A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity

Xudong Wu; Guangxu Yang; Penghui Yao

Paper

A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity

Abstract

We investigates a model of hybrid classical-quantum communication complexity, in which two parties first exchange classical messages and subsequently communicate using quantum messages. We study the trade-off between the classical and quantum communication for composed functions of the form

, where

and

is an inner product function of

bits. To prove the trade-off, we establish a novel lifting theorem for hybrid communication complexity. This theorem unifies two previously separate lifting paradigms: the query-to-communication lifting framework for classical communication complexity and the approximate-degree-to-generalized-discrepancy lifting methods for quantum communication complexity. Our hybrid lifting theorem therefore offers a new framework for proving lower bounds in hybrid classical-quantum communication models. As a corollary, we show that any hybrid protocol communicating

classical bits followed by

qubits to compute

must satisfy

, where

is the degree of

and

is the block sensitivity of

. For read-once formula

, this yields an almost tight trade-off: either they have to exchange

classical bits or

qubits, showing that classical pre-processing cannot significantly reduce the quantum communication required. To the best of our knowledge, this is the first non-trivial trade-off between classical and quantum communication in hybrid two-way communication complexity.