$f$-路由的计算界

中文摘要 AI 辅助

$f$-路由协议是量子位置验证的主要候选方案（Kent、Munro和Spiller，2011），但针对显式函数的安全性保证仍然有限。我们证明了针对均匀攻击者的无条件资源下界；我们的新技术绕过了先前工作中核心的通信复杂度下界，这些下界本质上至多随输入长度线性增长。我们证明，对于输入长度$n$和足够小的常数$\epsilon>0$，一个均匀生成的策略使用$q$个量子比特且描述长度为$\mathrm{poly}(q)$，在每个输入上成功概率至少为$1-\epsilon$，则蕴含关于$f$的以下计算界：1. 如果策略是任意量子信道，则$f\in\mathrm{QSZK}(\mathrm{poly}(nq))$，其中$\mathrm{QSZK}(T)$是拥有量子统计零知识证明的语言类，验证者运行时间为$T$（模拟器运行时间为$\mathrm{poly}(T)$）。2. 如果策略是$q$个量子比特上的显式泡利稀疏酉算子，且至多有$s$个非零泡利系数，则$f\in\mathrm{DTIME}(\mathrm{poly}(nqs))$。3. 如果策略是使用至多$t$个魔法门的Clifford+T电路，则$f\in\mathrm{DTIME}(\mathrm{poly}(nq2^t))$。时间和空间层级结构随后产生在计算限制下对多项式甚至拟多项式量子比特$q$安全的显式函数。这些界超过了Bluhm、Christandl和Speelman（2022）针对内积函数$f=\mathrm{IP}$的$q\le\log n$界，代价是限制对抗性计算并增加诚实求值复杂度。

英文摘要

The $f$-routing protocol is a leading candidate for quantum position verification (Kent, Munro, and Spiller, 2011), but security guarantees for explicit functions remain limited. We prove unconditional resource lower bounds for uniform attackers; our new techniques bypass communication-complexity bounds central to previous works, which are inherently at most linear in the input length. We show that, for input length $n$ and sufficiently small constant $ε>0$, a uniformly generated strategy using $q$ qubits and having description length $\mathrm{poly}(q)$, with success probability at least $1-ε$ on every input, implies the following computational bounds on $f$: 1. If the strategies are arbitrary quantum channels, then $f\in\mathrm{QSZK}(\mathrm{poly}(nq))$, where $\mathrm{QSZK}(T)$ is the class of languages having quantum statistical zero knowledge proofs in which the verifier runs in time $T$ (and the simulator in time $\mathrm{poly}(T)$). 2. If the strategies are explicit Pauli-sparse unitaries on $q$ qubits that have at most $s$ nonzero Pauli coefficients, then $f\in\mathrm{DTIME}(\mathrm{poly}(nqs))$. 3. If the strategies are Clifford+T circuits using at most $t$ magic gates, then $f\in\mathrm{DTIME}(\mathrm{poly}(nq2^t))$. Time and space hierarchies then yield explicit functions secure against polynomial and even quasipolynomial qubits $q$ under our computational restrictions. These bounds exceed the $q\le\log n$ bound of Bluhm, Christandl, and Speelman (2022) for inner product function $f=\mathrm{IP}$, at the cost of restricting adversarial computation and increasing honest evaluation complexity.

发表机构

• Cornell University（康奈尔大学）

机构由 AI 辅助整理，请以论文原文为准。