Quantum and Classical Communication Complexity of Permutation-Invariant Functions

2023-12-31

IEEE Transactions on Information Theory

AI Summary

This paper investigates the quantum and classical communication complexity of permutation-invariant Boolean functions. We establish that, for this class, the quantum communication complexity is at most a $mathrm{polylog}(n)$ factor larger than the randomized communication complexity—achieving quadratic equivalence up to logarithmic factors. Moreover, we confirm both the Log-rank Conjecture and its approximate-rank variant (the Log-approximate-rank Conjecture) for all nontrivial permutation-invariant functions, with bounds that are nearly tight. Our approach integrates spectral analysis of symmetric functions, approximate-rank theory, the quantum query-to-communication lifting framework, and refined techniques for proving communication lower bounds. The results reveal that permutation invariance fundamentally limits quantum speedup to at most polynomial, precluding exponential quantum advantage. Crucially, our work unifies the validity of the Log-approximate-rank Conjecture across both quantum and randomized models, up to a $mathrm{polylog}(n)$ factor.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper gives a nearly tight characterization of the quantum communication complexity of permutation-invariant Boolean functions. With such a characterization, we show that the quantum and randomized communication complexity of permutation-invariant Boolean functions are quadratically equivalent (up to a polylogarithmic factor of the input size). Our results extend a recent line of research regarding query complexity to communication complexity, showing symmetry prevents exponential quantum speedups. Furthermore, we show that the Log-rank Conjecture holds for any non-trivial total permutation-invariant Boolean function. Moreover, we establish a relationship between the quantum/classical communication complexity and the approximate rank of permutation-invariant Boolean functions. This implies the correctness of the Log-approximate-rank Conjecture for permutation-invariant Boolean functions in both randomized and quantum settings (up to a polylogarithmic factor of the input size).

Problem

• Characterizing quantum communication complexity of permutation-invariant Boolean functions
• Showing quadratic equivalence between quantum and classical complexity for symmetric functions
• Proving Log-rank Conjecture holds for total permutation-invariant Boolean functions

Innovation

• Characterizes quantum communication complexity of permutation-invariant functions
• Shows quadratic equivalence between quantum and classical complexities
• Establishes Log-rank Conjecture for permutation-invariant Boolean functions