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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.
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Keywords:
- neural network /
- quantum mixed state /
- local unitary operation /
- tensor product
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图 1 NNQMVS相匹配的人工神经网络
Figure 1. Artificial neural network encoding an NNQMVS.
图 2 含参数
$\varOmega = (0, b, W)$ 的NNQMVS对应的量子人工神经网络Figure 2. Quantum artificial neural network of NNQMVS with parameter
$\varOmega = (0, b, W)$ .
图 3 NNQMS相匹配的深度神经网络
Figure 3. Deep neural network encoding an NNQMS.
图 4
$ |0\rangle\langle 0 | $ 的两种神经网络表示Figure 4. Two neural network representation of the state
$ |0\rangle\langle 0 | $ .
图 5
$ |1\rangle\langle 1 | $ 的两种神经网络表示Figure 5. Two neural network representation of the state
$ |1\rangle\langle 1 | $ .
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Figure 6. Two neural network representation of the state
$p_1 |0\rangle\langle 0 |+p_2 |1\rangle\langle 1 |(p_1, p_2 > 0,\; p_1+p_2 = 1)$ .
图 7
$\rho$ 的反对偶的NNQMVS神经网络表示 (a) N = 2; (b) N = 3; (c) N = 4Figure 7. Anti-dual NNQMVS neural network representation of the state
$ \rho$ : (a) N = 2; (b) N = 3; (c) N = 4.
图 8 ρ的NNQMVS 神经网络表示
Figure 8. NNQMVS neural network representation of the state ρ.
图 9 ρ的NNQMS神经网络表示
Figure 9. NNQMS neural network representation of the state ρ
图 10
$ U\rho U^{\dagger} $ 的两种神经网络表示Figure 10. Two neural network representation of the state
$ U\rho U^{\dagger} $ .
图 11
$ U\rho U^{\dagger} $ 的两种神经网络表示Figure 11. Two neural network representation of the state
$ U\rho U^{\dagger} $ .
图 12
$ U\rho U^{\dagger} $ 的两种神经网络表示Figure 12. Two neural network representation of the state
$ U\rho U^{\dagger} $ .
图 13
$ U\rho U^{\dagger} $ 的两种神经网络表示Figure 13. Two neural network representation of the state
$ U\rho U^{\dagger} $ .
图 14
$ U\rho U^{\dagger} $ 的两种神经网络表示Figure 14. Two neural network representation of the state
$ U\rho U^{\dagger} $ .
图 15
$ U\rho U^{\dagger} $ 的两种神经网络表示Figure 15. Two neural network representation of the state
$ U\rho U^{\dagger} $ .
图 16
$ U\rho U^{\dagger} $ 的两种神经网络表示Figure 16. Two neural network representation of the state
$ U\rho U^{\dagger} $ .
图 17
$ U\rho U^{\dagger} $ 的两种神经网络表示Figure 17. Two neural network representation of the state
$ U\rho U^{\dagger} $ . -
[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 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 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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