A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form $$ F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n), $$ when the outer function $f$ is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function $f$, the bounded-error quantum and classical deterministic communication complexities of the function $f \circ \mathrm{AND}_2$ are polynomially related, up to polylogarithmic factors in $n$. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of $f$. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.

翻译：量子通信复杂度中的一个重大未解决问题是：对于计算全布尔函数，量子协议能否在效率上指数级优于经典协议？主流猜想认为不能。在开创性工作中，Razborov (2002) 针对形式为 $$ F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n) $$ 的AND函数（当外层函数 $f$ 为对称函数时），通过证明其有界误差量子与经典通信复杂度呈多项式相关，解决了该问题。此后，将此结果推广至所有AND函数的问题一直悬而未决，并被多位学者提出。本文以强形式解决了该问题。我们证明：对于任意布尔函数 $f$，函数 $f \circ \mathrm{AND}_2$ 的有界误差量子与经典确定性通信复杂度在 $n$ 的多对数因子范围内呈多项式相关。这一结论的推导基于两者——在多项式损失内——均由 $f$ 的德摩根稀疏性的对数所刻画。我们的结果建立在Chattopadhyay、Dahiya与Lovett (2025) 关于非稀疏布尔函数结构刻画的最新工作之上，并进一步将其扩展以解决一般AND函数的相关猜想。