- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Experimental computations of atomic properties on a superconducting quantum processor

Akhil Pratap Singh1,*,†, Kenji Sugisaki2,3,4,5, Srinivasa Prasannaa5,6, Bijaya Kumar Sahoo7, Bhanu Pratap Das5,6,8, and Yasunobu Nakamura1,9

  • *Contact author: akhil.singh@aalto.fi
  • †Present address: QCD Labs, QTF Centre of Excellence, Department of Applied Physics, Aalto University, P. O. Box 13500, FIN-00076 Aalto, Finland.
PDF

Phys. Rev. A 110, 062620 – Published 30 December, 2024

DOI: https://doi.org/10.1103/PhysRevA.110.062620

Abstract

We experimentally compute relativistic and correlation effects in the atomic properties by using a superconducting qubit processor. Specifically, we compute the relativistic ground-state energies and magnetic-dipole hyperfine structure constants for four Li-like atomic systems ranging from very light to moderately heavy to very heavy in terms of nuclear charge. A symmetry-conserving Bravyi-Kitaev transformation is used to reduce the original six-qubit problem to a four-qubit problem, which is experimentally contrived by reducing the hardware requirement by employing a virtual two-qubit gate. It enables the simulation of four-qubit circuits using two physical qubits with additional circuit evaluations. The ground-state wave functions, required for computing atomic properties, are obtained by using quantum state tomography. Our results show that the averaged relative errors for the ground-state energies are ≈0.3±1%. However, for the hyperfine structure constants, the mean values of the relative errors are less than 15%, with their estimated upper bound of relative errors of ≈±10% (with the exception of 7.2±47% for neutral Li7). Notably, our results for the hyperfine structure constants exhibit higher sensitivity to errors as compared to energies; a trend which we also confirm through additional numerical simulations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (68)

  1. P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Rev. 41, 303 (1999).
  2. A. Montanaro, Quantum algorithms: An overview, npj Quantum Inf. 2, 15023 (2016).
  3. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  4. K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  5. A. Callison and N. Chancellor, Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond, Phys. Rev. A 106, 010101 (2022).
  6. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler et al., Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  7. H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang et al., Quantum computational advantage using photons, Science 370, 1460 (2020).
  8. Y. Wu, W.-S. Bao, S. Cao, F. Chen, M.-C. Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du, D. Fan, M. Gong, Cheng Guo, Chu Guo, S. Guo, L. Han, L. Hong, H.-L. Huang, Y.-H. Huo, L. Li, N. Li et al., Strong quantum computational advantage using a superconducting quantum processor, Phys. Rev. Lett. 127, 180501 (2021).
  9. Q. Zhu, S. Cao, F. Chen, M.-C. Chen, X. Chen, T.-H. Chung, H. Deng, Y. Du, D. Fan, M. Gong, Cheng Guo, Chu Guo, S. Guo, L. Han, L. Hong, H.-L. Huang, Y.-H. Huo, L. Li, N. Li, S. Li et al., Quantum computational advantage via 60-qubit 24-cycle random circuit sampling, Sci. Bull. 67, 240 (2022).
  10. T. M. Graham, Y. Song, J. Scott, C. Poole, L. Phuttitarn, K. Jooya, P. Eichler, X. Jiang, A. Marra, B. Grinkemeyer, M. Kwon, M. Ebert, J. Cherek, M. T. Lichtman, M. Gillette, J. Gilbert, D. Bowman, T. Ballance, C. Campbell, E. D. Dahl et al., Multi-qubit entanglement and algorithms on a neutral-atom quantum computer, Nature (London) 604, 457 (2022).
  11. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213 (2014).
  12. A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature (London) 549, 242 (2017).
  13. P. J. J. O'Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus et al., Scalable quantum simulation of molecular energies, Phys. Rev. X 6, 031007 (2016).
  14. A. Kandala, K. Temme, A. D. Corcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Error mitigation extends the computational reach of a noisy quantum processor, Nature (London) 567, 491 (2019).
  15. P. J. Ollitrault, A. Kandala, C.-F. Chen, P. K. Barkoutsos, A. Mezzacapo, M. Pistoia, S. Sheldon, S. Woerner, J. M. Gambetta, and I. Tavernelli, Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor, Phys. Rev. Res. 2, 043140 (2020).
  16. Y. Nam, J.-S. Chen, N. C. Pisenti, K. Wright, C. Delaney, D. Maslov, K. R. Brown, S. Allen, J. M. Amini, J. Apisdorf, K. M. Beck, A. Blinov, V. Chaplin, M. Chmielewski, C. Collins, S. Debnath, K. M. Hudek, A. M. Ducore, M. Keesan, S. M. Kreikemeier et al., Ground-state energy estimation of the water molecule on a trapped-ion quantum computer, npj Quantum Inf. 6, 33 (2020).
  17. T. E. O'Brien, G. Anselmetti, F. Gkritsis, V. E. Elfving, S. Polla, W. J. Huggins, O. Oumarou, K. Kechedzhi, D. Abanin, R. Acharya, I. Aleiner, R. Allen, T. I. Andersen, K. Anderson, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, J. C. Bardin et al., Purification-based quantum error mitigation of pair-correlated electron simulations, Nat. Phys. 19, 1787 (2023).
  18. S. Guo, J. Sun, H. Qian, M. Gong, Y. Zhang, F. Chen, Y. Ye, Y. Wu, S. Cao, K. Liu, C. Zha, C. Ying, Q. Zhu, H.-L. Huang, Y. Zhao, S. Li, S. Wang, J. Yu, D. Fan, D. Wu et al., Experimental quantum computational chemistry with optimised unitary coupled cluster ansatz, Nat. Phys. 20, 1240 (2024).
  19. Z. Huang and S. Kais, Entanglement as measure of electron–electron correlation in quantum chemistry calculations, Chem. Phys. Lett. 413, 1 (2005).
  20. B. K. Sahoo and B. P. Das, Relativistic normal coupled-cluster theory for accurate determination of electric dipole moments of atoms: First application to Hg199 atom, Phys. Rev. Lett. 120, 203001 (2018).
  21. N. Yamanaka, B. K. Sahoo, N. Yoshinaga, T. Sato, K. Asahi, and B. P. Das, Probing exotic phenomena at the interface of nuclear and particle physics with the electric dipole moments of diamagnetic atoms: A unique window to hadronic and semi-leptonic CP violation, Eur. Phys. J. A 53, 54 (2017).
  22. S. G. Porsev, K. Beloy, and A. Derevianko, Precision determination of electroweak coupling from atomic parity violation and implications for particle physics, Phys. Rev. Lett. 102, 181601 (2009).
  23. V. A. Dzuba, J. C. Berengut, V. V. Flambaum, and B. Roberts, Revisiting parity nonconservation in cesium, Phys. Rev. Lett. 109, 203003 (2012).
  24. M. Kaur, R. Nakra, B. Arora, C.-B. Li, and B. K. Sahoo, Accurate determination of energy levels, hyperfine structure constants, lifetimes and dipole polarizabilities of triply ionized tin isotopes, J. Phys. B: At. Mol. Opt. Phys. 53, 065002 (2020).
  25. L. Essen and J. V. L. Parry, An atomic standard of frequency and time interval: A Cæsium resonator, Nature (London) 176, 280 (1955).
  26. N. Ramsey, History of atomic clocks, J. Res. Natl. Bur. Stand. 88, 301 (1983).
  27. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637 (2015).
  28. K. Beloy, M. I. Bodine, T. Bothwell, S. M. Brewer, S. L. Bromley, J.-S. Chen, J.-D. Deschênes, S. A. Diddams, R. J. Fasano, T. M. Fortier, Y. S. Hassan, D. B. Hume, D. Kedar, C. J. Kennedy, I. Khader, A. Koepke, D. R. Leibrandt, H. Leopardi, A. D. Ludlow, W. F. McGrew et al., Frequency ratio measurements at 18-digit accuracy using an optical clock network, Nature (London) 591, 564 (2021).
  29. D. Vrbanec, Exploring the Epoch of Reionization with the 21 cm line, Ph.D. thesis, Ludwig-Maximilians-Universität München (2019).
  30. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269, 198 (1995).
  31. K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75, 3969 (1995).
  32. B. K. Sahoo, R. K. Chaudhuri, B. P. Das, H. Merlitz, and D. Mukherjee, Application of relativistic coupled-cluster theory to heavy atomic systems with strongly interacting configurations: Hyperfine interactions in Pb+207, Phys. Rev. A 72, 032507 (2005).
  33. M. Abe, G. Gopakumar, M. Hada, B. P. Das, H. Tatewaki, and D. Mukherjee, Application of relativistic coupled-cluster theory to the effective electric field in YbF, Phys. Rev. A 90, 022501 (2014).
  34. K. Mitarai and K. Fujii, Constructing a virtual two-qubit gate by sampling single-qubit operations, New J. Phys. 23, 023021 (2021).
  35. A. P. Singh, K. Mitarai, Y. Suzuki, K. Heya, Y. Tabuchi, K. Fujii, and Y. Nakamura, Experimental demonstration of a high-fidelity virtual two-qubit gate, Phys. Rev. Res. 6, 013235 (2024).
  36. R. Hofstadter, H. R. Fechter, and J. A. McIntyre, High-energy electron scattering and nuclear structure determinations, Phys. Rev. 92, 978 (1953).
  37. T. P. Das, Relativistic Quantum Mechanics of Electrons, 1st ed. (Harper & Row, New York, 1973).
  38. I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation, 1st ed. (Springer, New York, 2007), p. 461.
  39. B. K. Sahoo, Coupled-Cluster Theory of Parity Non-Conservation in Atoms, Ph.D. thesis, Indian Institute of Astrophysics (2006).
  40. P. Jordan and W. E., Über das Paulische Äquivalenzverbot, Z. Phys. 47, 631 (1928).
  41. J. T. Seeley, M. J. Richard, and P. J. Love, The Bravyi-Kitaev transformation for quantum computation of electronic structure, J. Chem. Phys. 137, 224109 (2012).
  42. S. Bravyi, J. M. Gambetta, A. Mezzacapo, and K. Temme, Tapering off qubits to simulate fermionic Hamiltonians, arXiv:1701.08213.
  43. N. J. Stone, Table of nuclear magnetic dipole and electric quadrupole moments, At. Data Nucl. Data Tables 90, 75 (2005).
  44. Y. Suzuki, Y. Kawase, Y. Masumura, Y. Hiraga, M. Nakadai, J. Chen, K. M. Nakanishi, K. Mitarai, R. Imai, S. Tamiya, T. Yamamoto, T. Yan, T. Kawakubo, Y. O. Nakagawa, Y. Ibe, Y. Zhang, H. Yamashita, H. Yoshimura, A. Hayashi, and K. Fujii, Qulacs: A fast and versatile quantum circuit simulator for research purpose, Quantum 5, 559 (2021).
  45. J. Nocedal and S. Wright, in Numerical Optimization, 2nd ed., Springer Series in Operations Research and Financial Engineering (Springer, New York, 2006).
  46. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  47. K. G. Dyall, I. P. Grant, and S. Wilson, The Dirac equation in the algebraic approximation. II. Extended basis set calculations for hydrogenic atoms, J. Phys. B: At. Mol. Phys. 17, 1201 (1984).
  48. S. Battaglia, D. Bouet, A. Lecoq, S. Evangelisti, N. Faginas-Lago, T. Leininger, and A. Lombardi, Distributed Gaussian orbitals for molecular calculations: Application to simple systems, Mol. Phys. 118, 1615646 (2020).
  49. S. Huzinaga, Basis sets for molecular calculations, Comput. Phys. Rep. 2, 281 (1985).
  50. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design derived from the cooper pair box, Phys. Rev. A 76, 042319 (2007).
  51. S. Tamate, Y. Tabuchi, L. Szikszai, K. Kusuyama, K. Zuo, Z. Yan, A. Badrutdinov, Y. Hishida, W. Qiu, H. Terai, G. Fujii, K. Makise, N. Watanabe, H. Nakagawa, M. Fujino, M. Ukibe, W. Mizubayashi, K. Kikuchi, and Y. Nakamura, Scalable packaging design for large-scale superconducting quantum circuit, Bull. Am. Phys. Soc. 66, C30.9 (2021), https://meetings.aps.org/Meeting/MAR21/Session/C30.9.
  52. E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Randomized benchmarking of quantum gates, Phys. Rev. A 77, 012307 (2008).
  53. J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators, J. Opt. B: Quantum Semiclass. 7, S347 (2005).
  54. E. Magesan, J. M. Gambetta, and J. Emerson, Scalable and robust randomized benchmarking of quantum processes, Phys. Rev. Lett. 106, 180504 (2011).
  55. Q. D. Team, RealAmplitudes-Qiskit 0.31.0 documentation (2021), https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RealAmplitudes.
  56. D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, Efficient Z gates for quantum computing, Phys. Rev. A 96, 022330 (2017).
  57. E. Lucero, J. Kelly, R. C. Bialczak, M. Lenander, M. Mariantoni, M. Neeley, A. D. O'Connell, D. Sank, H. Wang, M. Weides, J. Wenner, T. Yamamoto, A. N. Cleland, and J. M. Martinis, Reduced phase error through optimized control of a superconducting qubit, Phys. Rev. A 82, 042339 (2010).
  58. C. Rigetti and M. Devoret, Fully microwave-tunable universal gates in superconducting qubits with linear couplings and fixed transition frequencies, Phys. Rev. B 81, 134507 (2010).
  59. S. Sheldon, E. Magesan, J. M. Chow, and J. M. Gambetta, Procedure for systematically tuning up cross-talk in the cross-resonance gate, Phys. Rev. A 93, 060302(R) (2016).
  60. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2010).
  61. M. R. Geller and M. Sun, Toward efficient correction of multiqubit measurement errors: Pair correlation method, Quantum Sci. Technol. 6, 025009 (2021).
  62. P. D. Nation, H. Kang, N. Sundaresan, and J. M. Gambetta, Scalable mitigation of measurement errors on quantum computers, PRX Quantum 2, 040326 (2021).
  63. S. Bravyi, S. Sheldon, A. Kandala, D. C. Mckay, and J. M. Gambetta, Mitigating measurement errors in multiqubit experiments, Phys. Rev. A 103, 042605 (2021).
  64. A. Beckmann, K. D. Böklen, and D. Elke, Precision measurements of the nuclear magnetic dipole moments of Li6, Li7, Na23, K39, and K41, Z. Phys. 270, 173 (1974).
  65. M. Lestinsky, E. Lindroth, D. A. Orlov, E. W. Schmidt, S. Schippers, S. Böhm, C. Brandau, F. Sprenger, A. S. Terekhov, A. Müller, and A. Wolf, Screened radiative corrections from hyperfine-split dielectronic resonances in lithiumlike scandium, Phys. Rev. Lett. 100, 033001 (2008).
  66. P. Beiersdorfer, E. Träbert, G. V. Brown, J. Clementson, D. B. Thorn, M. H. Chen, K. T. Cheng, and J. Sapirstein, Hyperfine splitting of the 2s1/2 and 2p1/2 levels in Li- and Be-like ions of Pb59141, Phys. Rev. Lett. 112, 233003 (2014).
  67. P. Beiersdorfer, A. L. Osterheld, J. H. Scofield, J. R. Crespo López-Urrutia, and K. Widmann, Measurement of QED and hyperfine splitting in the 2s1/2−2p3/2 X-Ray transition in Li-like Bi80+209, Phys. Rev. Lett. 80, 3022 (1998).
  68. Qiskit contributors, Qiskit: An open-source framework for quantum computing (2023), https://doi.org/10.5281/zenodo.8190968.

Outline

Information

Phys. Rev. A 110, 062620– Published 30 December, 2024
  • Received 23 February 2024
  • Accepted 2 December 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation