A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity

2025-11-21

AI Summary

This work investigates communication cost trade-offs for functions of the form $f circ G^n$, where $f$ is a Boolean function and $G$ is the inner-product gadget, in the hybrid classical–quantum communication model. Focusing on two-way “classical-then-quantum” hybrid protocols, we establish the first unified hybrid lifting theorem, integrating classical query complexity and quantum approximate degree via generalized discrepancy analysis under imbalanced distributions. We obtain the first nontrivial classical–quantum communication lower bound in the two-way hybrid setting: $c + q^2 = Omega(max{deg(f), mathrm{bs}(f)} cdot log n)$, where $c$ and $q$ denote classical and quantum communication costs, respectively. In particular, for read-once formulas $f$, we achieve nearly tight bounds: $Theta(n log n)$ classical bits or $widetilde{Theta}(sqrt{n} log n)$ quantum bits are necessary. Our results demonstrate that classical preprocessing cannot substantially reduce quantum communication requirements, revealing an inherent resource bottleneck in hybrid models.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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 $fcirc G^n$, where $f:{0,1}^n o{pm1}$ and $G$ is an inner product function of $Θ(log n)$ 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 $c$ classical bits followed by $q$ qubits to compute $fcirc G^n$ must satisfy $c+q^2=Ωig(max{mathrm{deg}(f),mathrm{bs}(f)}cdotlog nig)$, where $mathrm{deg}(f)$ is the degree of $f$ and $mathrm{bs}(f)$ is the block sensitivity of $f$. For read-once formula $f$, this yields an almost tight trade-off: either they have to exchange $Θig(ncdotlog nig)$ classical bits or $widetildeΘig(sqrt ncdotlog nig)$ 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.

Problem

• Establishes a lifting theorem for hybrid classical-quantum communication complexity models
• Studies trade-offs between classical and quantum communication for composed functions
• Proves first non-trivial classical-quantum communication trade-off in two-way protocols

Innovation

• Novel lifting theorem for hybrid communication complexity
• Unifies classical and quantum lifting paradigms
• Proves lower bounds in hybrid classical-quantum models