Robust measurement of wave function topology on NISQ quantum computers

Xiao Xiao; J. K. Freericks; A. F. Kemper

doi:10.22331/q-2023-04-27-987

Robust measurement of wave function topology on NISQ quantum computers

Xiao Xiao¹, J. K. Freericks², and A. F. Kemper¹

¹Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA
²Department of Physics, Georgetown University, 37th and O Sts. NW, Washington, DC 20057 USA

Published:	2023-04-27, volume 7, page 987
Eprint:	arXiv:2101.07283v6
Doi:	https://doi.org/10.22331/q-2023-04-27-987
Citation:	Quantum 7, 987 (2023).

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Abstract

Topological quantum phases of quantum materials are defined through their topological invariants. These topological invariants are quantities that characterize the global geometrical properties of the quantum wave functions and thus are immune to local noise. Here, we present a strategy to measure topological invariants on quantum computers. We show that our strategy can be easily integrated with the variational quantum eigensolver (VQE) so that the topological properties of generic quantum many-body states can be characterized on current quantum hardware. We demonstrate the robust nature of the method by measuring topological invariants for both non-interacting and interacting models, and map out interacting quantum phase diagrams on quantum simulators and IBM quantum hardware.

Featured image: A generic strategy allows accurate calculations of Chern numbers on IBM's quantum computers.

Popular summary

Calculations on current noisy intermediate-scale quantum (NISQ) hardware are susceptible to operation errors during the implementations of quantum circuits. Therefore, it is challenging to identify useful applications on which the present quantum hardware can be used. Here we provide a systematical strategy, which can be easily integrated with variational quantum algorithms, to characterize a key property of quantum matter: the wave function. Strikingly, we demonstrated that one important quantity that characterizes the wave function topology, the Chern number, can be calculated accurately on present quantum hardware without any error mitigation or error correction. Our work is an appealing demonstration that NISQ hardware might be a good platform to study quantum topological states and topological phenomena.

► BibTeX data

@article{Xiao2023robustmeasurementof,
  doi = {10.22331/q-2023-04-27-987},
  url = {https://doi.org/10.22331/q-2023-04-27-987},
  title = {Robust measurement of wave function topology on {NISQ} quantum computers},
  author = {Xiao, Xiao and Freericks, J. K. and Kemper, A. F.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {987},
  month = apr,
  year = {2023}
}

► References

[1] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
https://doi.org/10.1103/PhysRevLett.49.405

[2] Q. Niu, D. J. Thouless, and Y.-S. Wu, Phys. Rev. B 31, 3372 (1985).
https://doi.org/10.1103/PhysRevB.31.3372

[3] D. N. Sheng, L. Sheng, and Z. Y. Weng, Phys. Rev. B 73, 233406 (2006).
https://doi.org/10.1103/PhysRevB.73.233406

[4] H. Obuse, A. Furusaki, S. Ryu, and C. Mudry, Phys. Rev. B 76, 075301 (2007).
https://doi.org/10.1103/PhysRevB.76.075301

[5] J. Li, R.-L. Chu, J. K. Jain, and S.-Q. Shen, Phys. Rev. Lett. 102, 136806 (2009).
https://doi.org/10.1103/PhysRevLett.102.136806

[6] E. Prodan, J. Phys. A: Math. Theor. 44, 113001 (2011).
https://doi.org/10.1088/1751-8113/44/11/113001

[7] J. T. Chalker, M. Ortuno, and A. M. Somoza, Phys. Rev. B 83, 115317 (2011).
https://doi.org/10.1103/PhysRevB.83.115317

[8] J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Phys. Rev. Lett. 109, 267002 (2012).
https://doi.org/10.1103/PhysRevLett.109.267002

[9] A. M. Lobos, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. Lett. 109, 146403 (2012).
https://doi.org/10.1103/PhysRevLett.109.146403

[10] E. J. Konig, P. M. Ostrovsky, I. V. Protopopov, I. V. Gornyi, I. S. Burmistrov, and A. D. Mirlin Phys. Rev. B 88, 035106 (2013).
https://doi.org/10.1103/PhysRevB.88.035106

[11] A. Altland, D. Bagrets, L. Fritz, A. Kamenev, and H. Schmiedt, Phys. Rev. Lett. 112, 206602 (2014).
https://doi.org/10.1103/PhysRevLett.112.206602

[12] I. Mondragon-Shem, T. L. Hughes, J. Song, and E. Prodan, Phys. Rev. Lett. 113, 046802 (2014).
https://doi.org/10.1103/PhysRevLett.113.046802

[13] J. Song, and E. Prodan, Phys. Rev. B 89, 224203 (2014).
https://doi.org/10.1103/PhysRevB.89.224203

[14] M. S. Foster, H.-Y. Xie, and Y.-Z. Chou, Phys. Rev. B 89, 155140 (2014).
https://doi.org/10.1103/PhysRevB.89.155140

[15] J. Wang, B. Lian, and S.-C. Zhang, Phys. Rev. B 89, 085106 (2014).
https://doi.org/10.1103/PhysRevB.89.085106

[16] C. Liu, W. Gao, B. Yang, and S. Zhang, Phys. Rev. Lett. 119, 183901 (2017).
https://doi.org/10.1103/PhysRevLett.119.183901

[17] E. J. Meier, F. A. An, A. Dauphin, M. Maffei, P. Massignan, T. L. Hughes, B. Gadway, Science 362, 929 (2018).
https://doi.org/10.1126/science.aat3406

[18] S. Stutzer, Y. Plotnik, Y. Lumer, P. Titum, N. Linder, M. Segev, M. C. Rechtsman, and A. Szameit, Nature 560, 461 (2018).
https://doi.org/10.1038/s41586-018-0418-2

[19] X. Xiao, arXiv:1802.02687 (2018).
https://doi.org/10.48550/arXiv.1802.02687
arXiv:1802.02687

[20] O. Shtanko, and R. Movassagh, Phys. Rev. Lett. 121, 126803 (2018).
https://doi.org/10.1103/PhysRevLett.121.126803

[21] T. Okugawa, P. Tang, A. Rubio, and D. M. Kennes, Phys. Rev. B 102, 201405(R) (2020).
https://doi.org/10.1103/PhysRevB.102.201405

[22] P. Roushan, C. Neill, Yu Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland and J. M. Martinis Nature 515, 241 (2014).
https://doi.org/10.1038/nature13891

[23] K. Choo, C. W. von Keyserlingk, N. Regnault, and T. Neupert, Phys. Rev. Lett. 121, 086808 (2018).
https://doi.org/10.1103/PhysRevLett.121.086808

[24] A. Smith, B. Jobst, A. G. Green, and F. Pollmann, Phys. Rev. Research 4, L022020 (2022).
https://doi.org/10.1103/PhysRevResearch.4.L022020

[25] D. Azses, R. Haenel, Y. Naveh, R. Raussendorf, E. Sela, and E. G. DallaTorre, Phys. Rev. Lett. 125, 120502 (2020).
https://doi.org/10.1103/PhysRevLett.125.120502

[26] F. Mei, Q. Guo, Y.-F. Yu, L. Xiao, S.-L. Zhu, and S. Jia, Phys. Rev. Lett. 125, 160503 (2020).
https://doi.org/10.1103/PhysRevLett.125.160503

[27] X. Xiao, J. K. Freericks, and A. F. Kemper, Quantum 5, 553 (2021).
https://doi.org/10.22331/q-2021-09-28-553

[28] E. Flurin, V. V. Ramasesh, S. Hacohen-Gourgy, L. S. Martin, N. Y. Yao, and I. Siddiqi, Phys. Rev. X 7, 031023 (2017).
https://doi.org/10.1103/PhysRevX.7.031023

[29] X. Zhan, L. Xiao, Z. Bian, K. Wang, X. Qiu, B. C. Sanders, W. Yi, and P. Xue, Phys. Rev. Lett. 119, 130501 (2017).
https://doi.org/10.1103/PhysRevLett.119.130501

[30] X.-Y. Xu, Q.-Q. Wang, W.-W. Pan, K. Sun, J.-S. Xu, G. Chen, J.-S. Tang, M. Gong, Y.-J. Han, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 120, 260501 (2018).
https://doi.org/10.1103/PhysRevLett.120.260501

[31] A. Elben, J. Yu, G. Zhu, M. Hafezi, F. Pollmann, P. Zoller, and B. Vermersch, Sci. Adv. 6, eaaz3666 (2020).
https://doi.org/10.1126/sciadv.aaz3666

[32] J. Preskill, Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79

[33] A. Kandala, K. Temme, A. D. Corcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Nature 567, 491 (2019).
https://doi.org/10.1038/s41586-019-1040-7

[34] K. E. Hamilton, and R. C. Pooser, Quantum Machine Intelligence 2, 10 (2020).
https://doi.org/10.1007/s42484-020-00021-x

[35] A. Cervera-Lierta, Quantum 2, 114 (2018).
https://doi.org/10.22331/q-2018-12-21-114

[36] K. Yeter-Aydeniz, R. C. Pooser, and G. Siopsis, npj Quantum Inf. 6, 63 (2020).
https://doi.org/10.1038/s41534-020-00290-1

[37] G. E. Volovik, JETP Lett. 70, 609 (1999).
https://doi.org/10.1134/1.568223

[38] N. Read, and D. Green, Phys. Rev. B 61, 10267 (2000).
https://doi.org/10.1103/PhysRevB.61.10267

[39] T. Fukui, Y. Hutsugai, and H. Suzuki, J. Phys. Soc. Jpn. 74, 1674 (2005).
https://doi.org/10.1143/JPSJ.74.1674

[40] P. J. J. O'malley, et al. Phys. Rev. X 6, 031007 (2016).
https://doi.org/10.1103/PhysRevX.6.031007

[41] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Nature 549, 242 (2017).
https://doi.org/10.1038/nature23879

[42] C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, and P. Zoller, Nature 569, 355 (2019).
https://doi.org/10.1038/s41586-019-1177-4

[43] X. Yaun, S. Endo, Q. Zhao, Y. Li and S. C. Benjamin, Quantum 3, 191 (2019).
https://doi.org/10.22331/q-2019-10-07-191

[44] H. R. Grimsley, S. E. Economou, E. Barnes, and N. J. Mayhall Nat. Commun. 10, 3007 (2019).
https://doi.org/10.1038/s41467-019-10988-2

[45] Y. Hatsugai, and M. Kohmoto, J. Phys. Soc. Jpn 61, 2056 (1992).
https://doi.org/10.1143/JPSJ.61.2056

[46] K. Kudo, H. Watanabe, T. Kariyado, and Y. Hatsugai, Phys. Rev. Lett. 122, 146601 (2019).
https://doi.org/10.1103/PhysRevLett.122.146601

[47] P. W. Philips, L. Yeo, and E. W. Huang Nat. Phys. 16, 1175 (2020).
https://doi.org/10.1038/s41567-020-0988-4

[48] X.-L. Qi, Y. S. Wu and S. C. Zhang, Phys. Rev. B 74, 085308 (2006).
https://doi.org/10.1103/PhysRevB.74.085308

[49] H.-Q. Wu, Y.-Y. He, C. Fang, Z. Y. Meng and Z.-Y. Lu, Phys. Rev. Lett. 117, 066403 (2016).
https://doi.org/10.1103/PhysRevLett.117.066403

[50] C.-E. Bardyn, L. Wawer, A. Altland, M. Fleischhauer, and S. Diehl, Phys. Rev. X 8, 011035 (2018).
https://doi.org/10.1103/PhysRevX.8.011035

[51] D. Aharonov, V. Jones, and Z. Landau, Algorithmica 55, 395 (2009).
https://doi.org/10.1007/s00453-008-9168-0

[52] B. Murta, G. Catarina, J. Fernandez-Rossier Phys. Rev. A 101, 020302 (2020).
https://doi.org/10.1103/PhysRevA.101.020302

[53] G. Aleksandrowicz, et al. Qiskit: An open-source framework for quantum computing. (2019).
https://doi.org/10.5281/ZENODO.2562111

[54] E. Knill, Nature 434, 39 (2005).
https://doi.org/10.1038/nature03350

[55] A. Cross, D. P. Divincenzo, and B. M. Terhal, Quantum Info. Comput. 9, 541 (2009).
https://doi.org/10.48550/arXiv.0711.1556

[56] J. Lee, W. J. Huggins, M. Head-Gordon and K. B. Whaley, J. Chem. Theory Comput. 15, 311 (2019).
https://doi.org/10.1021/acs.jctc.8b01004

[57] J. Chen, H.-P. Cheng and J. K. Freericks, J. Chem. Theory Comput. 17, 841 (2021).
https://doi.org/10.1021/acs.jctc.0c01052

[58] Y. Yao, F. Zhang, C.-Z. Wang, K.-M. Ho, and P. P. Orth Phys. Rev. Research 3, 013184 (2021).
https://doi.org/10.1103/PhysRevResearch.3.013184

[59] K. Mitarai and K. Fujii, Phys. Rev. Reseach 1, 013006 (2019).
https://doi.org/10.1103/PhysRevResearch.1.013006

[60] Z.-P. Cian, H. Dehghani, A. Elben, B. Vermersch, G. Zhu, M. Barkeshli, P. Zoller, and M. Hafezi, Phys. Rev. Lett. 126, 050501 (2021).
https://doi.org/10.1103/PhysRevLett.126.050501

[61] I. G. Ryabinkin, T.-C. Yen, S. N. Genin, and A. F. Izmaylov, J. Chem. Theory Comput. 14, 6317 (2018).
https://doi.org/10.1021/acs.jctc.8b00932

[62] M. Motta, C. Sun, A. T. Tan, M. J. O'Rourke, E. Ye, A. J. Minnich, F. G. Brandao, G. K.-L. Chan, Nat. Phys. 16, 205 (2020).
https://doi.org/10.1038/s41567-019-0704-4

[63] D. A. Fedorov, B. Peng, N. Govind, and Y. Alexeev, arXiv: 2103.08505 (2021).
https://doi.org/10.48550/arXiv.2103.08505

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