# Query-to-Communication Lifting for BPP

## TL;DR

This paper establishes a fundamental link between query complexity and communication complexity for composed functions, enabling the transfer of complexity separations from query models to communication models.

## Contribution

It introduces a query-to-communication lifting theorem for BPP, connecting randomized decision tree complexity to communication complexity for functions composed with an index gadget.

## Key findings

- Communication complexity of f∘g^n matches the decision tree complexity of f
- Enables transfer of query complexity separations to communication complexity
- Provides a new tool for analyzing randomized communication protocols

## Abstract

For any $n$-bit boolean function $f$, we show that the randomized communication complexity of the composed function $f\circ g^n$, where $g$ is an index gadget, is characterized by the randomized decision tree complexity of $f$. In particular, this means that many query complexity separations involving randomized models (e.g., classical vs. quantum) automatically imply analogous separations in communication complexity.