Unbounded Quantum Advantage in Communication with Minimal Input Scaling

In communication complexity-like problems, previous studies have shown either an exponential quantum advantage or an unbounded quantum advantage with an exponentially large input set $\Theta(2^{n})$ bits with respect to classical communication $\Theta(n)$ bits. In the former, the quantum and classical separation grows exponentially in input while the latter's quantum communication resource is a constant. Remarkably, it was still open whether an unbounded quantum advantage exists while the inputs do not scale exponentially. Here we answer this question affirmatively using an input size of optimal order. Considering two variants as tasks: 1) distributed computation of relation and 2) {\it relation reconstruction}, we study the one-way zero-error communication complexity of a relation induced by a distributed clique labelling problem for orthogonality graphs. While we prove no quantum advantage in the first task, we show an {\it unbounded quantum advantage} in relation reconstruction without public coins. Specifically, for a class of graphs with order $m$, the quantum complexity is $\Theta(1)$ while the classical complexity is $\Theta(\log_2 m)$. Remarkably, the input size is $\Theta(\log_2 m)$ bits and the order of its scaling with respect to classical communication is {\it minimal}. This is exponentially better compared to previous works. Additionally, we prove a lower bound (linear in the number of maximum cliques) on the amount of classical public coin necessary to overcome the separation in the scenario of restricted communication and connect this to the existence of Orthogonal Arrays. Finally, we highlight some applications of this task to semi-device-independent dimension witnessing and the detection of Mutually Unbiased Bases.