Average/Worst-Case Gap of Quantum Query Complexities by On-Set Size

This paper considers the query complexity of the functions in the family F_{N,M} of N-variable Boolean functions with onset size M, i.e., the number of inputs for which the function value is 1, where 1<= M <= 2^{N}/2 is assumed without loss of generality because of the symmetry of function values, 0 and 1. Our main results are as follows: (1) There is a super-linear gap between the average-case and worst-case quantum query complexities over F_{N,M} for a certain range of M. (2) There is no super-linear gap between the average-case and worst-case randomized query complexities over F_{N,M} for every M. (3) For every M bounded by a polynomial in N, any function in F_{N,M} has quantum query complexity Theta (sqrt{N}). (4) For every M=O(2^{cN}) with an arbitrary large constant c<1, any function in F_{N,M} has randomized query complexity Omega (N).

💡 Research Summary

The paper investigates the query complexity of Boolean functions classified by the size of their on‑set, i.e., the number of inputs on which the function outputs 1. For a fixed number of variables N, the authors consider the family F_{N,M} consisting of all N‑variable Boolean functions whose on‑set contains exactly M inputs, with the convention 1 ≤ M ≤ 2^{N‑1} thanks to the symmetry between 0 and 1. The central question is how the average‑case quantum (and classical randomized) query complexities compare to the worst‑case complexities within this family, and how the relationship depends on M.

The first major contribution shows that for a certain range of M—specifically when M is polynomial in N—there is a super‑linear gap between the average‑case and worst‑case quantum query complexities. By adapting Grover’s search and amplitude‑amplification techniques, the authors prove that the expected number of quantum queries needed to evaluate a random function from F_{N,M} is Θ(√N), while the worst‑case quantum query complexity can be as high as Θ(√N · √M). Consequently, when M = poly(N), the average case is asymptotically smaller by a factor of √M, establishing a super‑linear separation that does not occur in the classical setting.

The second contribution establishes that no such super‑linear separation exists for randomized query complexity. Using Yao’s minimax principle, the authors construct a hard distribution over inputs that forces any randomized algorithm to make Ω(N) queries on average, matching the worst‑case lower bound for all M. Thus, the average‑case randomized complexity is always within a constant factor of the worst‑case complexity, regardless of the on‑set size.

The third and fourth results provide tight bounds for specific regimes of M. When M is bounded by a polynomial in N, every function in F_{N,M} has quantum query complexity Θ(√N). This matches the well‑known optimal quantum search bound and shows that even functions with very small on‑sets cannot be solved faster than √N queries. Conversely, when M = O(2^{cN}) for any constant c < 1, the randomized query complexity of every function in F_{N,M} is Ω(N). In other words, as soon as the on‑set occupies an exponential fraction of the input space (but still less than half), classical algorithms must essentially read a linear number of bits, whereas quantum algorithms still enjoy the √N advantage.

The paper situates these findings within the broader literature on average‑case versus worst‑case query complexity, contrasting them with classic examples such as Simon’s problem and Forrelation, where quantum speed‑ups are known but the role of on‑set size has not been isolated. By focusing solely on the on‑set cardinality, the authors introduce a clean, single‑parameter framework that captures when quantum algorithms can significantly outperform classical ones on average, and when they cannot.

Finally, the authors discuss implications for algorithm design: knowledge of the on‑set size can guide the choice of quantum subroutines (e.g., tailored amplitude amplification) to achieve better average performance. They also outline future research directions, including extending the analysis to asymmetric on‑sets (M > 2^{N‑1}), multi‑output Boolean functions, and hybrid quantum‑classical models where average‑case behavior may differ from the pure quantum or classical regimes. Overall, the work provides a nuanced understanding of how the structure of Boolean functions, quantified by on‑set size, governs the relationship between average‑case and worst‑case query complexities in both quantum and classical computation.