On the communication complexity of XOR functions

On the Communication Complexity of XOR Functions

The paper "On the communication complexity of XOR functions" by Ashley Montanaro and Tobias J. Osborne provides a comprehensive investigation into the classical and quantum communication complexities of XOR functions. The study explore both deterministic and randomized protocols, offering significant contributions to our understanding of communication complexity in various settings.

Overview

An XOR function is defined as ( g(x,y) = f(x \oplus y) ), where ( f ) is a boolean function on ( n ) bits and ( \oplus ) denotes the bitwise XOR operation. The authors thoroughly characterize the one-way communication complexity for XOR functions within quantum and classical contexts under exact protocols. Additionally, they establish insights into the complexity differences when ( f ) is a monotone function or a linear threshold function (LTF).

Key Concepts and Results

One-Way Communication Complexity

For any XOR function, the paper demonstrates that both classical and quantum one-way deterministic communication complexities are equal to the Fourier dimension of ( f ), specifically ( D^{cc,1}(g) = Q_E^{cc,1}(g) = \dim f ). This result leverages the Fourier analysis of boolean functions, aligning with the concept of Fourier dimension which represents the minimal subspace dimension where the Fourier spectrum of ( f ) resides.

Separation Between One-Way and Two-Way Communication Complexity

The authors establish an exponential separation between one-way quantum and two-way classical communication complexities for certain XOR functions. For instance, an XOR function derived from an addressing function demonstrates that while two-way deterministic complexity can be ( O(m) ), its one-way quantum complexity can be as high as ( \Omega(2^m) ).

Parity Decision Trees and Fourier Spectra

The paper posits that deterministic two-way communication complexity of XOR functions can be upper bounded using parity decision tree complexity. They introduce a conjecture that, if proven, would suggest the Fourier spectra of boolean function subsets intersect significantly, which in turn would indicate that classical and quantum communication complexities for XOR functions are asymptotically equivalent.

Randomized Protocols

Two key results stand out in the discussion of randomized protocols for XOR functions:

1. Functions with low spectral norms have efficient randomized communication protocols with ( R^{|,pub}(g) = O(|\hat{f}|_1²⁾ ).
2. Functions closely resembling parity functions, or those taking a specific value on limited inputs, have protocols with ( R^{|,pub}(g) = O(\log m) ).

Monotone Functions and LTFs

The paper highlights that for monotone XOR functions, classical deterministic communication complexity is quadratically related to quantum complexity, formally ( D^{cc}(g) = O(Q_E^{cc}(g)2) ). For XOR functions derived from LTFs that depend on ( n ) bits, the deterministic communication complexities scale linearly with ( n ).

Implications and Future Work

Understanding the communication complexity characteristics of XOR functions has profound implications for theoretical computer science, particularly in the context of differentiating between classical and quantum computation resources. The structural conjectures presented in the paper, once proven, could significantly advance our understanding of the Fourier analysis of boolean functions.

The introduction of parity decision tree complexity provides a new lens for examining communication complexity problems. Future work could include proving the mentioned conjecture about Fourier spectra structure, which would resolve questions about polynomial relationships between quantum and classical complexities.

In practice, optimizing communication protocols for specific functions, especially LTFs, could see applications in distributed computing and information theory, where efficient communication is critical.

In conclusion, Montanaro and Osborne's work on the communication complexity of XOR functions opens up several research avenues and provides a solid foundation for further exploration within the landscape of both classical and quantum communication complexity.