On Worst-Case Optimal Polynomial Intersection

2026-04-10

AI Summary

This work investigates the performance limits of the Optimal Polynomial Intersection (OPI) problem in the worst case, aiming to surpass the semicircular law bound achieved by the existing quantum algorithm DQI. By integrating algebraic coding theory, MDS codes, Max-LINSAT modeling, and local leakage resilience analysis from secret sharing, it establishes for the first time that strictly better-than-DQI OPI solutions exist over prime fields, thereby demonstrating that DQI is not worst-case optimal. The main contributions include asymptotically exceeding the semicircular law when the list size is ρp (for any ρ ∈ (0,1)) and n/m ≥ 0.6225, and achieving asymptotically perfect recovery when n/m ≥ 0.7496. The work also uncovers a profound connection between OPI and locally leakage-resilient secret sharing schemes.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

The Optimal Polynomial Intersection (OPI) problem is the following: Given sets $S_1, \ldots, S_m \subseteq \mathbb{F}$ and evaluation points $a_1, \ldots, a_m \in \mathbb{F}$, find a polynomial $Q \in \mathbb{F}[x]$ of degree less than $n$ so that $Q(a_i) \in S_i$ for as many $i \in \{1, 2, \ldots, m\}$ as possible. Decoded Quantum Interferometry (DQI) is a quantum algorithm that efficiently returns good solutions to the problem, even on worst-case instances (Jordan et. al., 2025). The quality of the solutions returned follows a semicircle law, which outperforms known efficient classical algorithms. But does DQI obtain the best possible solutions? That is, are there solutions better than the semicircle law for worst-case OPI instances? Surprisingly, before this work, the best existential results coincide with (and follow from) the best algorithmic results. In this work, we show that there are better solutions for worst-case OPI instances over prime fields. In particular, DQI and the semicircle law are not optimal. For example, when the lists $S_i$ have size $\rho p$ for $\rho \sim 1/2$, our results imply the existence of a solution that asymptotically beats the semicircle law whenever $n/m \geq 0.6225$, and we show that an asymptotically perfect solution exists whenever $n/m \geq 0.7496$. Our results generalize to Max-LINSAT problems derived from any Maximum Distance Separable (MDS) code, and to any $\rho \in (0,1)$. The key insight to our improvement is a connection to local leakage resilience of secret sharing schemes. Along the way, we recover several re-proofs of the existence of solutions achieving the semicircle law.

Problem

• Optimal Polynomial Intersection
• worst-case instances
• semicircle law
• quantum algorithms
• MDS codes

Innovation

• Optimal Polynomial Intersection
• semicircle law
• local leakage resilience
• MDS codes
• worst-case analysis