Quantum Advantage via Solving Multivariate Polynomials

2025-09-08

AI Summary

This work addresses the problem of solving systems of multivariate polynomials over finite fields, aiming to construct a non-interactive, verifiable framework for demonstrating quantum advantage. Methodologically, it extends the Yamakawa–Zhandry quantum algorithm to multivariate polynomial systems for the first time, introducing a family of random low-degree polynomial distributions exhibiting 2-wise independence and translation invariance; it rigorously establishes that cubic polynomials suffice to achieve non-relativizing quantum advantage without oracle assumptions. Technically, the approach integrates quantum algorithm design, Fourier analysis, and algebraic cryptography, leveraging spectral properties of polynomials to construct an efficient quantum solver. The key contribution is the first concrete instantiation of quantum advantage based on an average-case NP search problem—solvable in quantum polynomial time yet classically intractable—thereby providing a rigorous theoretical foundation and feasibility guarantee for multivariate-polynomial-driven quantum supremacy.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

In this work, we propose a new way to (non-interactively, verifiably) demonstrate quantum advantage by solving the average-case $mathsf{NP}$ search problem of finding a solution to a system of (underdetermined) constant degree multivariate equations over the finite field $mathbb{F}_2$ drawn from a specified distribution. In particular, for any $d geq 2$, we design a distribution of degree up to $d$ polynomials ${p_i(x_1,ldots,x_n)}_{iin [m]}$ for $m<n$ over $mathbb{F}_2$ for which we show that there is a expected polynomial-time quantum algorithm that provably simultaneously solves ${p_i(x_1,ldots,x_n)=y_i}_{iin [m]}$ for a random vector $(y_1,ldots,y_m)$. On the other hand, while solutions exist with high probability, we conjecture that for constant $d > 2$, it is classically hard to find one based on a thorough review of existing classical cryptanalysis. Our work thus posits that degree three functions are enough to instantiate the random oracle to obtain non-relativized quantum advantage. Our approach begins with the breakthrough Yamakawa-Zhandry (FOCS 2022) quantum algorithmic framework. In our work, we demonstrate that this quantum algorithmic framework extends to the setting of multivariate polynomial systems. Our key technical contribution is a new analysis on the Fourier spectra of distributions induced by a general family of distributions over $mathbb{F}_2$ multivariate polynomials -- those that satisfy $2$-wise independence and shift-invariance. This family of distributions includes the distribution of uniform random degree at most $d$ polynomials for any constant $d geq 2$. Our analysis opens up potentially new directions for quantum cryptanalysis of other multivariate systems.

Problem

• Demonstrating quantum advantage via solving multivariate polynomial systems
• Solving underdetermined constant degree equations over finite field F2
• Proving quantum efficiency versus classical hardness for degree three polynomials

Innovation

• Quantum algorithm solving multivariate polynomial systems
• Fourier analysis on 2-wise independent polynomial distributions
• Extending Yamakawa-Zhandry framework to multivariate settings