分布式量子性质检验与量子信鸽
Distributed Quantum Property Testing with Quantum Carrier Pigeons
AI总结：
本文提出通信约束下的分布式量子推断框架，针对量子态认证问题，给出共享随机与私密硬币下的副本复杂度上下界，并证明共享随机性的必要性及私密硬币算法的近最优性。
AI中文摘要：
我们提出了一个在通信约束下的分布式量子推断框架。在我们的模型中，$m$个分布式节点各自接收一个未知的$d$维量子态$\rho$的副本，然后通过受约束的单向通信信道与一个中心节点通信，该中心节点旨在推断$\rho$的某个性质。该框架推广了Acharya、Canonne和Tyagi [COLT 2019]提出的经典分布式推断框架，允许使用量子资源，如量子通信和共享纠缠。在此设置下，我们关注量子态认证这一基本问题：给定某个态$\sigma$的完整描述，判定$\rho=\sigma$还是$\\|\rho-\sigma\\|_1\geq \epsilon$。此外，我们关注分布式节点与中心节点之间通信受限的情况：我们假设每个通信信道仅限$n_c$比特和$n_q$量子比特，且$n_c + n_q \leq \log d$。当所有节点可以使用共享随机源时，我们证明分布式态认证的副本复杂度为$\Theta(\frac{d^2}{2^{n_q} 2^{n_c/2}\epsilon^2})$。我们进一步证明共享随机性对于实现上述复杂度是必要的，通过在$\textit{私密硬币}$设置下证明一个$\Omega(\frac{d^3}{4^{n_q} 2^{n_c} \epsilon^2})$的下界。此外，我们开发了一个私密硬币算法，该算法匹配此下界直至$\sqrt{\log d}$因子，表明该复杂度接近最优。总之，我们的工作为通信约束下的分布式量子推断建立了一个通用框架，并刻画了有限通信下分布式态认证的复杂度。
英文摘要：
We introduce a framework for distributed quantum inference under communication constraints. In our model, $m$ distributed nodes each receive one copy of an unknown $d$-dimensional quantum state $ρ$, before communicating via a constrained one-way communication channel with a central node, which aims to infer some property of $ρ$. This framework generalizes the classical distributed inference framework introduced by Acharya, Canonne, and Tyagi [COLT 2019], by allowing quantum resources such as quantum communication and shared entanglement. Within this setting, we focus on the fundamental problem of quantum state certification: Given a complete description of some state $σ$, decide whether $ρ=σ$ or $\|ρ-σ\|_1\geq ε$. Additionally, we focus on the case of limited communication between distributed nodes and the central node: we assume each communication channel is limited to only $n_c$ bits and $n_q$ qubits with $n_c + n_q \leq \log d$. When all nodes can make use of a shared source of randomness, we show that the copy complexity of distributed state certification is $Θ(\frac{d^2}{2^{n_q} 2^{n_c/2}ε^2})$. We further demonstrate that shared randomness is necessary to achieve the above complexity, by proving an $Ω(\frac{d^3}{4^{n_q} 2^{n_c} ε^2})$ lower bound in the $\textit{private-coin}$ setting. Moreover, we develop a private-coin algorithm that matches this bound up to a $\sqrt{\log d}$ factor, showing this complexity is near-optimal. Together, our work establishes a general framework for distributed quantum inference with communication constraints and characterizes the complexity of distributed state certification with limited communication.
