Digraph states and their neural network representations

Ying Yang(杨莹)^{1} and Huaixin Cao(曹怀信)^{2,†}

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

Abstract With the rapid development of machine learning, artificial neural networks provide a powerful tool to represent or approximate many-body quantum states. It was proved that every graph state can be generated by a neural network. Here, we introduce digraph states and explore their neural network representations (NNRs). Based on some discussions about digraph states and neural network quantum states (NNQSs), we construct explicitly an NNR for any digraph state, implying every digraph state is an NNQS. The obtained results will provide a theoretical foundation for solving the quantum many-body problem with machine learning method whenever the wave-function is known as an unknown digraph state or it can be approximated by digraph states.

Fund: This work was supported by the National Natural Science Foundation of China (Grant Nos. 12001480 and 11871318), the Applied Basic Research Program of Shanxi Province (Grant Nos. 201901D211461 and 201901D211462), the Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2020L0554), the Excellent Doctoral Research Project of Shanxi Province (Grant No. QZX-2020001), and the PhD Start-up Project of Yuncheng University (Grant No. YQ-2019021).

Corresponding Authors:
Huaixin Cao
E-mail: caohx@snnu.edu.cn

Cite this article:

Ying Yang(杨莹) and Huaixin Cao(曹怀信) Digraph states and their neural network representations 2022 Chin. Phys. B 31 060303

[1] Li Y, Ye X J and Chen J L 2016 Chin. Phys. Lett.33 080301 [2] Gao J and Zhang M C 2016 Chin. Phys. Lett.33 010303 [3] Du L, Wang Y L, Liang G H, Kang G Z and Zong H S 2016 Chin. Phys. Lett.33 030301 [4] Sun J and Lu S F 2020 Chin. Phys. B29 100303 [5] Klein A and Zemach C 1957 Phys. Rev.108 126 [6] Schweigler T, Kasper V, Erne S, Mazets I, Rauer B, Cataldini F, Langen T, Gasenzer T, Berges J and Schmiedmayer J 2017 Nature545 323 [7] Hodgman S S, Khakimov R I, Lewis-Swan R J, Truscott A G and Kheruntsyan K V 2017 Phys. Rev. Lett.118 240402 [8] Schollwöck U 2011 Ann. Phys.326 96 [9] Kolmogorov A N 1957 Dokl. Akad. Nauk SSSR114 953 [10] Schuch N, Wolf M M, Verstraete F and Cirac J I 2008 Phys. Rev. Lett.100 040501 [11] Ceperley D and Alder B 1986 Science231 555 [12] Schuch N, Wolf M M, Verstraete F and Cirac J I 2007 Phys. Rev. Lett.98 140506 [13] Verstraete F, Wolf M M, Garcia D P and Cirac J I 2006 Phys. Rev. Lett.96 220601 [14] Loh E Y, Gubernatis J E, Scalettar R T, White S R, Scalapino D J and Sugar R L 1990 Phys. Rev. B41 9301 [15] Cybenko G 1989 Math. Control Signal Syst.2 303 [16] Funahashi K 1989 Neural Networks2 183 [17] Hornik K, Stinchcombe M and White H 1989 Neural Networks2 359 [18] Hornik K 1991 Neural Networks4 251 [19] Kolmogorov A N 1957 Dokl. Akad. Nauk SSSR114 953 [20] Roux N L and Bengio Y 2008 Neural Comput.20 1631 [21] LeCun Y, Bengio Y and Hinton G 2015 Nature521 436 [22] Cao L Z, Wang P J, Sai L W, Fu J and Duan X M 2020 Chin. Phys. B29 117304 [23] Hinton G E and Salakhutdinov R R 2006 Science313 504 [24] Salakhutdinov R R, Mnih A and Hinton G 2007 ICML24 791 [25] Larochelle H and Bengio Y 2008 ICML25 536 [26] Amin M H, Andriyash E, Rolfe J, Kulchytskyy B and Melko R 2016 Phys. Rev. X8 021050 [27] Lu S, Gao X and Duan L M 2019 Phys. Rev. B99 155136 [28] Sarma S D, Deng D L and Duan L M 2019 Phys. Today72 48 [29] Carleo G, Cirac I, Cranmer K, Daudet L, Schuld M, Tishby N, Maranto L V and Zdeborová L 2019 Rev. Mod. Phys.91 045002 [30] Carleo G and Troyer M 2017 Science355 602 [31] Chen J, Cheng S, Xie H D, Wang L and Xiang T 2018 Phys. Rev. B97 085104 [32] Robeva E and Seigal A 2019 Inf. Inference8 273 [33] Clark S R 2018 J. Phys. A: Math. Theor.51 135301 [34] Huang Y C and Moore J E 2021 Phys. Rev. Lett.127 170601 [35] Lei S J, Bai D, Ren Z Z and Lyu M J 2021 Chin. Phys. Lett.38 051101 [36] Yin Q, Xiang G Y, Li C F and Guo G C 2017 Chin. Phys. Lett.34 030301 [37] Yang Y, Zhang C Y and Cao H X 2019 Entropy21 82 [38] Nomura Y, Darmawan A S, Yamaji Y and Imada M 2017 Phys. Rev. B96 205152 [39] Deng D L, Li X P and Sarma S D 2017 Phys. Rev. B96 195145 [40] Rocchetto A, Grant E, Strelchuk S, Carleo G and Severini S 2018 npj Quantum Inf.4 28 [41] Glasser I, Pancotti N, August M, Rodriguez I D and Cirac J I 2017 Phys. Rev. X8 011006 [42] Kaubruegger R, Pastori L and Budich J C 2018 Phys. Rev. B97 195136 [43] Cai Z 2018 Phys. Rev. B97 035116 [44] Saito H and Kato M 2017 J. Phys. Soc. Jpn.87 014001 [45] Jia Z A, Yi B, Zhai R, Wu Y C, Guo G C and Guo G P 2019 Adv. Quantum Technol.2 1800077 [46] Rao W J 2020 Chin. Phys. Lett.37 080501 [47] Yang Y, Cao H X and Zhang Z J 2020 Sci. China-Phys. Mech. Astron.63 210312 [48] Raussendorf R and Briegel H J 2001 Phys. Rev. Lett.86 5188 [49] Schlingemann D and Werner R F 2002 Phys. Rev. A65 012308 [50] Mandel O, Greiner M, Widera A, Rom T, Haensch T W and Bloch I 2003 Nature425 937 [51] Walther P, Resch J K, Rudolph T, Schenck E and Weinfurter H 2005 Nature434 169 [52] Gao X and Duan L M 2017 Nat. Commun.8 662 [53] Kottos T and Smilansky U 1997 Phys. Rev. Lett.79 4794 [54] Kottos T and Smilansky U 1999 Ann. Phys. N.Y.274 76 [55] Tanner G 2000 J. Phys. A: Math. Gen.33 3567 [56] Tanner G 2001 J. Phys. A: Math. Gen.34 8485 [57] Pakoński P, Życzkowski K and Kuś M 2001 J. Phys. A: Math. Gen.34 9303 [58] Chang C T and Hwang H C 1992 Ind. Eng. Chem. Res.31 1490 [59] Palmer C and Chung P W H 2000 Ind. Eng. Chem. Res.39 2548 [60] Bhushan M and Rengaswamy R 2000 Ind. Eng. Chem. Res.39 999 [61] Deng D L, Li X P and Sarma S D 2017 Phys. Rev. B96 195145 [62] Kay E 1977 J. Oper. Res. Soc.28 237

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