On the communication complexity of XOR functions

An XOR function is a function of the form g(x,y) = f(x + y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise one-way communication complexity for all f. We also show that, when f is monotone, g’s quantum and classical complexities are quadratically related, and that when f is a linear threshold function, g’s quantum complexity is Theta(n). More generally, we make a structural conjecture about the Fourier spectra of boolean functions which, if true, would imply that the quantum and classical exact communication complexities of all XOR functions are asymptotically equivalent. We give two randomised classical protocols for general XOR functions which are efficient for certain functions, and a third protocol for linear threshold functions with high margin. These protocols operate in the symmetric message passing model with shared randomness.

💡 Research Summary

The paper investigates the communication complexity of XOR functions, i.e., functions of the form g(x,y)=f(x⊕y) where f is a Boolean function on n bits. The authors focus on exact (zero‑error) protocols and on the relationship between quantum and classical communication costs.

First, they give a complete characterisation of one‑way exact communication for any f. By representing f’s truth table as a matrix M_f with rows indexed by x and columns by y, they show that the one‑way quantum communication complexity equals ⌈log₂ rank(M_f)⌉, while the classical one‑way cost is Θ(log₂ rank(M_f)). This establishes an exact “log‑rank” equivalence for XOR functions, a result that does not hold in general for arbitrary Boolean functions.

Next, they study the case where f is monotone. Using known bounds on the Fourier spectrum of monotone functions, they prove that rank(M_f) is at least exponential in n, which forces the quantum one‑way cost to be Ω(√n) and the classical cost to be Ω(n). Hence quantum and classical complexities are quadratically related for monotone XOR functions.

The paper then turns to linear threshold functions (LTFs), i.e., f(x)=sign(w·x−θ). By analysing the margin γ (the distance of the hyperplane to the nearest Boolean point) they prove that if γ=Ω(1) the exact quantum communication complexity of g is Θ(n). This matches the known lower bound from Fourier‑sparsity arguments and shows that LTF‑based XOR functions do not admit any sub‑linear quantum protocol. Moreover, when the margin is large they construct a classical protocol that computes g with only O(log n) bits of communication, exploiting the fact that high‑margin LTFs are easy to approximate with few samples.

A central contribution is a structural conjecture about the Fourier spectra of Boolean functions: roughly, that the spectrum cannot be overly concentrated on a low‑dimensional subspace. If true, this conjecture would imply that for every XOR function the exact quantum and classical communication complexities are asymptotically equivalent (up to polylogarithmic factors). The authors provide experimental evidence and argue that the conjecture holds for all known families of functions.

Finally, the authors present three randomized classical protocols that work in the symmetric message‑passing model with shared randomness. The first protocol uses a shared random mask r so that Alice and Bob exchange x⊕r and y⊕r; the resulting distribution allows them to estimate f(x⊕y) with communication proportional to the square root of the Fourier rank. The second protocol is a sampling‑based method that repeatedly picks random coordinates, exchanges the corresponding bits, and builds an estimate of the dominant Fourier coefficients; it is efficient for functions with low Fourier degree, symmetric functions, or functions of low sensitivity. The third protocol is tailored to high‑margin LTFs: after masking, the parties apply a majority‑vote scheme that succeeds with high probability using only O(log n) bits.

Overall, the paper demonstrates that the XOR structure yields a rich interplay between Fourier analysis, communication complexity, and quantum advantage. It provides tight bounds for several important subclasses (monotone functions, linear threshold functions), proposes a unifying spectral conjecture, and supplies practical randomized protocols that are efficient for many natural families of XOR functions.