Quantum advantage in zero-error function computation with side information

2024-02-02

arXiv.org

AI Summary

This paper investigates zero-error function computation with side information: Alice and Bob hold correlated random variables (X) and (Y), respectively, and Bob must compute (f(X,Y)) perfectly using a finite-length message sent over a classical or quantum channel. The authors introduce an (m)-block communication model based on the confusability graph (G), and establish necessary and sufficient conditions for the (m)-fold strong product (G^{oxtimes m}) or OR product (G^{vee m}). They demonstrate a fundamental separation between single-shot and asymptotic quantum advantages—exhibiting explicit examples where quantum protocols outperform classical ones in the single-shot regime but offer no asymptotic rate advantage. Leveraging graph coloring numbers, orthogonal rank, zero-error information theory, and quantum communication complexity, they precisely characterize the ratio of quantum to classical zero-error communication rates across multiple graph families, thereby decoupling single-shot and asymptotic advantages and confirming the intrinsic finite-length dependence of quantum advantage.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Multiagent Systems: Agent Communication

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the problem of zero-error function computation with side information. Alice and Bob have correlated sources $X,Y$ with joint p.m.f. $p_{XY}(cdot, cdot)$. Bob wants to calculate $f(X,Y)$ with zero error. Alice communicates $m$-length blocks $(m geq 1)$ with Bob over error-free channels: classical or quantum. In the classical setting, the minimum communication rate depends on the asymptotic growth of the chromatic number of an appropriately defined $m$-instance ``confusion graph'' $G^{(m)}$. In the quantum setting, it depends on the asymptotic growth of the orthogonal rank of the complement of $G^{(m)}$. The behavior of the quantum advantage (ratio of classical and quantum rates) depends critically on $G^{(m)}$ which is shown to be sandwiched between $G^{oxtimes m}$ ($m$-times strong product) and $G^{lor m}$ ($m$-times OR product) respectively. Our work presents necessary and sufficient conditions on the function $f(cdot, cdot)$ and joint p.m.f. $p_{XY}(cdot,cdot)$ such that $G^{(m)}$ equals either $G^{oxtimes m}$ or $G^{lor m}$. We study the behavior of the quantum advantage in the single-instance case and the asymptotic (in $m$) case for several classes of confusion graphs and demonstrate, e.g., that there exist problems where there is a quantum advantage in the single-instance rate but no quantum advantage in the asymptotic rate.

Problem

• Quantum Computing
• Function Evaluation
• Advantage Conditions

Innovation

• Quantum Advantage
• Error-free Computation
• Single-shot vs Multiple-shot Communication