Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas

We show lower bounds of $\Omega(\sqrt{n})$ and $\Omega(n^{1/4})$ on the randomized and quantum communication complexity, respectively, of all $n$-variable read-once Boolean formulas. Our results complement the recent lower bound of $\Omega(n/8^d)$ by Leonardos and Saks and $\Omega(n/2^{\Omega(d\log d)})$ by Jayram, Kopparty and Raghavendra for randomized communication complexity of read-once Boolean formulas with depth $d$. We obtain our result by “embedding” either the Disjointness problem or its complement in any given read-once Boolean formula.

💡 Research Summary

The paper investigates the communication complexity of read‑once Boolean formulas—circuits in which each input variable appears exactly once and the computation is performed by a binary tree of AND, OR, and NOT gates. While earlier works established depth‑dependent lower bounds (Leonardos and Saks proved a randomized lower bound of Ω(n/8^d), and Jayram, Kopparty and Raghavendra obtained Ω(n/2^{Ω(d log d)}) for depth‑d formulas), no depth‑independent bound was known. This work fills that gap by proving that every n‑variable read‑once formula Φ satisfies a randomized communication lower bound of Ω(√n) and a quantum lower bound of Ω(n^{1/4}), regardless of its depth.

The central technical contribution is an “embedding” technique. The authors show that any read‑once formula contains a large subtree whose leaf set has size at least Θ(√n). By carefully selecting two disjoint subsets of variables, one held by Alice and the other by Bob, they embed an instance of the Disjointness problem (or its complement, Intersection) into that subtree. Because Disjointness is a canonical hard problem in communication complexity—requiring Ω(n) randomized bits and Ω(√n) quantum bits—the embedded instance forces the whole formula to inherit a comparable lower bound after scaling by the size of the subtree. In the randomized setting the scaling yields Ω(√n), and in the quantum setting it yields Ω(n^{1/4}).

The embedding construction works uniformly for all shapes of read‑once formulas: pure AND‑trees, pure OR‑trees, or any mixture of the two, with or without NOT gates. The proof proceeds in two main steps. First, a combinatorial lemma guarantees the existence of a sufficiently large subtree regardless of depth. Second, the authors describe how to replace the logical operation at the root of this subtree with a Disjointness gadget, preserving the overall output of the formula while ensuring that any protocol solving Φ must also solve the embedded Disjointness instance. Error probabilities are controlled so that the success probability of the overall protocol matches that required for the original problem.

The paper also discusses the broader implications of this technique. By demonstrating that a hard communication problem can be hidden inside any read‑once formula, the authors show that depth alone does not mitigate communication difficulty for this class. This result complements the earlier depth‑dependent bounds, providing a unified picture: shallow formulas may have stronger (larger) lower bounds, but even deep formulas cannot drop below Ω(√n) (randomized) and Ω(n^{1/4}) (quantum). The authors suggest that similar embedding ideas could be applied to other restricted circuit classes such as constant‑depth AC⁰ or TC⁰, and they highlight open questions including tightening the quantum bound, extending the approach to deterministic communication, and exploring whether even larger depth‑independent bounds are possible.

In summary, the paper delivers a clean, depth‑independent communication lower bound for read‑once Boolean formulas by embedding the Disjointness problem into any such formula. The results advance our understanding of the intrinsic communication hardness of structured Boolean functions and open new avenues for applying embedding techniques to broader classes of computational models.