Communication Complexities of XOR functions

We call $F:{0, 1}^n\times {0, 1}^n\to{0, 1}$ a symmetric XOR function if for a function $S:{0, 1, …, n}\to{0, 1}$, $F(x, y)=S(|x\oplus y|)$, for any $x, y\in{0, 1}^n$, where $|x\oplus y|$ is the Hamming weight of the bit-wise XOR of $x$ and $y$. We show that for any such function, (a) the deterministic communication complexity is always $\Theta(n)$ except for four simple functions that have a constant complexity, and (b) up to a polylog factor, the error-bounded randomized and quantum communication complexities are $\Theta(r_0+r_1)$, where $r_0$ and $r_1$ are the minimum integers such that $r_0, r_1\leq n/2$ and $S(k)=S(k+2)$ for all $k\in[r_0, n-r_1)$.

💡 Research Summary

The paper studies a broad class of two‑party communication problems defined by a symmetric XOR function. Formally, for a Boolean function S : {0,…,n}→{0,1}, the communication task is to compute
 F(x,y)=S(|x⊕y|) for inputs x,y∈{0,1}ⁿ,
where |x⊕y| denotes the Hamming weight of the bit‑wise XOR of the two n‑bit strings. The authors completely characterize the deterministic, bounded‑error randomized, and bounded‑error quantum communication complexities of every such function.

Deterministic complexity.
The communication matrix of F has a block‑constant structure: all entries corresponding to pairs (x,y) with the same Hamming distance k share the same value S(k). By analyzing the rank of this matrix and applying known rank‑vs‑deterministic‑complexity lower bounds, the authors show that unless S is one of four trivial functions, the matrix has full rank Θ(n). Consequently, the deterministic communication complexity D(F) is Θ(n) for all but the following four cases: (1) S(k)=0 for all k, (2) S(k)=1 for all k, (3) S(k)=k mod 2 (parity of the distance), and (4) S(k)=1−(k mod 2). In each of these exceptional cases the matrix collapses to a constant or a simple checkerboard pattern, and a constant‑bit protocol suffices. Thus D(F)=Θ(n) for the overwhelming majority of symmetric XOR functions.

Randomized and quantum complexity.
The core of the second result is a pair of parameters r₀ and r₁ that capture where S becomes 2‑periodic. Define r₀ (resp. r₁) as the smallest integer ≤ n/2 such that S(k)=S(k+2) holds for every k in the interval