Quantum advantage in zero-error function computation with side information

• URL: http://arxiv.org/abs/2402.01549v3
• Date: Tue, 14 Jan 2025 23:45:09 GMT
• Title: Quantum advantage in zero-error function computation with side information
• Authors: Ruoyu Meng, Aditya Ramamoorthy,
• Abstract summary: We consider the problem of zero-error function computation with side information.<n>Alice and Bob have correlated sources $X,Y$ with joint p.m.f. $p_XY(cdot, cdot)$.<n>We study the behavior of the quantum advantage in the single-instance case and the case for several classes of confusion graphs.
• Score: 10.0060346233449
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $G^{\boxtimes m}$ ($m$-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 $G^{\boxtimes m}$ 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.

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