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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
[1] Shor P W 1994 Proceedings 35th Annual Symposium on Foundations of Computer Science Santa Fe, NM, USA, IEEE p124
[2] Grover L K 1997 Phys. Rev. Lett. 79 325
Google Scholar
[3] Spring J B, Metcalf B J, Humphreys P C, Kolthammer W S, Jin X M, Barbieri M, Datta A., Thomas N, Langford N K, Kundys D, Gates J C, Smith B J, Smith P G, Walmsley I A 2013 Science 339 798
Google Scholar
[4] Zhong H S, Wang H, Deng Y H, Chen M C, Peng L C, Luo Y H, Qin J, Wu D, Ding X, Hu Y H, Yang X Y, Zhang W J, Li H, Li Y X, Jiang X, Gan L, Yang G G, You L X, Wang Z, Li L, Liu N L, Lu C Y, Pan J W 2020 Science 370 1460
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[5] Zhong H S, Li Y, Li W, Peng L C, Su Z E, Hu Y, He Y M, Ding X, Zhang W J, Li H, Zhang L, Wang Z, You L X, Wang X L, Jiang X, Li L, Chen Y A, Liu N L, Lu C Y, Pan J W 2018 Phys. Rev. Lett. 121 250505
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[6] Wang H, Li W, Jiang X, He Y M, Li Y H, Ding X, Chen M C, Qin J, Peng C Z, Schneider C, Kamp M, Zhang W J, Li H, You LX, Wang Z, Dowling J P, Höfling S, Lu C Y, Pan J W 2018 Phys. Rev. Lett. 120 230502
Google Scholar
[7] Peropadre B, Guerreschi G G, Huh J, Aspuru-Guzik A 2016 Phys. Rev. Lett. 117 140505
Google Scholar
[8] Aaronson S, Arkhipov A 2011 Proceedings of the forty-third annual ACM symposium on Theory of computing San Jose, California, June 6–8, 2011 p333
[9] Spagnolo N, Vitelli C, Bentivegna M, Brod D J, Crespi A, Flamini F, Giacomini S, Milani G, Ramponi R, Mataloni P, Osellame R, Galvão E F, Sciarrino F 2014 Nat. Photonics 8 615
Google Scholar
[10] Bentivegna M, Spagnolo N, Vitelli C, Brod D J, Crespi A, Flamini F, Ramponi R, Mataloni P, Osellame R, Galvão E F, Sciarrino F 2015 Int. J. Quantum. Inf. 12 1560028
Google Scholar
[11] Agresti I, Viggianiello N, Flamini F, Spagnolo N, Crespi A, Osellame R, Wiebe N, Sciarrino F 2017 Phys. Rev. X 9 2160
Google Scholar
[12] Huh J, Guerreschi G G, Peropadre B, Mcclean J R, Aspuru-Guzik A 2015 Nat. Photonics 9 615
Google Scholar
[13] Renema1 J J, Shchesnovich V, Garcia Patron R 2019 arXiv: 1809.01953 v2 [quant-ph]
[14] Tichy C M 2015 Phys. Rev. A 91 022316
Google Scholar
[15] Renema J J, Menssen A, Clements W R, Triginer G, Kolthammer W S, Walmsley I A 2018 Phys. Rev. Lett. 120 220502
Google Scholar
[16] Wang H, He Y, Li Y H, Su Z E, Li B, Huang H L, Ding X, Chen M C, Liu C, Qin J, Li J P, He Y M, Schneider C, Kamp M, Peng C Z, Höfling S, Lu C Y, Pan J W 2017 Nat. Photonics 11 361
Google Scholar
[17] García-Patrón R, Renema J J, Shchesnovich V 2019 Quantum 3 169
Google Scholar
[18] Oszmaniec M, Brod D J 2018 New J. Phys. 20 092002
Google Scholar
[19] Clements W R, Humphreys P C, Metcalf B J, Steven K W, Walsmley I A 2016 Optica 3 1460
Google Scholar
[20] Reck M, Zeilinger A, Bernstein H J, Bertani P 1994 Phys. Rev. Lett. 73 58
Google Scholar
[21] Brod D J, Galvão E F, Crespi A, Osellame R, Spagnolo N, Sciarrino F 2019 Adv. Photon. 1 034001
Google Scholar
[22] Neville A, Sparrow C, Clifford R, Johnston E, Birchall P M, Montanaro A, Laing A 2017 Nat. Phys. 13 1153
Google Scholar
[23] Clifford P, Clifford R 2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms New Orleans Louisiana January 7–10, 2018 p146
[24] Aaronson S, Brod D J 2015 Phys. Rev. A 93 012335
Google Scholar
[25] Miller D A B 2015 Optica 2 8
Google Scholar
[26] Wang H, Qin J, Ding X, Chen M C, Chen S, You X, He Y M, Jiang X, You L, Wang Z, Schneider C, Renema J J, Höfling S, Lu C Y, Pan J W 2019 Phys. Rev. Lett. 123 250503
Google Scholar
[27] Zhang R, Liu L Z, Li Z D, Fei Y Y, Yin X F, Li L, Liu N L, Mao Y, Chen Y A, Pan J W 2022 Optica 9 2
Google Scholar
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图 1 光学线性网络示意图: (a) Reck结构; (b) Clements结构
Figure 1. Sketches of optical linear networks: (a) the Reck structure; (b) the Clements structure.
图 2 (a) 考虑光子损失的玻色采样输出组合概率的总变差距离随光子损失概率Ploss的变化关系; (b) Ploss为0.1时, 两种光学网络结构随着模式数增加对应玻色采样结果总变差距离的变化关系. 图中空心图形代表实验结果平均值, 误差棒代表标准差
Figure 2. Plots between the total variation distance of boson sampling output combination probabilities considering photon losses and (a) the photon loss probability Ploss or (b) mode numbers when Ploss is 0.1. Open shapes represent mean values and error bars represent standard deviations.
图 3 考虑光子损失的玻色采样输出组合概率的总变差距离平均值随光子损失概率的标准差σPloss的变化关系
Figure 3. Plots between the mean values of total variation distance of boson sampling output combination probabilities considering photon losses and the standard deviation σPloss of photon loss probability.
图 4 玻色采样固定光子个数的输出组合概率之和随着Ploss的变化关系. (a) Reck结构; (b) Clements结构光学网络. 图中空心图形代表实验结果平均值, 误差棒代表标准差
Figure 4. Plots between the sum of boson sampling output combination probabilities with fixed photon numbers and Ploss: (a) the Reck structure; (b) the Clements structure. Open shapes represent mean values and error bars represent standard deviations.
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: for k ← 1 to (m – 1) do //m为U的维数 3: for t ← 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| ≠ 0 then 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: for k ← 1 to (m – 1) do //m为U的维数 3: for t← 0 to (k – 1) do 4: if k mod 2 ≠ 0 then 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| ≠ 0 then 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: for k ← (m – 1) to 1 by –1 do 20: for t ← (k – 1) to 0 by –1 do 21: if k mod 2 = 0 then 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| ≠ 0 then 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
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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 $Em(m – 1), M1∈M_list //将对角矩阵的维度扩展为m2维 3: for k ← 2 to (m(m – 1)/2 + 1) do 4: Mk ← Mk$\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
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[1] Shor P W 1994 Proceedings 35th Annual Symposium on Foundations of Computer Science Santa Fe, NM, USA, IEEE p124
[2] Grover L K 1997 Phys. Rev. Lett. 79 325
Google Scholar
[3] Spring J B, Metcalf B J, Humphreys P C, Kolthammer W S, Jin X M, Barbieri M, Datta A., Thomas N, Langford N K, Kundys D, Gates J C, Smith B J, Smith P G, Walmsley I A 2013 Science 339 798
Google Scholar
[4] Zhong H S, Wang H, Deng Y H, Chen M C, Peng L C, Luo Y H, Qin J, Wu D, Ding X, Hu Y H, Yang X Y, Zhang W J, Li H, Li Y X, Jiang X, Gan L, Yang G G, You L X, Wang Z, Li L, Liu N L, Lu C Y, Pan J W 2020 Science 370 1460
Google Scholar
[5] Zhong H S, Li Y, Li W, Peng L C, Su Z E, Hu Y, He Y M, Ding X, Zhang W J, Li H, Zhang L, Wang Z, You L X, Wang X L, Jiang X, Li L, Chen Y A, Liu N L, Lu C Y, Pan J W 2018 Phys. Rev. Lett. 121 250505
Google Scholar
[6] Wang H, Li W, Jiang X, He Y M, Li Y H, Ding X, Chen M C, Qin J, Peng C Z, Schneider C, Kamp M, Zhang W J, Li H, You LX, Wang Z, Dowling J P, Höfling S, Lu C Y, Pan J W 2018 Phys. Rev. Lett. 120 230502
Google Scholar
[7] Peropadre B, Guerreschi G G, Huh J, Aspuru-Guzik A 2016 Phys. Rev. Lett. 117 140505
Google Scholar
[8] Aaronson S, Arkhipov A 2011 Proceedings of the forty-third annual ACM symposium on Theory of computing San Jose, California, June 6–8, 2011 p333
[9] Spagnolo N, Vitelli C, Bentivegna M, Brod D J, Crespi A, Flamini F, Giacomini S, Milani G, Ramponi R, Mataloni P, Osellame R, Galvão E F, Sciarrino F 2014 Nat. Photonics 8 615
Google Scholar
[10] Bentivegna M, Spagnolo N, Vitelli C, Brod D J, Crespi A, Flamini F, Ramponi R, Mataloni P, Osellame R, Galvão E F, Sciarrino F 2015 Int. J. Quantum. Inf. 12 1560028
Google Scholar
[11] Agresti I, Viggianiello N, Flamini F, Spagnolo N, Crespi A, Osellame R, Wiebe N, Sciarrino F 2017 Phys. Rev. X 9 2160
Google Scholar
[12] Huh J, Guerreschi G G, Peropadre B, Mcclean J R, Aspuru-Guzik A 2015 Nat. Photonics 9 615
Google Scholar
[13] Renema1 J J, Shchesnovich V, Garcia Patron R 2019 arXiv: 1809.01953 v2 [quant-ph]
[14] Tichy C M 2015 Phys. Rev. A 91 022316
Google Scholar
[15] Renema J J, Menssen A, Clements W R, Triginer G, Kolthammer W S, Walmsley I A 2018 Phys. Rev. Lett. 120 220502
Google Scholar
[16] Wang H, He Y, Li Y H, Su Z E, Li B, Huang H L, Ding X, Chen M C, Liu C, Qin J, Li J P, He Y M, Schneider C, Kamp M, Peng C Z, Höfling S, Lu C Y, Pan J W 2017 Nat. Photonics 11 361
Google Scholar
[17] García-Patrón R, Renema J J, Shchesnovich V 2019 Quantum 3 169
Google Scholar
[18] Oszmaniec M, Brod D J 2018 New J. Phys. 20 092002
Google Scholar
[19] Clements W R, Humphreys P C, Metcalf B J, Steven K W, Walsmley I A 2016 Optica 3 1460
Google Scholar
[20] Reck M, Zeilinger A, Bernstein H J, Bertani P 1994 Phys. Rev. Lett. 73 58
Google Scholar
[21] Brod D J, Galvão E F, Crespi A, Osellame R, Spagnolo N, Sciarrino F 2019 Adv. Photon. 1 034001
Google Scholar
[22] Neville A, Sparrow C, Clifford R, Johnston E, Birchall P M, Montanaro A, Laing A 2017 Nat. Phys. 13 1153
Google Scholar
[23] Clifford P, Clifford R 2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms New Orleans Louisiana January 7–10, 2018 p146
[24] Aaronson S, Brod D J 2015 Phys. Rev. A 93 012335
Google Scholar
[25] Miller D A B 2015 Optica 2 8
Google Scholar
[26] Wang H, Qin J, Ding X, Chen M C, Chen S, You X, He Y M, Jiang X, You L, Wang Z, Schneider C, Renema J J, Höfling S, Lu C Y, Pan J W 2019 Phys. Rev. Lett. 123 250503
Google Scholar
[27] Zhang R, Liu L Z, Li Z D, Fei Y Y, Yin X F, Li L, Liu N L, Mao Y, Chen Y A, Pan J W 2022 Optica 9 2
Google Scholar
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