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From Promises to Totality: A Framework for Ruling Out Quantum Speedups

Thomas Huffstutler, Upendra Kapshikar, David Miloschewsky, Supartha Podder

Abstract

We study when partial Boolean functions can (and cannot) exhibit superpolynomial quantum query speedups, and develop a general framework for ruling out such speedups via two complementary lenses: promise-aware complexity measures and function completions. First, we introduce promise versions of standard combinatorial measures (including block sensitivity and related variants) and prove that if the relevant promise and completion measures collapse, then deterministic and quantum query complexities are necessarily polynomially related, i.e., $D(f)=poly(Q(f))$. We then analyze structured families of promises, including symmetric partial functions and promises supported on Hamming slices, obtaining sharp (up to polynomial factors) characterizations in terms of a single gap parameter for the symmetric case and refined slice-dependent bounds for $k$-slice domains. Next, we formalize completion complexity as the minimum of a measure over total completions of a partial function, and show that completability of a measure captures the possibility of superpolynomial quantum speedups. Finally, we apply this viewpoint to derive broad non-speedup criteria for some classes of functions admitting well-behaved completions, such as functions with low maximum influence on both the standard and $p$-biased hypercubes and functions with efficiently identifiable domains, and then show some hardness results for general completion techniques.

From Promises to Totality: A Framework for Ruling Out Quantum Speedups

Abstract

We study when partial Boolean functions can (and cannot) exhibit superpolynomial quantum query speedups, and develop a general framework for ruling out such speedups via two complementary lenses: promise-aware complexity measures and function completions. First, we introduce promise versions of standard combinatorial measures (including block sensitivity and related variants) and prove that if the relevant promise and completion measures collapse, then deterministic and quantum query complexities are necessarily polynomially related, i.e., . We then analyze structured families of promises, including symmetric partial functions and promises supported on Hamming slices, obtaining sharp (up to polynomial factors) characterizations in terms of a single gap parameter for the symmetric case and refined slice-dependent bounds for -slice domains. Next, we formalize completion complexity as the minimum of a measure over total completions of a partial function, and show that completability of a measure captures the possibility of superpolynomial quantum speedups. Finally, we apply this viewpoint to derive broad non-speedup criteria for some classes of functions admitting well-behaved completions, such as functions with low maximum influence on both the standard and -biased hypercubes and functions with efficiently identifiable domains, and then show some hardness results for general completion techniques.

Paper Structure

This paper contains 22 sections, 27 theorems, 92 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Our techniques
  4. Discussion and further work
  5. Organization of the paper
  6. Preliminaries
  7. Query complexity
  8. Definition of measures for partial functions
  9. Fourier analysis
  10. Promise measures
  11. Functions with $S_n$ symmetry
  12. Symmetric functions
  13. Functions defined on a slice
  14. Quantum speedups through completions
  15. Completion complexity
  16. ...and 7 more sections

Key Result

Lemma 1

For every Boolean function $f$, we have, $\mathrm{D}(f) \geq \mathrm{R}(f) \geq \mathrm{Q}(f)\geq \Omega\left({\widetilde{\mathrm{deg}}(f)}\right)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: The orange points represent the original inputs in the domain, and the red dotted polynomial $p(x)$ is a polynomial that approximates the function $f$ on these points. The solid black polynomial $P(x)$ is a completion that provides an approximation across the entire Boolean hypercube.
  • Figure 2: Relationships between the measures discussed. Measures introduced in this work are in orange and the relationships proved in this work are in green.
  • Figure 3: The ball here has radius $R\leq poly(d)$, and we see two example $\epsilon$-slabs which would correspond to two points $x\notin \textup{Dom} \!\left(f\right)$. The vector $\Delta$ shown here avoids these $\epsilon$-slabs while simultaneously being bounded in magnitude within the volume of the ball. This is what we will then choose as our perturbation for the polynomial $p_{\Delta}(x)$.

Theorems & Definitions (86)

  • Definition 1: Degree
  • Definition 2: Approximate degree
  • Lemma 1: beals2001quantum
  • Definition 3: Partial assignment
  • Definition 4: Certificate, chakraborty2022certificate
  • Definition 5: Certificate measure, chakraborty2022certificate
  • Definition 6: Block notation
  • Definition 7: Sensitivity and block sensitivity
  • Definition 8: $\mathcal{A}$-sensitive blocks
  • Definition 9: Critical Block Sensitivity huynh2012virtue
  • ...and 76 more