A Lifting Theorem for Hybrid Classical-Quantum Communication Complexity

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.

💡 Research Summary

The paper introduces a hybrid communication model in which two parties first exchange classical messages and then communicate using quantum messages. The authors focus on composed functions of the form (F = f \circ G^{n}), where (f:{0,1}^{n}\to{\pm1}) is an outer Boolean function and (G) is an inner‑product gadget that operates on (\Theta(\log n)) bits. The central contribution is a hybrid lifting theorem that unifies two previously separate lifting paradigms: (i) query‑to‑communication lifting for classical communication, and (ii) approximate‑degree‑to‑generalized‑discrepancy lifting for quantum communication.

In the hybrid protocol, the first (classical) phase transmits (c) bits, thereby partitioning the input space into (2^{c}) rectangles. The second (quantum) phase then uses (q) qubits to finish the computation. The authors prove that for any rectangle (R) produced by the classical phase that satisfies a “density” condition (both sides are 0.99‑dense), the quantum communication cost of computing (F) restricted to (R) is at least (\Omega(\deg_{\varepsilon}(f)\cdot\log n)). The density condition is guaranteed by a standard argument from classical lifting: as each classical bit is sent, the current rectangle is split, and the larger piece is kept; if the rectangle ever becomes sparse, a small set of coordinates (I) is fixed to make the gadget constant on a sub‑rectangle, restoring density. After (c) bits, at most (O(c/\log n)) coordinates are fixed, so the degree of (f) drops by at most that amount.

Combining the density guarantee with the approximate‑degree lifting yields the main trade‑off:

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