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On query complexity measures and their relations for symmetric functions

Rajat Mittal, Sanjay S Nair, Sunayana Patro

TL;DR

The paper investigates how key quantum query lower-bound techniques—notably positive adversary and certificate-game based bounds—relate to decision-tree measures within the class of total symmetric Boolean functions. It provides an explicit dual construction showing Adv^+(f) = Θ(√(t_f·n)) = Θ(√(CG(f))) and proves that spectral sensitivity λ(f) matches this same bound, thereby unifying several lower-bound caps in the symmetric setting. For GapMaj_n, the authors establish Q_ε(GapMaj_n) = Θ(√n) via quantum counting and an explicit adversary-based lower bound, and derive a bound linking noisy randomized query complexity to quantum query complexity for total functions. They also study separations among s(f), bs(f), and C(f) for symmetric functions, obtaining tight constants in these relationships. Overall, the results show that, for total symmetric functions, a broad class of lower-bound measures collapse to a common scale, clarifying the landscape of quantum-query lower bounds and their composition properties in a natural function class.

Abstract

The main reason for query model's prominence in complexity theory and quantum computing is the presence of concrete lower bounding techniques: polynomial and adversary method. There have been considerable efforts to give lower bounds using these methods, and to compare/relate them with other measures based on the decision tree. We explore the value of these lower bounds on quantum query complexity and their relation with other decision tree based complexity measures for the class of symmetric functions, arguably one of the most natural and basic sets of Boolean functions. We show an explicit construction for the dual of the positive adversary method and also of the square root of private coin certificate game complexity for any total symmetric function. This shows that the two values can't be distinguished for any symmetric function. Additionally, we show that the recently introduced measure of spectral sensitivity gives the same value as both positive adversary and approximate degree for every total symmetric Boolean function. Further, we look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. Finally, we study how large certificate complexity and block sensitivity can be as compared to sensitivity for symmetric functions (even up to constant factors). We show tight separations, i.e., give upper bounds on possible separations and construct functions achieving the same.

On query complexity measures and their relations for symmetric functions

TL;DR

The paper investigates how key quantum query lower-bound techniques—notably positive adversary and certificate-game based bounds—relate to decision-tree measures within the class of total symmetric Boolean functions. It provides an explicit dual construction showing Adv^+(f) = Θ(√(t_f·n)) = Θ(√(CG(f))) and proves that spectral sensitivity λ(f) matches this same bound, thereby unifying several lower-bound caps in the symmetric setting. For GapMaj_n, the authors establish Q_ε(GapMaj_n) = Θ(√n) via quantum counting and an explicit adversary-based lower bound, and derive a bound linking noisy randomized query complexity to quantum query complexity for total functions. They also study separations among s(f), bs(f), and C(f) for symmetric functions, obtaining tight constants in these relationships. Overall, the results show that, for total symmetric functions, a broad class of lower-bound measures collapse to a common scale, clarifying the landscape of quantum-query lower bounds and their composition properties in a natural function class.

Abstract

The main reason for query model's prominence in complexity theory and quantum computing is the presence of concrete lower bounding techniques: polynomial and adversary method. There have been considerable efforts to give lower bounds using these methods, and to compare/relate them with other measures based on the decision tree. We explore the value of these lower bounds on quantum query complexity and their relation with other decision tree based complexity measures for the class of symmetric functions, arguably one of the most natural and basic sets of Boolean functions. We show an explicit construction for the dual of the positive adversary method and also of the square root of private coin certificate game complexity for any total symmetric function. This shows that the two values can't be distinguished for any symmetric function. Additionally, we show that the recently introduced measure of spectral sensitivity gives the same value as both positive adversary and approximate degree for every total symmetric Boolean function. Further, we look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. Finally, we study how large certificate complexity and block sensitivity can be as compared to sensitivity for symmetric functions (even up to constant factors). We show tight separations, i.e., give upper bounds on possible separations and construct functions achieving the same.

Paper Structure

This paper contains 22 sections, 24 theorems, 46 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our results
  3. Lower bounds on quantum query complexity for total symmetric functions
  4. for total symmetric Boolean functions:
  5. Certificate game complexity for symmetric functions:
  6. Quantum query complexity for Gap majority function
  7. Proof of Theorem \ref{['gapMajQuery']}
  8. Acknowledgements
  9. Preliminaries
  10. Positive adversary
  11. Spectral Sensitivity
  12. Approximate degree
  13. Sensitivity Measures and Certificate Complexity
  14. Expectational certificate complexity
  15. Appendix
  16. ...and 7 more sections

Key Result

theorem thmcountertheorem

Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be a total symmetric Boolean function. Here $t_f$ is the minimum $t$ such that $f$ is constant for Hamming weights between $t$ and $n-t$

Figures (2)

  • Figure 1: The asymptotic relations on complexity measures of total symmetric functions. We show, 1) $\sqrt{\mathrm{CG}(f)}$ and $\lambda(f)$ are asymptotically identical to $\mathrm{Adv}^+(f)$; 2) $Q_{\epsilon}(f) = O(\mathrm{noisyR}_{\epsilon}(f)\cdot\sqrt{n})$; 3) The relation between $\mathrm{bs}(f)$ ($\mathrm{C}(f)$) and $\mathrm{s}(f)$ are obtained up to constants.
  • Figure 2: A general total symmetric Boolean function viewed on Hamming weights, it is constant in the range $[t_f, n-t_f]$ by definition.

Theorems & Definitions (53)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • ...and 43 more