发表机构
Center on Frontiers of Computing Studies, Peking University; School of Computer Science, Peking University; State Key Laboratory of Novel Software Technology, Nanjing University; Hefei National Laboratory; Tencent Quantum Laboratory（北京大学前沿计算研究中心; 北京大学计算机学院; 南京大学软件新技术国家重点实验室; 合肥国家实验室; 腾讯量子实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对稀疏QROM的T计数问题，证明了渐近最优T计数界，推导了稀疏态制备和稀疏矩阵块编码的匹配T计数界，为相关量子电路设计提供了理论支撑。
AI 中文摘要
许多量子算法需要对经典数据进行相干访问，通常用量子只读存储器（QROM）建模。本文启动对稀疏QROM的T计数研究，其中2ⁿ个地址里仅s个存储非零数据。我们证明了渐近最优T计数界为Θ(√(sm)+√(sn))，其对支撑大小s和消息长度m呈平方根依赖。我们的上界采用多级哈希方案，而下界将稀疏QROM归约为态制备，并使用自适应Clifford+T电路的计数论证，因此下界即使允许 mid-circuit 测量和经典控制操作也成立。作为应用，我们得到s-稀疏态制备的匹配T计数界为Θ(√(sn)+√(s log(1/ε))+log(1/ε))，以及s-稀疏矩阵块编码的匹配T计数界为Θ(√(2ⁿ sn)+√(2ⁿ s log(s/ε_BE))+log(s/ε_BE))，其中ε和ε_BE分别为态制备和块编码的精度。
英文摘要
Many quantum algorithms require coherent access to classical data, often modeled by quantum read-only memory (QROM). We initiate the study of the $\mathrm{T}$ count of sparse QROM, in which only $s$ of the $2^n$ addresses store nonzero data. We prove that the optimal $\mathrm{T}$ count is $Θ\left(n+\min\left\{s,\sqrt{s\left(m+\log(2^{n+1}/s)\right)}\right\}\right)$. Our upper bounds use a multilevel hashing scheme, while our lower bounds reduce sparse QROM to state preparation and use counting arguments for adaptive Clifford+$\mathrm{T}$ circuits. The lower bounds thus hold even when mid-circuit measurements and classically controlled operations are allowed. As applications, we obtain matching $\mathrm{T}$-count bounds $Θ\left(\min\left\{s,\sqrt{s\log(2^{n+1}/s)}\right\} +\sqrt{s\log(1/\varepsilon)}+\log(1/\varepsilon)\right)$ for $s$-sparse state preparation and $Θ\left(\sqrt{2^n s\left(n+\log(1/\varepsilon_{\rm BE})\right)} +\log(1/\varepsilon_{\rm BE})\right)$ for block encoding of $s$-sparse matrices, where $\varepsilon$ and $\varepsilon_{\rm BE}$ are the precision of state preparation and block encoding, respectively.
Comments50 pages, 2 tables
