Unbounded-error One-way Classical and Quantum Communication Complexity

TL;DR

This paper establishes a precise relationship between quantum and classical one-way communication complexities under unbounded-error conditions, and applies this to determine bounds for quantum random access coding.

Contribution

It proves that for any Boolean function, quantum one-way complexity is exactly half of the classical complexity in the unbounded-error setting, and derives bounds for quantum random access codes.

Findings

Quantum one-way complexity is exactly half of classical complexity for any Boolean function.

Established the exact bounds for the existence of (m,n,p)-QRAC with p > 1/2.

Connected complexity results to the existence and non-existence of specific quantum random access codes.

Abstract

This paper studies the gap between quantum one-way communication complexity $Q(f)$ and its classical counterpart $C(f)$, under the {\em unbounded-error} setting, i.e., it is enough that the success probability is strictly greater than 1/2. It is proved that for {\em any} (total or partial) Boolean function $f$, $Q(f)=\lceil C(f)/2 \rceil$, i.e., the former is always exactly one half as large as the latter. The result has an application to obtaining (again an exact) bound for the existence of $(m,n,p)$-QRAC which is the $n$-qubit random access coding that can recover any one of $m$ original bits with success probability $\geq p$. We can prove that $(m,n,>1/2)$-QRAC exists if and only if $m\leq 2^{2n}-1$. Previously, only the construction of QRAC using one qubit, the existence of $(O(n),n,>1/2)$-RAC, and the non-existence of $(2^{2n},n,>1/2)$-QRAC were known.