Quantum-Classical Equivalence for AND-Functions
AND函数的量子-经典等价性
Sreejata Kishor Bhattacharya, Farzan Byramji, Arkadev Chattopadhyay, Yogesh Dahiya, Shachar Lovett
AI总结 通过证明任意布尔函数f的AND组合f∘AND_2的有界误差量子通信复杂性与经典确定性通信复杂性多项式相关(至多对数因子),解决了关于AND函数量子通信优势的长期猜想。
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量子通信复杂性中的一个主要开放问题是,对于计算全布尔函数,量子协议是否可能比经典协议指数级更高效;普遍猜想认为不可能。在一项开创性工作中,Razborov (2002) 通过证明当外层函数$f$对称时,形如$$ F(x,y) = f(x_1 \land y_1, \ldots, x_n \land y_n) $$的AND函数的有界误差量子与经典通信复杂性是多项式相关的,从而解决了该问题。此后,将此结果推广到所有AND函数一直悬而未决,并被多位作者提出。在本工作中,我们以强有力的方式解决了这个问题。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性是多项式相关的,至多相差$n$的多对数因子。我们通过证明两者——至多多项式损失——均由$f$的德摩根稀疏性的对数刻画来证明这一点。我们的结果建立在Chattopadhyay、Dahiya和Lovett (2025) 关于非稀疏布尔函数结构刻画的最新工作之上,我们将其推广以解决一般AND函数的猜想。
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.