Document Open Access Logo

Improved Separation Between Quantum and Classical Computers for Sampling and Functional Tasks

Authors Simon C. Marshal , Scott Aaronson , Vedran Dunjko

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.CCC.2025.5.pdf
  • Filesize: 0.78 MB
  • 14 pages
HTML5 Logo HTML (experimental)
LIPIcs.CCC.2025.5.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum advantage
  • Approximate counting
  • Boson sampling

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

This paper furthers existing evidence that quantum computers are capable of computations beyond classical computers. Specifically, we strengthen the collapse of the polynomial hierarchy to the second level if: (i) Quantum computers with postselection are as powerful as classical computers with postselection (PostBQP = PostBPP), (ii) any one of several quantum sampling experiments (BosonSampling, IQP, DQC1) can be approximately performed by a classical computer (contingent on existing assumptions). This last result implies that if any of these experiment’s hardness conjectures hold, then quantum computers can implement functions classical computers cannot (FBQP≠ FBPP) unless the polynomial hierarchy collapses to its 2nd level. These results are an improvement over previous work which either achieved a collapse to the third level or were concerned with exact sampling, a physically impractical case.
The workhorse of these results is a new technical complexity-theoretic result which we believe could have value beyond quantum computation. In particular, we prove that if there exists an equivalence between problems solvable with an exact counting oracle and problems solvable with an approximate counting oracle, then the polynomial hierarchy collapses to its second level, indeed to ZPP^NP.

Cite As Get BibTex

Simon C. Marshal, Scott Aaronson, and Vedran Dunjko. Improved Separation Between Quantum and Classical Computers for Sampling and Functional Tasks. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.5

Author Details

Simon C. Marshal
  • Leiden University, The Netherlands
Scott Aaronson
  • University of Texas at Austin, TX, USA
Vedran Dunjko
  • Leiden University, The Netherlands

Funding

  • Marshal, Simon C.: supported by the project NEASQC funded from the European Union’s Horizon 2020 research and innovation programme (grant agreement No 951821) and by an unrestricted gift from Google Quantum AI.
  • Aaronson, Scott: supported by a Vannevar Bush Fellowship from the US Department of Defense, the Berkeley NSF-QLCI CIQC Center, and a Simons Investigator Award.
  • Dunjko, Vedran: supported by the European Union’s Horizon Europe program through the ERC CoG BeMAIQuantum (Grant No. 101124342) and the Dutch National Growth Fund (NGF), as part of the Quantum Delta NL program.

Acknowledgements

SCM thanks Jan H. Kirchner for their insightful comments.

References

  1. Scott Aaronson. Quantum computing, postselection, and probabilistic polynomial-time. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2063):3473-3482, 2005. Google Scholar
  2. Scott Aaronson. The equivalence of sampling and searching. Theory of Computing Systems, 55(2):281-298, 2014. URL: https://doi.org/10.1007/S00224-013-9527-3.
  3. Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 333-342, 2011. URL: https://doi.org/10.1145/1993636.1993682.
  4. Scott Aaronson, Greg Kuperberg, and Christopher Granade. The complexity zoo, 2005. Google Scholar
  5. Dorit Aharonov, Xun Gao, Zeph Landau, Yunchao Liu, and Umesh Vazirani. A polynomial-time classical algorithm for noisy random circuit sampling. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 945-957, 2023. URL: https://doi.org/10.1145/3564246.3585234.
  6. V. Arvind and Johannes Köbler. New lowness results for zppnp and other complexity classes. Journal of Computer and System Sciences, 65(2):257-277, 2002. URL: https://doi.org/10.1006/jcss.2002.1835.
  7. Manuel Blum and Sampath Kannan. Designing programs that check their work. Journal of the ACM (JACM), 42(1):269-291, 1995. URL: https://doi.org/10.1145/200836.200880.
  8. Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. Average-case complexity versus approximate simulation of commuting quantum computations. Physical review letters, 117(8):080501, 2016. Google Scholar
  9. J Feigenbaum and L Fortnow. On the random-self-reducibility of complete sets, university of chicago technical report 90-22. Computer Science Department, 20, 1990. Google Scholar
  10. Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. Power of quantum computation with few clean qubits. arXiv preprint, 2015. URL: https://arxiv.org/abs/1509.07276.
  11. Yenjo Han, Lane A Hemaspaandra, and Thomas Thierauf. Threshold computation and cryptographic security. SIAM Journal on Computing, 26(1):59-78, 1997. URL: https://doi.org/10.1137/S0097539792240467.
  12. Tomoyuki Morimae. Hardness of classically sampling the one-clean-qubit model with constant total variation distance error. Physical Review A, 96(4):040302, 2017. Google Scholar
  13. Ryan O'Donnell and AC Cem Say. The weakness of ctc qubits and the power of approximate counting. ACM Transactions on Computation Theory (TOCT), 10(2):1-22, 2018. URL: https://doi.org/10.1145/3196832.
  14. Larry Stockmeyer. On approximation algorithms for#p. SIAM Journal on Computing, 14(4):849-861, 1985. URL: https://doi.org/10.1137/0214060.
  15. Seinosuke Toda. Pp is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865-877, 1991. URL: https://doi.org/10.1137/0220053.
  16. Leslie G Valiant and Vijay V Vazirani. Np is as easy as detecting unique solutions. In Proceedings of the seventeenth annual ACM symposium on Theory of computing, pages 458-463, 1985. URL: https://doi.org/10.1145/22145.22196.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact