AI 中文总结
本文针对Clifford+T模型下的稀疏量子态制备问题，提出了T门计数的亚线性上界，证明了小支撑区域内线性依赖不可避免，缩小了该领域的上下界差距。
AI 中文摘要
本文研究容错量子计算中制备稀疏量子态的代价，以Clifford+T模型中的T门计数衡量。n量子比特态若最多支撑在s个计算基态上，则称为s-稀疏态。对于任意n量子比特态，最优T门计数为Θ(√(2ⁿlog(1/ε))+log(1/ε))，但此前s-稀疏态的最优上界仍为s的线性函数。本文证明，任意n量子比特s-稀疏态可在误差ε内制备，所需T门计数为Õ(min{s, n^(3/4)√s}+√(s log(1/ε))+log(1/ε))，首次实现s足够大时对s的亚线性依赖。该方法基于稀疏布尔函数的支撑感知合成定理，本身也具独立研究价值。本文还证明，对所有0<ε≤1/6且2≤s≤2^(n/2)，稀疏态制备需要Ω(min{s,√(ns)})个T门，表明小支撑区域内对s的线性依赖不可避免，且在该参数范围内大幅缩小了已知上下界的差距。
英文摘要
We study the fault-tolerant cost of preparing sparse quantum states, measured by $T$-count in the Clifford+$T$ model. Here an $n$-qubit state is called $s$-sparse if it is supported on at most $s$ computational-basis states. For arbitrary $n$-qubit states, the optimal $T$-count is $Θ(\sqrt{2^n\log(1/ε)}+\log(1/ε))$, but for $s$-sparse states the best previous upper bounds remained linear in $s$. We show that any $n$-qubit $s$-sparse state can be prepared up to error $ε$ using $\widetilde{O}(\min\{s,\ n^{3/4}\sqrt{s}\}+\sqrt{s\log(1/ε)}+\log(1/ε))$ $T$ gates, giving the first sublinear dependence on $s$ once the support is sufficiently large. Our approach is based on a support-aware synthesis theorem for sparse Boolean functions, which may be of independent interest. We also prove that, for every $0<ε\le 1/6$ and $2\le s\le 2^{n/2}$, sparse-state preparation requires $Ω(\min\{s,\sqrt{ns}\})$ $T$ gates, showing that linear dependence on $s$ is unavoidable in the small-support regime and substantially narrowing the gap between the known upper and lower bounds within this parameter range.
