Perturbation theory with quantum signal processing

Kosuke Mitarai; Kiichiro Toyoizumi; Wataru Mizukami

doi:10.22331/q-2023-05-12-1000

Perturbation theory with quantum signal processing

Kosuke Mitarai^1,2, 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.

Published:	2023-05-12, volume 7, page 1000
Eprint:	arXiv:2210.00718v3
Doi:	https://doi.org/10.22331/q-2023-05-12-1000
Citation:	Quantum 7, 1000 (2023).

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

Abstract

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

@article{Mitarai2023perturbationtheory,
  doi = {10.22331/q-2023-05-12-1000},
  url = {https://doi.org/10.22331/q-2023-05-12-1000},
  title = {Perturbation theory with quantum signal processing},
  author = {Mitarai, Kosuke and Toyoizumi, Kiichiro and Mizukami, Wataru},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1000},
  month = may,
  year = {2023}
}

► References

[1] G. H. Low and I. L. Chuang, Phys. Rev. Lett. 118, 010501 (2017a).
https://doi.org/10.1103/PhysRevLett.118.010501

[2] J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, PRX Quantum 2, 040203 (2021).
https://doi.org/10.1103/PRXQuantum.2.040203

[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.
https://doi.org/10.1145/3313276.3316366

[4] T. J. Yoder, G. H. Low, and I. L. Chuang, Phys. Rev. Lett. 113, 210501 (2014).
https://doi.org/10.1103/PhysRevLett.113.210501

[5] G. Wang, D. E. Koh, P. D. Johnson, and Y. Cao, PRX Quantum 2, 010346 (2021).
https://doi.org/10.1103/PRXQuantum.2.010346

[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).
https://doi.org/10.48550/arXiv.2203.07275
arXiv:2203.07275

[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).
https://doi.org/10.48550/arXiv.2110.10664
arXiv:2110.10664

[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.
https://doi.org/10.1002/9781119019572.ch14

[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.
https://doi.org/10.4230/LIPIcs.ICALP.2019.33

[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).
https://doi.org/10.1103/PhysRevX.8.041015

[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).
https://doi.org/10.1090/conm/305/05215
arXiv:0005055

[13] D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, npj Quantum Information 7, 52 (2021).
https://doi.org/10.1038/s41534-021-00379-1

[14] P. Rall and B. Fuller, Quantum 7, 937 (2023).
https://doi.org/10.22331/q-2023-03-02-937

[15] L. Lin and Y. Tong, Quantum 4, 372 (2020).
https://doi.org/10.22331/q-2020-12-14-372

[16] L. Lin and Y. Tong, PRX Quantum 3, 010318 (2022).
https://doi.org/10.1103/PRXQuantum.3.010318

[17] Y. Ge, J. Tura, and J. I. Cirac, Journal of Mathematical Physics 60, 022202 (2019).
https://doi.org/10.1063/1.5027484

[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).
https://doi.org/10.1103/PhysRevResearch.4.043210

[19] Y. Tong, D. An, N. Wiebe, and L. Lin, Phys. Rev. A 104, 032422 (2021).
https://doi.org/10.1103/PhysRevA.104.032422

[20] J. Sun, S. Endo, H. Lin, P. Hayden, V. Vedral, and X. Yuan, Phys. Rev. Lett. 129, 120505 (2022).
https://doi.org/10.1103/PhysRevLett.129.120505

[21] G. H. Low and I. L. Chuang, ``Hamiltonian Simulation by Uniform Spectral Amplification'', arXiv: 1707.05391 (2017b).
https://doi.org/10.48550/arXiv.1707.05391

[22] A. M. Childs, R. Kothari, and R. D. Somma, SIAM Journal on Computing 46, 1920 (2017).
https://doi.org/10.1137/16M1087072

[23] D. J. Wales and M. P. Hodges, Chemical Physics Letters 286, 65 (1998).
https://doi.org/10.1016/S0009-2614(98)00065-7

[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).
https://doi.org/10.1002/wcms.1340

[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).
https://doi.org/10.1063/5.0006074

[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).
https://doi.org/10.1088/2058-9565/ab8ebc

[27] P.-O. Löwdin, The Journal of Chemical Physics 18, 365 (1950).
https://doi.org/10.1063/1.1747632

[28] P. Jordan and E. Wigner, Zeitschrift für Physik 47, 631 (1928).
https://doi.org/10.1007/BF01331938

[29] J. Hachmann, J. J. Dorando, M. Avilés, and G. K.-L. Chan, The Journal of Chemical Physics 127, 134309 (2007).
https://doi.org/10.1063/1.2768362

[30] E. R. Sayfutyarova and S. Hammes-Schiffer, Journal of Chemical Theory and Computation 15, 1679 (2019).
https://doi.org/10.1021/acs.jctc.8b01196

[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).
https://doi.org/10.1021/acs.jctc.9b00532

[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).
https://doi.org/10.1063/5.0004835

[33] D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, Quantum 3, 208 (2019).
https://doi.org/10.22331/q-2019-12-02-208

[34] J. F. Gonthier, M. D. Radin, C. Buda, E. J. Doskocil, C. M. Abuan, and J. Romero, Phys. Rev. Research 4, 033154 (2022).
https://doi.org/10.1103/PhysRevResearch.4.033154

[35] Y. Dong, X. Meng, K. B. Whaley, and L. Lin, Phys. Rev. A 103, 042419 (2021).
https://doi.org/10.1103/PhysRevA.103.042419

[36] E. Koridon, S. Yalouz, B. Senjean, F. Buda, T. E. O'Brien, and L. Visscher, Phys. Rev. Research 3, 033127 (2021).
https://doi.org/10.1103/PhysRevResearch.3.033127

Cited by

[1] Bartosz Tyrcha, Filip Brzęk, and Piotr S. Żuchowski, "Second quantization-based symmetry-adapted perturbation theory: Generalizing exchange beyond single electron pair approximation", The Journal of Chemical Physics 160 4, 044118 (2024).

[2] Sapthagiri Miriyala and Venkata Ramireddy Chirra, "From Quantum Algorithm Primitives to Design Patterns: A Unified Framework for NISQ and Fault‐Tolerant Regimes", Advanced Quantum Technologies 9 6, e70310 (2026).

[3] Jakob Günther, Freek Witteveen, Alexander Schmidhuber, Marek Miller, Matthias Christandl, and Aram W. Harrow, "Phase Estimation with Partially Randomized Time Evolution", PRX Quantum 7 2, 020332 (2026).

[4] T. M. C. Priyanka and Rui Liu, "Multifractal analysis on the attractors of perturbed iterated function systems", The European Physical Journal Special Topics 234 29, 9531 (2026).

[5] Leonardo Lavagna, Sandra Carillo, and Massimo Panella, "A topical review on time-independent perturbation theory in one-dimensional quantum systems", Physica Scripta 100 10, 102001 (2025).

[6] Viktor Khinevich and Wataru Mizukami, "Enhancing quantum computations with the synergy of auxiliary field quantum Monte Carlo and computational basis tomography", Physical Review Research 7 4, 043344 (2025).

[7] Zhiyao Li, Dorota M. Grabowska, and Martin J. Savage, "Sequency Hierarchy Truncation (SeqHT) for Adiabatic State Preparation and Time Evolution in Quantum Simulations", Quantum 9, 1865 (2025).

[8] Takeru Utsumi and Yoshifumi Nakata, "Explicit decoders using fixed-point amplitude amplification based on QSVT", Quantum 10, 2024 (2026).

[9] Kiichiro Toyoizumi, Naoki Yamamoto, and Kazuo Hoshino, "Hamiltonian simulation using the quantum singular-value transformation: Complexity analysis and application to the linearized Vlasov-Poisson equation", Physical Review A 109 1, 012430 (2024).

[10] Hayata Morisaki, Kosuke Mitarai, Keisuke Fujii, and Yuya O. Nakagawa, "Classical variational optimization of a PREPARE circuit for quantum phase estimation of quantum chemistry Hamiltonians", Physical Review Research 6 4, 043186 (2024).

[11] Jakob Günther, Alberto Baiardi, Markus Reiher, and Matthias Christandl, "More quantum chemistry with fewer qubits", Physical Review Research 6 4, 043021 (2024).

[12] 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", Physical Review Research 6 1, 013096 (2024).

[13] Raffaele Santagati, Alan Aspuru-Guzik, Ryan Babbush, Matthias Degroote, Leticia González, 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", Nature Physics 20 4, 549 (2024).

[14] Davide Castaldo and Markus Reiher, "Utility-Scale Quantum Computational Chemistry", The Journal of Physical Chemistry Letters 17 29, 8140 (2026).

[15] Cristian L. Cortes, Matthias Loipersberger, Robert M. Parrish, Sam Morley-Short, William Pol, Sukin Sim, Mark Steudtner, Christofer S. Tautermann, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Fault-Tolerant Quantum Algorithm for Symmetry-Adapted Perturbation Theory", PRX Quantum 5 1, 010336 (2024).

[16] 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", Quantum 7, 1164 (2023).

[17] Yun-Zhuo Fan and Dan-Bo Zhang, "Full counting statistics of particle distribution on a digital quantum computer", Physical Review A 109 1, 012412 (2024).

[18] Yusuke Ino, Misaki Yonekawa, Hideto Yuzawa, Yuichiro Minato, and Kenji Sugisaki, "Workflow for practical quantum chemical calculations with a quantum phase estimation algorithm: electronic ground and π–π* excited states of benzene and its derivatives", Physical Chemistry Chemical Physics 26 48, 30044 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 14:19:56) and SAO/NASA ADS (last updated successfully 2026-08-09 14:19:57). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.