Document Open Access Logo

Lower Bounds for XOR of Forrelations

Authors Uma Girish, Ran Raz, Wei Zhan



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2021.52.pdf
  • Filesize: 0.79 MB
  • 14 pages

Document Identifiers

Author Details

Uma Girish
  • Department of Computer Science, Princeton University, NJ, USA
Ran Raz
  • Department of Computer Science, Princeton University, NJ, USA
Wei Zhan
  • Department of Computer Science, Princeton University, NJ, USA

Acknowledgements

We would like to thank Avishay Tal for very helpful conversations. We would also like to thank Chin Ho Lee for pointing out a simpler proof of Lemma 28.

Cite AsGet BibTex

Uma Girish, Ran Raz, and Wei Zhan. Lower Bounds for XOR of Forrelations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 52:1-52:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.52

Abstract

The Forrelation problem, first introduced by Aaronson [Scott Aaronson, 2010] and Aaronson and Ambainis [Scott Aaronson and Andris Ambainis, 2015], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [Scott Aaronson and Andris Ambainis, 2015]; the first separation between poly-logarithmic quantum query complexity and bounded-depth circuits of super-polynomial size, a result that also implied an oracle separation of the classes BQP and PH [Ran Raz and Avishay Tal, 2019]; and improved separations between quantum and classical communication complexity [Uma Girish et al., 2021]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√N, that is, the success probability is larger than ≈ 1/2 + 1/√N. This is unavoidable as ≈ 1/√N is the correlation between two coordinates of an input that is sampled from the Forrelation distribution, and hence there are simple classical protocols that achieve advantage ≈ 1/√N, in all these models. To achieve separations when the classical protocol has smaller advantage, we study in this work the xor of k independent copies of (a variant of) the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k (that is, the sum of the absolute values of all Fourier coefficients at level 2k is bounded by α^k), cannot compute the xor of k independent copies of the Forrelation function with advantage better than O((α^k)/(N^{k/2})). This is a strengthening of a result of [Eshan Chattopadhyay et al., 2019], that gave a similar statement for k = 1, using the technique of [Ran Raz and Avishay Tal, 2019]. We give several applications of our result. In particular, we obtain the following separations: Quantum versus Classical Communication Complexity. We give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol with communication complexity polylog(N) (where Alice and Bob also share polylog(N) EPR pairs), and such that, any classical randomized protocol of communication complexity at most õ(N^{1/4}), with any number of rounds, has quasipolynomially small advantage over a random guess. Previously, only separations where the classical protocol has polynomially small advantage were known between these models [Dmitry Gavinsky, 2016; Uma Girish et al., 2021]. Quantum Query Complexity versus Bounded Depth Circuits. We give the first example of a partial Boolean function that has a quantum query algorithm with query complexity polylog(N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess. Previously, only separations where the constant-depth circuit has polynomially small advantage were known [Ran Raz and Avishay Tal, 2019].

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Oracles and decision trees
Keywords
  • Forrelation
  • Quasipolynomial
  • Separation
  • Quantum versus Classical
  • Xor

Metrics

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

References

  1. Scott Aaronson. BQP and the Polynomial Hierarchy. In STOC 2010, 2010. Google Scholar
  2. Scott Aaronson and Andris Ambainis. Forrelation: A Problem That Optimally Separates Quantum from Classical Computing. In STOC 2015, 2015. Google Scholar
  3. Ziv Bar-Yossef, T. S. Jayram, and Iordanis Kerenidis. Exponential Separation of Quantum and Classical One-Way Communication Complexity. In STOC 2004, 2004. Google Scholar
  4. Harry Buhrman, Richard Cleve, and Avi Wigderson. Quantum vs. Classical Communication and Computation. In STOC 1998, 1998. Google Scholar
  5. Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi. Query-To-Communication Lifting for BPP Using Inner Product. In ICALP 2019, 2019. Google Scholar
  6. Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, and Abhishek Shetty. Fractional Pseudorandom Generators from Any Fourier Level. CoRR, abs/2008.01316, 2020. Google Scholar
  7. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom Generators from Polarizing Random Walks. In CCC 2018, 2018. Google Scholar
  8. Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal. Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates. In ITCS 2019, 2019. Google Scholar
  9. Andrew Drucker. Improved Direct Product Theorems for Randomized Query Complexity. In CCC 2011, 2011. Google Scholar
  10. Dmitry Gavinsky. Entangled Simultaneity versus Classical Interactivity in Communication Complexity. In STOC 2016, 2016. Google Scholar
  11. Dmitry Gavinsky, Julia Kempe, Iordanis Kerenidis, Ran Raz, and Ronald de Wolf. Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography. In STOC 2007, 2007. Google Scholar
  12. Uma Girish, Ran Raz, and Avishay Tal. Quantum versus Randomized Communication Complexity, with Efficient Players. In ITCS 2021, 2021. Google Scholar
  13. Ran Raz. Fourier Analysis for Probabilistic Communication Complexity. In Computational Complexity Journal 1995, 1995. Google Scholar
  14. Ran Raz. Exponential Separation of Quantum and Classical Communication Complexity. In STOC 1999, 1999. Google Scholar
  15. Ran Raz and Avishay Tal. Oracle separation of BQP and PH . In STOC 2019, 2019. Google Scholar
  16. Oded Regev and Boàz Klartag:. Quantum One-Way Communication can be Exponentially Stronger than Classical Communication. In STOC 2011, 2011. Google Scholar
  17. Avishay Tal. Tight Bounds on the Fourier Spectrum of AC0. In CCC 2017, 2017. Google Scholar
  18. Avishay Tal. Towards Optimal Separations between Quantum and Randomized Query Complexities. In FOCS 2020, 2020. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail