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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.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 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
$\varOmega = (0, b, W)$ .
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 | $ .
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 | $ .
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
$p_1 |0\rangle\langle 0 |+p_2 |1\rangle\langle 1 |(p_1, p_2 > 0,\; p_1+p_2 = 1)$ .
图 7
$\rho$ 的反对偶的NNQMVS神经网络表示 (a) N = 2; (b) N = 3; (c) N = 4Fig. 7. Anti-dual NNQMVS neural network representation of the state
$ \rho$ : (a) N = 2; (b) N = 3; (c) N = 4.
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} $ .
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} $ .
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} $ .
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} $ .
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} $ .
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} $ .
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} $ .
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} $ . -
[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
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Google Scholar
[4] Schollwöck U 2011 Ann. Phys. 326 96
Google Scholar
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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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Google Scholar
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Google Scholar
Cheng S, Chen J, Wang L 2017 Physics 46 416
Google Scholar
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Google Scholar
Cai Z 2017 Physics 46 590
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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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
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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
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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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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
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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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