Characterizing Quantum Advantage for Generalizations of the Boolean Hidden Matching Problem

2026-10-05

AI Summary

This study investigates the one-way communication complexity of the f-Boolean Hidden Partition problem, aiming to precisely characterize quantum advantage. By integrating communication complexity theory, sign-degree analysis, and randomized protocol design, it rigorously establishes classical and quantum communication complexity bounds based on sign degree. The primary contributions include a complete characterization of function classes exhibiting polynomial quantum advantage, proving that such an advantage exists if and only if the sign degree d ≥ 2, thereby confirming a related conjecture. Furthermore, this work identifies a novel infinite family of exponential separations. Overall, these findings provide a solid theoretical foundation for understanding quantum computational supremacy.

Abstract

We study the one-way communication complexity of the $f$-Boolean Hidden Partition problem. Here, Alice is given an $n$-bit string and Bob is given $\Omega(n)$ disjoint blocks of its indices together with a string of labels. Under the promise that the evaluations of $f$ on these blocks either agree with all of the labels or disagree with all of them, Bob must determine which is the case using a single message from Alice. This problem generalizes the Boolean Hidden Matching and Hidden Hypermatching problems, which capture the special case where $f$ is the parity function. We establish classical and quantum communication bounds for $f$-Boolean Hidden Partition in terms of the sign degree $d$ of $f$, proving a conjecture of Doriguello and Montanaro (TQC 2020). Their prior work gave logarithmic communication upper bounds for classical protocols when $d \le 1$ and quantum protocols when $d \le 2$, as well as polynomial lower bounds for certain structured functions with larger sign degree. We show that for every $f$ of sign degree $d \ge 2$, the randomized classical communication complexity of this problem is $\Theta(n^{1-1/d})$ while its quantum communication complexity lies between $\Omega(n^{1-2/d})$ and $\bO{n^{1-1/\lceil d/2 \rceil} \log n}$. This completely characterizes the functions $f$ admitting polynomial quantum advantage as those for which $d \ge 2$, with a new infinite family of exponential separations given by the case $d = 2$.

Problem

• Quantum Advantage
• Communication Complexity
• Boolean Hidden Partition
• Sign Degree

Innovation

• Boolean Hidden Partition
• Communication Complexity
• Quantum Advantage
• Sign Degree
• Exponential Separation