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 $f\circ G^n$, where $f:\{0,1\}^n\to\{\pm1\}$ 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 $f\circ G^n$ must satisfy $c+q^2=Ω\big(\max\{\mathrm{deg}(f),\mathrm{bs}(f)\}\cdot\log n\big)$, 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 $Θ\big(n\cdot\log n\big)$ classical bits or $\widetildeΘ\big(\sqrt n\cdot\log n\big)$ 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.

翻译：本文研究了一种混合经典量子通信复杂性模型，其中通信双方首先交换经典信息，随后使用量子信息进行通信。针对形如$f\circ G^n$的复合函数（其中$f:\{0,1\}^n\to\{\pm1\}$，$G$为长度为$\Theta(\log n)$比特的内积函数），我们研究了经典通信与量子通信之间的权衡关系。为证明该权衡，我们建立了混合通信复杂性的新型提升定理。该定理统一了此前两个独立的提升范式：经典通信复杂性的查询到通信提升框架，以及量子通信复杂性的近似度到广义差异提升方法。因此，我们的混合提升定理为混合经典量子通信模型的下界证明提供了新框架。作为推论，我们证明任何计算$f\circ G^n$的混合协议若传输$c$个经典比特后传输$q$个量子比特，则必须满足$c+q^2=\Omega\big(\max\{\mathrm{deg}(f),\mathrm{bs}(f)\}\cdot\log n\big)$，其中$\mathrm{deg}(f)$为$f$的度数，$\mathrm{bs}(f)$为$f$的块灵敏度。对于只读公式$f$，该结果给出了近乎紧致的权衡：要么交换$\Theta\big(n\cdot\log n\big)$个经典比特，要么交换$\widetilde{\Theta}\big(\sqrt n\cdot\log n\big)$个量子比特，这表明经典预处理无法显著减少所需的量子通信。据我们所知，这是混合双向通信复杂性中经典通信与量子通信之间的首个非平凡权衡结果。