量子混合态的两种神经网络表示

杨莹; 曹怀信

doi:10.7498/aps.72.20221905

摘要

量子信息与人工智能是近年来的两个前沿研究领域, 取得了诸多改变传统科学的进展, 实现这两个领域的交叉融合成为科学家关注的热点问题. 尽管学者们在这方面已进行了许多探索, 借助它们模拟开放多体量子系统的稳态和动力学性质, 但是量子混合态的神经网络的精确表示问题仍待研究. 本文致力于量子混合态的神经网络表示问题. 借助两种神经网络架构, 构建了具有一般输入可观测量的神经网络量子混合拟态(NNQMVS) 与神经网络量子混合态(NNQMS), 分别探讨了它们的性质, 得到了张量积运算、局部酉运算下NNQMVS与NNQMS的相关结论. 为了量化给定混合态分别由规范化的NNQMVS与NNQMS逼近的能力, 分别定义了它由规范化NNQMVS 与NNQMS逼近的最佳逼近度, 给出了一般混合态能被规范化的NNQMVS与NNQMS表示的充要条件, 并探究了能用这两种神经网络架构表示的混合态的类型, 给出了相应的神经网络表示.

关键词:

Abstract

Quantum information and artificial intelligence are the two most cutting-edge research fields in recent years, which have made a lot of progress in changing the traditional science. It has become a hot topic of research to realize the cross fusion of the two fields. Scholars have made many explorations in this field. For example, they have simulated the steady state and the dynamics of open quantum many-body systems. However, little attention has been paid to the problem of accurate representation of neural networks. In this paper, we focus on neural network representations of quantum mixed states. We first propose neural network quantum mixed virtual states (NNQMVS) and neural network quantum mixed states (NNQMS) with general input observables by using two neural network architectures, respectively. Then we explore their properties and obtain the related conclusions of NNQMVS and NNQMS under tensor product operation and local unitary operation.To quantify the approximation degree of normalized NNQMVS and NNQMS for a given mixed state, we define the best approximation degree by using normalized NNQMVS and NNQMS, and obtain the necessary and sufficient conditions for the representability of a general mixed state by using normalized NNQMVS and NNQMS. Moreover, we explore the types of mixed states that can be represented by these two neural network architectures and show their accurate neural network representations.

Keywords:

作者及机构信息

杨莹¹,
曹怀信²

1.
运城学院数学与信息技术学院, 运城　044000

2.
陕西师范大学数学与统计学院, 西安　710119

通信作者: 曹怀信, caohx@snnu.edu.cn

基金项目: 国家自然科学基金(批准号: 12001480, 11871318, 12271325, 11971283)、山西省应用基础研究项目(批准号: 201901D211461, 20210302123082)、山西省优秀博士来晋科研专项(批准号: QZX-2020001)和运城学院博士启动项目(批准号: YQ-2019021) 资助的课题

Authors and contacts

Yang Ying¹,
Cao Huai-Xin²

1.
School of Mathematics and Information Technology, Yuncheng University, Yuncheng 044000, China

2.
School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China

Corresponding author: Cao Huai-Xin, caohx@snnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12001480, 11871318, 12271325, 11971283), the Applied Basic Research Program of Shanxi Province, China (Grant Nos. 201901D211461, 20210302123082), the Excellent Doctoral Research Project of Shanxi Province, China (Grant No. QZX-2020001), and the PhD Start-up Project of Yuncheng University (Grant No. YQ-2019021)

文章全文 : translate this paragraph

参考文献

[1]	LeCun Y, Bengio Y, Hinton G 2015 Nature 521 436 Google Scholar
[2]	Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N, Lloyd S 2017 Nature 549 195 Google Scholar
[3]	Ciliberto C, Herbster M, Ialongo A D, Pontil M, Rocchetto A, Severini S, Wossnig L 2018 Proc. R. Soc. A 474 20170551 Google Scholar
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[5]	Verstraete F, Murg V, Cirac J I 2008 Adv. Phys. 57 143 Google Scholar
[6]	Schuch N, Wolf M M, Verstraete F, Cirac J I 2008 Phys. Rev. Lett. 100 040501 Google Scholar
[7]	Ceperley D, Alder B 1986 Science 231 555 Google Scholar
[8]	Loh E Y, Gubernatis J E, Scalettar R T, White S R, Scalapino D J, Sugar R L 1990 Phys. Rev. B 41 9301
[9]	Schuch N, Wolf M M, Verstraete F, Cirac J I 2007 Phys. Rev. Lett. 98 140506 Google Scholar
[10]	Verstraete F, Wolf M M, Perez-Garcia D, Cirac J I 2006 Phys. Rev. Lett. 96 220601 Google Scholar
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[12]	程嵩, 陈靖, 王磊 2017 物理 46 416 Google Scholar Cheng S, Chen J, Wang L 2017 Physics 46 416 Google Scholar
[13]	蔡子 2017 物理 46 590 Google Scholar Cai Z 2017 Physics 46 590 Google Scholar
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[29]	Carleo G, Cirac I, Cranmer K, Daudet L, Schuld M, Tishby N, Maranto L V, Zdeborová L 2019 Rev. Mod. Phys. 91 045002 Google Scholar
[30]	Lu S R, Duan L M, Deng D L 2020 Phys. Rev. Res. 2 033212 Google Scholar
[31]	杨光, 刘琦, 聂敏, 刘原华, 张美玲 2022 物理学报 71 100301 Google Scholar Yang G, Liu Q, Nie M, Liu Y H, Zhang M L 2022 Acta Phys. Sin. 71 100301 Google Scholar
[32]	陈以鹏, 刘靖阳, 朱佳莉, 方伟, 王琴 2022 物理学报 71 220301 Google Scholar Chen Y P, Liu J Y, Zhu J L, Fang W, Wang Q 2022 Acta Phys. Sin. 71 220301 Google Scholar
[33]	Li W K, Lu S, Deng D L 2021 Sci. China-Phys. Mech. Astron. 64 100312 Google Scholar
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施引文献

图 1 NNQMVS相匹配的人工神经网络

Fig. 1. Artificial neural network encoding an NNQMVS.

下载: 全尺寸图片幻灯片

图 2 含参数$\varOmega = (0, b, W)$的NNQMVS对应的量子人工神经网络

Fig. 2. Quantum artificial neural network of NNQMVS with parameter $\varOmega = (0, b, W)$.

下载: 全尺寸图片幻灯片

图 3 NNQMS相匹配的深度神经网络

Fig. 3. Deep neural network encoding an NNQMS.

下载: 全尺寸图片幻灯片

图 4 $ |0\rangle\langle 0 | $ 的两种神经网络表示

Fig. 4. Two neural network representation of the state $ |0\rangle\langle 0 | $.

下载: 全尺寸图片幻灯片

图 5 $ |1\rangle\langle 1 | $ 的两种神经网络表示

Fig. 5. Two neural network representation of the state $ |1\rangle\langle 1 | $.

下载: 全尺寸图片幻灯片

图 6 $p_1 |0\rangle\langle 0 |+p_2 |1\rangle\langle 1 |(p_1, p_2 > 0,\; p_1+p_2 = 1)$的两种神经网络表示

Fig. 6. Two neural network representation of the state $p_1 |0\rangle\langle 0 |+p_2 |1\rangle\langle 1 |(p_1, p_2 > 0,\; p_1+p_2 = 1)$.

下载: 全尺寸图片幻灯片

图 7 $\rho$ 的反对偶的NNQMVS神经网络表示　(a) N = 2; (b) N = 3; (c) N = 4

Fig. 7. Anti-dual NNQMVS neural network representation of the state $ \rho$: (a) N = 2; (b) N = 3; (c) N = 4.

下载: 全尺寸图片幻灯片

图 8 ρ的NNQMVS 神经网络表示

Fig. 8. NNQMVS neural network representation of the state ρ.

下载: 全尺寸图片幻灯片

图 9 ρ的NNQMS神经网络表示

Fig. 9. NNQMS neural network representation of the state ρ

下载: 全尺寸图片幻灯片

图 10 $ U\rho U^{\dagger} $的两种神经网络表示

Fig. 10. Two neural network representation of the state $ U\rho U^{\dagger} $.

下载: 全尺寸图片幻灯片

图 11 $ U\rho U^{\dagger} $的两种神经网络表示

Fig. 11. Two neural network representation of the state $ U\rho U^{\dagger} $.

下载: 全尺寸图片幻灯片

图 12 $ U\rho U^{\dagger} $ 的两种神经网络表示

Fig. 12. Two neural network representation of the state $ U\rho U^{\dagger} $.

下载: 全尺寸图片幻灯片

图 13 $ U\rho U^{\dagger} $ 的两种神经网络表示

Fig. 13. Two neural network representation of the state $ U\rho U^{\dagger} $.

下载: 全尺寸图片幻灯片

图 14 $ U\rho U^{\dagger} $的两种神经网络表示

Fig. 14. Two neural network representation of the state $ U\rho U^{\dagger} $.

下载: 全尺寸图片幻灯片

图 15 $ U\rho U^{\dagger} $的两种神经网络表示

Fig. 15. Two neural network representation of the state $ U\rho U^{\dagger} $.

下载: 全尺寸图片幻灯片

图 16 $ U\rho U^{\dagger} $的两种神经网络表示

Fig. 16. Two neural network representation of the state $ U\rho U^{\dagger} $.

下载: 全尺寸图片幻灯片

图 17 $ U\rho U^{\dagger} $的两种神经网络表示

Fig. 17. Two neural network representation of the state $ U\rho U^{\dagger} $.

下载: 全尺寸图片幻灯片

[1]	LeCun Y, Bengio Y, Hinton G 2015 Nature 521 436 Google Scholar
[2]	Biamonte J, Wittek P, Pancotti N, Rebentrost P, Wiebe N, Lloyd S 2017 Nature 549 195 Google Scholar
[3]	Ciliberto C, Herbster M, Ialongo A D, Pontil M, Rocchetto A, Severini S, Wossnig L 2018 Proc. R. Soc. A 474 20170551 Google Scholar
[4]	Schollwöck U 2011 Ann. Phys. 326 96 Google Scholar
[5]	Verstraete F, Murg V, Cirac J I 2008 Adv. Phys. 57 143 Google Scholar
[6]	Schuch N, Wolf M M, Verstraete F, Cirac J I 2008 Phys. Rev. Lett. 100 040501 Google Scholar
[7]	Ceperley D, Alder B 1986 Science 231 555 Google Scholar
[8]	Loh E Y, Gubernatis J E, Scalettar R T, White S R, Scalapino D J, Sugar R L 1990 Phys. Rev. B 41 9301
[9]	Schuch N, Wolf M M, Verstraete F, Cirac J I 2007 Phys. Rev. Lett. 98 140506 Google Scholar
[10]	Verstraete F, Wolf M M, Perez-Garcia D, Cirac J I 2006 Phys. Rev. Lett. 96 220601 Google Scholar
[11]	Carleo G, Troyer M 2017 Science 355 602 Google Scholar
[12]	程嵩, 陈靖, 王磊 2017 物理 46 416 Google Scholar Cheng S, Chen J, Wang L 2017 Physics 46 416 Google Scholar
[13]	蔡子 2017 物理 46 590 Google Scholar Cai Z 2017 Physics 46 590 Google Scholar
[14]	Ma Y C, Yung M H 2018 Npj Quantum Inform. 4 34 Google Scholar
[15]	Gao J, Qiao L F, Jiao Z Q, Ma Y C, Hu C Q, Ren R J, Yang A L, Tang H, Yung M H, Jin X M 2018 Phys. Rev. Lett. 120 240501 Google Scholar
[16]	Qiu P H, Chen X G, Shi Y W 2019 IEEE Access 7 94310 Google Scholar
[17]	Deng D L, Li X P, Sarma S D 2017 Phys. Rev. B 96 195145 Google Scholar
[18]	Deng D L, Li X P, Sarma S D 2017 Phys. Rev. X 7 021021
[19]	Deng D L 2018 Phys. Rev. Lett. 120 240402 Google Scholar
[20]	Gao X, Duan L M 2017 Nat. Commun. 8 662 Google Scholar
[21]	Lu S, Gao X, Duan L M 2019 Phys. Rev. B 99 155136
[22]	Gao X, Zhang Z Y, Duan L M 2018 Sci. Adv. 4 eaat9004 Google Scholar
[23]	Carleo G, Nomura Y, Imada M 2018 Nat. Commun. 9 5322 Google Scholar
[24]	Glasser I, Pancotti N, August M, Rodriguez I D, Cirac J I 2018 Phys. Rev. X 8 011006
[25]	Jia Z A, Yi B, Zhai R, Wu Y C, Guo G C, Guo G P 2019 Adv. Quantum. Technol. 2 1800077 Google Scholar
[26]	Hu L, Wu S H, Cai W Z, Ma Y W, Mu X J, Xu Y, Wang H Y, Song Y P, Deng D L, Zou C L, Sun L Y 2019 Sci. Adv. 5 2761 Google Scholar
[27]	Yang Y, Cao H X, Zhang Z J 2020 Sci. China-Phys. Mech. Astron. 63 210312 Google Scholar
[28]	Sarma S D, Deng D L, Duan L M 2019 Phys. Today 72 48
[29]	Carleo G, Cirac I, Cranmer K, Daudet L, Schuld M, Tishby N, Maranto L V, Zdeborová L 2019 Rev. Mod. Phys. 91 045002 Google Scholar
[30]	Lu S R, Duan L M, Deng D L 2020 Phys. Rev. Res. 2 033212 Google Scholar
[31]	杨光, 刘琦, 聂敏, 刘原华, 张美玲 2022 物理学报 71 100301 Google Scholar Yang G, Liu Q, Nie M, Liu Y H, Zhang M L 2022 Acta Phys. Sin. 71 100301 Google Scholar
[32]	陈以鹏, 刘靖阳, 朱佳莉, 方伟, 王琴 2022 物理学报 71 220301 Google Scholar Chen Y P, Liu J Y, Zhu J L, Fang W, Wang Q 2022 Acta Phys. Sin. 71 220301 Google Scholar
[33]	Li W K, Lu S, Deng D L 2021 Sci. China-Phys. Mech. Astron. 64 100312 Google Scholar
[34]	Li W K, Deng D L 2022 Sci. China-Phys. Mech. Astron. 65 220301 Google Scholar
[35]	Torlai G, Melko R G 2018 Phys. Rev. Lett. 120 240503 Google Scholar
[36]	Hartmann M J, Carleo G 2019 Phys. Rev. Lett. 122 250502 Google Scholar
[37]	Vicentini F, Biella A, Regnault N, Ciuti C 2019 Phys. Rev. Lett. 122 250503 Google Scholar
[38]	Nagy A, Savona V 2019 Phys. Rev. Lett. 122 250501
[39]	Yoshioka N, Hamazaki R 2019 Phys. Rev. B 99 214306 Google Scholar

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量子混合态的两种神经网络表示