Perturbation theory with quantum signal processing

Kosuke Mitarai1,2, Kiichiro Toyoizumi3, and Wataru Mizukami2

1Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.
2Center for Quantum Information and Quantum Biology, Osaka University, Japan.
3Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan.

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Perturbation theory is an important technique for reducing computational cost and providing physical insights in simulating quantum systems with classical computers. Here, we provide a quantum algorithm to obtain perturbative energies on quantum computers. The benefit of using quantum computers is that we can start the perturbation from a Hamiltonian that is classically hard to solve. The proposed algorithm uses quantum signal processing (QSP) to achieve this goal. Along with the perturbation theory, we construct a technique for ground state preparation with detailed computational cost analysis, which can be of independent interest. We also estimate a rough computational cost of the algorithm for simple chemical systems such as water clusters and polyacene molecules. To the best of our knowledge, this is the first of such estimates for practical applications of QSP. Unfortunately, we find that the proposed algorithm, at least in its current form, does not exhibit practical numbers despite of the efficiency of QSP compared to conventional quantum algorithms. However, perturbation theory itself is an attractive direction to explore because of its physical interpretability; it provides us insights about what interaction gives an important contribution to the properties of systems. This is in sharp contrast to the conventional approaches based on the quantum phase estimation algorithm, where we can only obtain values of energy. From this aspect, this work is a first step towards “explainable'' quantum simulation on fault-tolerant quantum computers.

► BibTeX data

► References

[1] G. H. Low and I. L. Chuang, Phys. Rev. Lett. 118, 010501 (2017a).

[2] J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, PRX Quantum 2, 040203 (2021).

[3] A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 (Association for Computing Machinery, New York, NY, USA, 2019) p. 193–204.

[4] T. J. Yoder, G. H. Low, and I. L. Chuang, Phys. Rev. Lett. 113, 210501 (2014).

[5] G. Wang, D. E. Koh, P. D. Johnson, and Y. Cao, PRX Quantum 2, 010346 (2021).

[6] P. D. Johnson, A. A. Kunitsa, J. F. Gonthier, M. D. Radin, C. Buda, E. J. Doskocil, C. M. Abuan, and J. Romero, ``Reducing the cost of energy estimation in the variational quantum eigensolver algorithm with robust amplitude estimation'', arXiv:2203.07275 (2022).

[7] A. Katabarwa, A. Kunitsa, B. Peropadre, and P. Johnson, ``Reducing runtime and error in VQE using deeper and noisier quantum circuits'', arXiv:2110.10664 (2021).

[8] T. Helgaker, P. Jørgensen, and J. Olsen, Perturbation theory, in Molecular Electronic‐Structure Theory (John Wiley & Sons, Ltd, 2000) Chap. 14, pp. 724–816.

[9] S. Chakraborty, A. Gilyén, and S. Jeffery, in 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 132, edited by C. Baier, I. Chatzigiannakis, P. Flocchini, and S. Leonardi (Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2019) pp. 33:1–33:14.

[10] R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Phys. Rev. X 8, 041015 (2018).

[11] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2010).

[12] G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, ``Quantum Amplitude Amplification and Estimation'', arXiv:0005055 (2000).

[13] D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, npj Quantum Information 7, 52 (2021).

[14] P. Rall and B. Fuller, Quantum 7, 937 (2023).

[15] L. Lin and Y. Tong, Quantum 4, 372 (2020).

[16] L. Lin and Y. Tong, PRX Quantum 3, 010318 (2022).

[17] Y. Ge, J. Tura, and J. I. Cirac, Journal of Mathematical Physics 60, 022202 (2019).

[18] T. E. O'Brien, M. Streif, N. C. Rubin, R. Santagati, Y. Su, W. J. Huggins, J. J. Goings, N. Moll, E. Kyoseva, M. Degroote, C. S. Tautermann, J. Lee, D. W. Berry, N. Wiebe, and R. Babbush, Phys. Rev. Res. 4, 043210 (2022).

[19] Y. Tong, D. An, N. Wiebe, and L. Lin, Phys. Rev. A 104, 032422 (2021).

[20] J. Sun, S. Endo, H. Lin, P. Hayden, V. Vedral, and X. Yuan, Phys. Rev. Lett. 129, 120505 (2022).

[21] G. H. Low and I. L. Chuang, ``Hamiltonian Simulation by Uniform Spectral Amplification'', arXiv: 1707.05391 (2017b).

[22] A. M. Childs, R. Kothari, and R. D. Somma, SIAM Journal on Computing 46, 1920 (2017).

[23] D. J. Wales and M. P. Hodges, Chemical Physics Letters 286, 65 (1998).

[24] Q. Sun, T. C. Berkelbach, N. S. Blunt, G. H. Booth, S. Guo, Z. Li, J. Liu, J. D. McClain, E. R. Sayfutyarova, S. Sharma, S. Wouters, and G. K.-L. Chan, WIREs Computational Molecular Science 8, e1340 (2018).

[25] Q. Sun, X. Zhang, S. Banerjee, P. Bao, M. Barbry, N. S. Blunt, N. A. Bogdanov, G. H. Booth, J. Chen, Z.-H. Cui, J. J. Eriksen, Y. Gao, S. Guo, J. Hermann, M. R. Hermes, K. Koh, P. Koval, S. Lehtola, Z. Li, J. Liu, N. Mardirossian, J. D. McClain, M. Motta, B. Mussard, H. Q. Pham, A. Pulkin, W. Purwanto, P. J. Robinson, E. Ronca, E. R. Sayfutyarova, M. Scheurer, H. F. Schurkus, J. E. T. Smith, C. Sun, S.-N. Sun, S. Upadhyay, L. K. Wagner, X. Wang, A. White, J. D. Whitfield, M. J. Williamson, S. Wouters, J. Yang, J. M. Yu, T. Zhu, T. C. Berkelbach, S. Sharma, A. Y. Sokolov, and G. K.-L. Chan, The Journal of Chemical Physics 153, 024109 (2020).

[26] J. R. McClean, N. C. Rubin, K. J. Sung, I. D. Kivlichan, X. Bonet-Monroig, Y. Cao, C. Dai, E. S. Fried, C. Gidney, B. Gimby, P. Gokhale, T. Häner, T. Hardikar, V. Havlíček, O. Higgott, C. Huang, J. Izaac, Z. Jiang, X. Liu, S. McArdle, M. Neeley, T. O'Brien, B. O'Gorman, I. Ozfidan, M. D. Radin, J. Romero, N. P. D. Sawaya, B. Senjean, K. Setia, S. Sim, D. S. Steiger, M. Steudtner, Q. Sun, W. Sun, D. Wang, F. Zhang, and R. Babbush, Quantum Science and Technology 5, 034014 (2020).

[27] P.-O. Löwdin, The Journal of Chemical Physics 18, 365 (1950).

[28] P. Jordan and E. Wigner, Zeitschrift für Physik 47, 631 (1928).

[29] J. Hachmann, J. J. Dorando, M. Avilés, and G. K.-L. Chan, The Journal of Chemical Physics 127, 134309 (2007).

[30] E. R. Sayfutyarova and S. Hammes-Schiffer, Journal of Chemical Theory and Computation 15, 1679 (2019).

[31] I. Fdez. Galván, M. Vacher, A. Alavi, C. Angeli, F. Aquilante, J. Autschbach, J. J. Bao, S. I. Bokarev, N. A. Bogdanov, R. K. Carlson, L. F. Chibotaru, J. Creutzberg, N. Dattani, M. G. Delcey, S. S. Dong, A. Dreuw, L. Freitag, L. M. Frutos, L. Gagliardi, F. Gendron, A. Giussani, L. González, G. Grell, M. Guo, C. E. Hoyer, M. Johansson, S. Keller, S. Knecht, G. Kovačević, E. Källman, G. Li Manni, M. Lundberg, Y. Ma, S. Mai, J. a. P. Malhado, P. Å. Malmqvist, P. Marquetand, S. A. Mewes, J. Norell, M. Olivucci, M. Oppel, Q. M. Phung, K. Pierloot, F. Plasser, M. Reiher, A. M. Sand, I. Schapiro, P. Sharma, C. J. Stein, L. K. Sørensen, D. G. Truhlar, M. Ugandi, L. Ungur, A. Valentini, S. Vancoillie, V. Veryazov, O. Weser, T. A. Wesołowski, P.-O. Widmark, S. Wouters, A. Zech, J. P. Zobel, and R. Lindh, Journal of Chemical Theory and Computation 15, 5925 (2019).

[32] F. Aquilante, J. Autschbach, A. Baiardi, S. Battaglia, V. A. Borin, L. F. Chibotaru, I. Conti, L. De Vico, M. Delcey, I. Fdez. Galván, N. Ferré, L. Freitag, M. Garavelli, X. Gong, S. Knecht, E. D. Larsson, R. Lindh, M. Lundberg, P. Å. Malmqvist, A. Nenov, J. Norell, M. Odelius, M. Olivucci, T. B. Pedersen, L. Pedraza-González, Q. M. Phung, K. Pierloot, M. Reiher, I. Schapiro, J. Segarra-Martí, F. Segatta, L. Seijo, S. Sen, D.-C. Sergentu, C. J. Stein, L. Ungur, M. Vacher, A. Valentini, and V. Veryazov, The Journal of Chemical Physics 152, 214117 (2020).

[33] D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, Quantum 3, 208 (2019).

[34] J. F. Gonthier, M. D. Radin, C. Buda, E. J. Doskocil, C. M. Abuan, and J. Romero, Phys. Rev. Research 4, 033154 (2022).

[35] Y. Dong, X. Meng, K. B. Whaley, and L. Lin, Phys. Rev. A 103, 042419 (2021).

[36] E. Koridon, S. Yalouz, B. Senjean, F. Buda, T. E. O'Brien, and L. Visscher, Phys. Rev. Research 3, 033127 (2021).

Cited by

[1] Raffaele Santagati, Alan Aspuru-Guzik, Ryan Babbush, Matthias Degroote, Leticia Gonzalez, Elica Kyoseva, Nikolaj Moll, Markus Oppel, Robert M. Parrish, Nicholas C. Rubin, Michael Streif, Christofer S. Tautermann, Horst Weiss, Nathan Wiebe, and Clemens Utschig-Utschig, "Drug design on quantum computers", arXiv:2301.04114, (2023).

[2] Mark Steudtner, Sam Morley-Short, William Pol, Sukin Sim, Cristian L. Cortes, Matthias Loipersberger, Robert M. Parrish, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Fault-tolerant quantum computation of molecular observables", arXiv:2303.14118, (2023).

[3] Kiichiro Toyoizumi, Naoki Yamamoto, and Kazuo Hoshino, "Hamiltonian simulation using quantum singular value transformation: complexity analysis and application to the linearized Vlasov-Poisson equation", arXiv:2304.08937, (2023).

[4] Wataru Inoue, Koki Aoyama, Yusuke Teranishi, Keita Kanno, Yuya O. Nakagawa, and Kosuke Mitarai, "Almost optimal measurement scheduling of molecular Hamiltonian via finite projective plane", arXiv:2301.07335, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-06-09 00:26:04). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2023-06-09 00:26:03).