Unbounded-Error One-Way Classical and Quantum Communication Complexity

This paper studies the gap between quantum one-way communication complexity Q(f) and its classical counterpart C(f), under the unbounded-error setting, i.e., it is enough that the success probability is strictly greater than 1/2. It is proved that for any (total or partial) Boolean function f , Q(f) = ⌈C(f)/2⌉, 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 ≥ p. We can prove that (m,n, > 1/2)-QRAC exists if and only if m ≤ 2 − 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, n, > 1/2)-QRAC were known.