Exponential quantum space advantage in random data streams

2026-10-02

AI Summary

This study addresses the space bottleneck encountered by classical streaming models when handling code intersections and Ordered Polynomial Intersections (OPI), as well as the absence of rigorous proofs for quantum advantage in these settings. To overcome these limitations, this work extends the Decoded Quantum Interferometry (DQI) algorithm to the random-order streaming model, employing density recovery partitioning and convex potential functions for theoretical analysis. The primary contribution is establishing the first unconditional exponential quantum space advantage for the Yamakawa-Zhandry problem. Furthermore, it proves that OPI requires only O(d log n) qubits, achieving an exponential compression compared to the classical Ω(n) lower bound and thereby surpassing traditional complexity limitations.

Abstract

We show two unconditional quantum space advantages in the random-order streaming model. First, we show the Yamakawa--Zhandry Code Intersection problem admits exponential quantum advantage in the streaming model when its inputs are streamed in random order. This means quantum computers exhibit exponential space advantage even when simply receiving $(x,f(x))$ pairs for a uniformly random function $f$ in a uniformly random order. Our lower bound is shown using density-restoring partitions as in the work of Göös, Gur, Jain, and Li (STOC 2025) combined with a convex potential, similar to the work of Raz (J.ACM 2018) on parity learning and its generalization by Garg, Raz, and Tal (STOC 2018). Second, we use our framework to show quantum space advantage for the Optimal Polynomial Intersection (OPI) problem in certain regimes via a streaming version of the Decoded Quantum Interferometry algorithm (Nature 2025; arXiv:2510.10967). In particular, we show that for degree $d$ and $n$ evaluation points, attaining $1/2 + Ω(\sqrt{d/n})$ fraction of satisfied OPI constraints via streaming requires $Ω(n)$ classical bits of memory but only $O(d\log n)$ qubits. This yields provable quantum advantage in a "low-rate" regime when the number of evaluation points is much larger than the degree, with a space advantage that can be as large as exponential in certain parameter settings.

Problem

• quantum space advantage
• random-order streaming model
• Yamakawa-Zhandry Code Intersection
• Optimal Polynomial Intersection

Innovation

• Quantum space advantage
• Random-order streaming model
• Decoded Quantum Interferometry
• Optimal Polynomial Intersection
• Density-restoring partitions