Analyzing XOR-Forrelation Through Stochastic Calculus

Author Xinyu Wu



PDF
Thumbnail PDF

File

LIPIcs.STACS.2022.60.pdf
  • Filesize: 0.75 MB
  • 7 pages

Document Identifiers

Author Details

Xinyu Wu
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

Acknowledgements

I would like to thank Ryan O'Donnell for many helpful comments on this paper.

Cite AsGet BibTex

Xinyu Wu. Analyzing XOR-Forrelation Through Stochastic Calculus. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 60:1-60:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.STACS.2022.60

Abstract

In this note we present a simplified analysis of the quantum and classical complexity of the k-XOR Forrelation problem (introduced in the paper of Girish, Raz and Zhan [Uma Girish et al., 2020]) by a stochastic interpretation of the Forrelation distribution.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum complexity
  • Brownian motion

Metrics

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

References

  1. Scott Aaronson and Andris Ambainis. Forrelation: a problem that optimally separates quantum from classical computing. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, pages 307-316, 2015. Google Scholar
  2. Nikhil Bansal and Makrand Sinha. k-Forrelation optimally separates quantum and classical query complexity. Technical Report 2008.07003, arXiv, 2020. Google Scholar
  3. F. Barthe and B. Maurey. Some remarks on isoperimetry of Gaussian type. Ann. Inst. H. Poincaré Probab. Statist., 36(4):419-434, 2000. URL: https://doi.org/10.1016/S0246-0203(00)00131-X.
  4. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom generators from polarizing random walks. Theory Comput., 15:Paper No. 10, 26, 2019. URL: https://doi.org/10.4086/toc.2019.v015a010.
  5. Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal. Pseudorandom generators from the second Fourier level and applications to AC0 with parity gates. In Proceedings of the 10th Annual Innovations in Theoretical Computer Science Conference, pages 22:1-22:15, 2018. Google Scholar
  6. Ronen Eldan and Renan Gross. Concentration on the Boolean hypercube via pathwise stochastic analysis. In Proceedings of the 52nd Annual ACM Symposium on Theory of Computing, pages 208-221. ACM, New York, 2020. Google Scholar
  7. Uma Girish, Ran Raz, and Wei Zhan. Lower bounds for XOR of forrelations. Technical report, arXiv, 2020. URL: http://arxiv.org/abs/2007.03631.
  8. Bernt Øksendal. Stochastic differential equations. Universitext. Springer-Verlag, Berlin, sixth edition, 2003. An introduction with applications. URL: https://doi.org/10.1007/978-3-642-14394-6.
  9. Ran Raz and Avishay Tal. Oracle separation of BQP and PH. In Proceedings of the 51st Annual ACM Symposium on Theory of Computing, pages 13-23. ACM, New York, 2019. URL: https://doi.org/10.1145/3313276.3316315.
  10. Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999. URL: https://doi.org/10.1007/978-3-662-06400-9.
  11. Avishay Tal. Towards optimal separations between quantum and randomized query complexities. In Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science, pages 228-239. IEEE Computer Soc., Los Alamitos, CA, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00030.
  12. Martin J. Wainwright. High-dimensional statistics, volume 48 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2019. A non-asymptotic viewpoint. URL: https://doi.org/10.1017/9781108627771.
  13. Xinyu Wu. A stochastic calculus approach to the oracle separation of BQP and PH. Technical report, arXiv, 2020. URL: http://arxiv.org/abs/2007.02431.