通信复杂度中最优且可验证的量子优势
Optimal and Verifiable Quantum Advantages in Communication Complexity
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中文总结 AI 辅助
我们建立了搜索问题中量子通信复杂度与经典的最优分离，引入可验证的 Pelagic Fourier Fishing 变体，并构造了 Clifford 层级协议族，实现渐近最优的量子优势，且可适应于可验证的近期实验。
中文摘要 AI 辅助
我们针对搜索问题建立了通信复杂度中量子与经典的最优分离。我们引入了一个名为 Pelagic Fourier Fishing 的全搜索问题，并证明它允许一个 $n$ 量子比特的量子单向协议，而任何随机双向协议需要 $\u03a9(2^n)$ 比特的通信。随后，我们引入了该问题的一个变体，其解可以在多项式时间内验证。该变体同样允许一个 $n$ 量子比特的量子单向协议，而任何随机单向协议需要 $\u03a9(2^n)$ 比特的通信。我们还构造了一个高效可验证的全搜索问题族，对于每个固定的 $d \ge 2$，实现了量子单向与随机单向通信之间 $n$ 对 $\Omega_d(n^d)$ 的分离。在量子协议中，Alice 使用 Clifford 层级第 $d$ 层的单个酉算子准备她的消息，Bob 执行 Clifford 测量。在此对 Alice 消息的限制下，该分离是渐近最优的。对于 $d = 2$，Alice 的消息是稳定子态，Bob 的测量是 Clifford 的，因此该协议不使用魔法态，却实现了最优的二次量子优势。最后，我们讨论了如何将这些分离适应于近期量子优势实验，其中所展示的优势既无条件又可高效验证。
英文摘要
We establish optimal quantum-classical separations in communication complexity for search problems. We introduce a total search problem called Pelagic Fourier Fishing and show that it admits an $n$-qubit quantum one-way protocol, whereas every randomized two-way protocol requires $Ω(2^n)$ bits of communication. We then introduce a variant of this problem whose solutions can be verified in polynomial time. This variant also admits an $n$-qubit quantum one-way protocol, while every randomized one-way protocol requires $Ω(2^n)$ bits of communication. We also construct a family of efficiently verifiable total search problems achieving an $n$ versus $Ω_d(n^d)$ separation between quantum one-way and randomized one-way communication for every fixed $d \ge 2$. In the quantum protocol, Alice prepares her message using a single unitary from the $d$th level of the Clifford hierarchy, and Bob performs a Clifford measurement. This separation is asymptotically optimal under this restriction on Alice's message. For $d = 2$, Alice's message is a stabilizer state and Bob's measurement is Clifford, so the protocol uses no magic, yet achieves an optimal quadratic quantum advantage. Finally, we discuss how these separations can be adapted to near-term quantum advantage experiments in which the demonstrated advantage is both unconditional and efficiently verifiable.
发表机构
- UT Austin（德克萨斯大学奥斯汀分校）
- Google Quantum AI（谷歌量子人工智能）
- IBM Quantum（IBM 量子）
- Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。
