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A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity

Authors Xudong Wu , Guangxu Yang , Penghui Yao

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LIPIcs.ICALP.2026.155.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Quantum communication complexity
Keywords
  • Hybrid communication
  • Generalized discrepancy bound
  • Dual polynomial
  • Query-to-communication lifting

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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 f∘Gⁿ, where f:{0,1}ⁿ → {±1} 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 f∘Gⁿ must satisfy c+q² = Ω(max{deg(f), bs(f)}⋅log n), where deg(f) is the degree of f and {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 Θ(n⋅log n) classical bits or Θ(√n ⋅ log n) 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.

Cite As Get BibTex

Xudong Wu, Guangxu Yang, and Penghui Yao. A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 155:1-155:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.155

Author Details

Xudong Wu
  • State Key Laboratory of Novel Software Technology, Nanjing University, China
Guangxu Yang
  • University of Southern California, Los Angeles, CA, USA
Penghui Yao
  • State Key Laboratory of Novel Software Technology, Nanjing University, China
  • Hefei National Laboratory, China

Funding

Xudong Wu and Penghui Yao were supported by the National Natural Science Foundation of China (Grant No. 62332009 and 12347104), the Quantum Science and Technology-National Science and Technology Major Project (Grant No. 2021ZD0302901), the NSFC/RGC Joint Research Scheme (Grant No. 12461160276), the Natural Science Foundation of Jiangsu Province (No. BK20243060), the Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (No. JYB2025XDXM118), the "111 Center" (No. B26023), and the Fundamental Research Funds for the Central Universities (Grant No. 2026300376).

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