发表机构
Universidad Complutense de Madrid; Instituto de Ciencias Matematicas（马德里康普顿斯大学; 数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对非局域量子计算中的对角酉算子族，证明了若低度多项式策略可实现恒定精度，则至少一个算子的实现需要指数级资源，从而为基于位置的量子密码学提供了更强的下界。
AI 中文摘要
非局域量子计算（NLQC）是指两个合作的远距离玩家通过两轮协议（中间只有一轮同时通信）来实现一个二分酉算子 $U_{AB}$。一个重要的开放问题是：是否存在一个酉算子 $U_{AB}$，其NLQC实现需要额外量子系统的维度为 ${\rm exp}(n^{\Omega(1)})$，其中 $n$ 是系统 $A$ 和 $B$ 的维度。这种指数下界对于基于位置的量子密码学的安全性至关重要，因为对手的能力正是由NLQC描述的。它还与许多其他问题相关，例如通用量子模拟器的最优性，或全息量子引力的预期性质。目前最好的下界仅关于 $n$ 是次线性的。在本文中，我们考虑一族对角取值为 $\pm 1$ 的酉算子 $U_{\varepsilon}$，由布尔超立方体中的元素 $\varepsilon \in \{\pm 1\}^{n^2}$ 索引，即 $U_{\varepsilon}|i\rangle_A|j\rangle_B =\varepsilon_{ij} |i\rangle_A|j\rangle_B$。我们证明：如果对于所有 $\varepsilon$，$U_{\varepsilon}$ 能在NLQC中以恒定精度实现，且相关策略第一轮中对 $\varepsilon$ 的依赖是次数为 $O(n^{\frac{1}{4}-\delta})$ 的多项式，那么至少一个 $U_{\varepsilon}$ 的NLQC实现所需的资源规模为 $\exp\left(\Omega\left(\frac{n^{2\delta}}{(\log n)^2}\right)\right)$。证明基于特定Banach空间的几何性质，即它们的类型常数，以及在其上取值的布尔函数的随机估计。
英文摘要
Non-local quantum computation (NLQC) consists on the implementation of a bipartite unitary $U_{AB}$ by two cooperating distant players by means of a two-round protocol with a single intermediate round of simultaneous communication. It is a major open question to know whether there exists a unitary $U_{AB}$ whose implementation in NLQC requires extra quantum systems with dimensions ${\rm exp}(n^{Ω(1)})$, where $n$ is the dimension of systems $A$ and $B$. This type of exponential lower bound is essential for security of position-based quantum cryptography, since the capabilities of the adversaries are precisely described by NLQC. It has also been connected to many other problems, such as the optimality of universal quantum simulators, or the expected properties of holographic quantum gravity. Currently the best lower bounds are only sublinear in $n$. In this paper we consider the family of diagonal ${\pm 1}$ valued unitaries $U_{\varepsilon}$, indexed by an element of the Boolean hypercube $\varepsilon \in \{\pm 1\}^{n^2}$, i.e. $U_{\varepsilon}|i\rangle_A|j\rangle_B =\varepsilon_{ij} |i\rangle_A|j\rangle_B$. We show that if $U_{\varepsilon}$ can be implemented in NLQC with constant accuracy for all $\varepsilon$, and the dependency on $\varepsilon$ in the first round of the associated strategy is a polynomial of degree $O(n^{\frac{1}{4}-δ})$, then the implementation of at least one $U_{\varepsilon}$ in NLQC requires resources scaling as $\exp\left(Ω\left(\frac{n^{2δ}}{(\log n)^2}\right)\right)$. The proof is based on geometric properties of particular Banach spaces, namely their type constants, together with random estimates of Boolean functions valued on them.
Comments28 pages, 1 figure. No artificial intelligence tools were used in the preparation of this manuscript
