和Real分别表示temp的虚部和实部 7: if|um–t,k–t+1sinω –e–iφum–t,k–tcosω|≠ 0then 8: ω←–ω //修正ω符号, 使得步骤10的消元能够顺利进行 9: M← OpticalUnit(m,k – t,k – t+ 1,ω,φ)
//利用(1)式计算m维光学单元矩阵, 其中分束器所在通道数为(k–t)和(k–t+ 1), 相移器所在通道数为(k–t)10:M_list ← (M_list,M),U←UM–1 //对矩阵U进行消元 11:end for 12:end for 13:M_list←(M_list,U) //将消元后得到的对角矩阵放入M_list 14:M_list←Reverse(M_list) //反向排列M_list, 使得M_list中所有元素乘积为待分解矩阵 
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算法2 任意幺正矩阵符合Clements结构的分解算法[19] 输入 任意幺正矩阵U 输出 符合Clements结构的矩阵列表M_list 1:M_list ←M_list1←M_list2←M_list3←ω_list←$\phi$
//初始化M_list 为空集, 并初始化辅助集合M_list1,M_ list2,M_list3以及ω_list 为空集 2:fork← 1 to (m– 1)do //m为U的维数 3: fort← 0 to (k– 1)do 4: ifkmod 2 ≠ 0then 5: Compute(ω,φ) //根据算法1计算ω,φ 6: M← OpticalUnit(m,k–t,k–t+ 1,ω,φ) 7: M_list1← (M_list1,M),U←UM–1 //若k为奇数, 则利用M–1右乘U, 对U进行消元 8: else 9: ω← –arctan(|um–k+t+1,t+1/um–k+t,t+1|) 10: temp ← –(um–k+t+1,t+1/um–k+t,t+1)cotω 11: φ← arctan(Imag(temp)/Real(temp)) 12: if|um–k+t+1,t+1cosω+ eiφum–k+t,t+1sinω| ≠ 0then 13: ω← –ω //修正ω符号, 使得步骤15的消元能够顺利进行 14: M← OpticalUnit(m,m–k+t,m–k+t+ 1,ω,φ) 15: ω_ list←(ω_list ,ω),M_list2←(M_list2,M),U←MU //若k为偶数, 则利用M左乘U, 对U进行消元 16: end for 17:end for 18:ω_list←Reverse(ω_list ),M_list2←Reverse(M_list2),D←U,p← 1
//U经过消元变换为对角矩阵D, 待分解矩阵可以表示为形如M–1M–1···M–1DMM···M的形式19:fork← (m– 1) to 1 by –1do 20: fort← (k– 1) to 0 by –1do 21: ifkmod 2 = 0then 22: ω ←|ωp| //计算M'的分束器参数ω,ωp为ω_list的第p个元素 23: temp ← –tanωcotωpdm–k+t/dm–k+t+1 //d表示D的对角元素, 下标表示d所在行(列)数 24: φ← arctan(Imag(temp)/Real(temp)) //计算M'的相移器参数φ 25: if|dm–k+tsinωp+ eiφdm–k+t+1cosωp| ≠ 0then 26: ω ← –ω //修正ω符号 27: Compute(φ) //在得到修正的ω后, 根据步骤23和24计算φ, 使得$ \boldsymbol D' \boldsymbol M'= \boldsymbol M_p^{-1} \boldsymbol D $, 其中Mp为M_list2的第p个元素 28: M←M'← OpticalUnit(m,m–k+t,m–k+t+ 1,ω,φ) //计算M'并赋值给M 29: M_list3←(M_list3,M),$ \boldsymbol D← \boldsymbol D'← \boldsymbol M_p^{-1} \boldsymbol D \boldsymbol M^{-1} $ ,p←p+ 1 //计算D'并赋值给D 30: end for 31:end for 32:M_list3←(M_list3,D) //将对角矩阵放入M_list3 33:M_list1←Reverse(M_list1),M_list3←Reverse(M_list3) 34:M_list←(M_list3,M_list1) //M_list中所有元素乘积为待分解矩阵 DownLoad:
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算法3 考虑光子损失的玻色采样输出组合概率算法 输入 输入组合S, 输出组合T,m维光学网络矩阵U, 单条实际光路在单个光学单元处的光子损失概率Ploss 输出 输出组合概率P out 1:M_list ← Decompose(U) //按照Reck或Clements结构分解矩阵U 2:A←M1$\oplus $Em(m–1),M1∈M_list //将对角矩阵的维度扩展为m2维 3:fork← 2 to (m(m– 1)/2 + 1)do 4: Mk←Mk$\oplus $Em(m– 1) , Mk∈M_ list //将光学单元矩阵的维度扩展为m2维 5: ω← arccos(1 – (1 –Ploss)1/2) //计算虚构分束器参数ω 6: Compute(ch) //确定原始光学单元所在通道数ch和(ch+ 1) 7: B← OpticalUnit(m2,ch,m+ 2k– 3,ω, 0),B'← OpticalUnit(m2,ch+ 1,m+ 2k– 2,ω, 0)
//构造单个光学单元中的2个虚构分束器矩阵8: A←AMkBB'//构造考虑光子损失的等效光学网络矩阵 9:end for 10:S← (S, [0]m(m–1)) //将输入组合维度变为m2 11:Pout← ∑T'| Perm(AS, (T,T'))|2,T' ∈{0, 1}m(m–1) //Perm()表示矩阵积和式函数,T'中所有元素之和为损失光子数 DownLoad:
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