发表机构
United Arab Emirates University; University of Edinburgh（阿联酋大学; 爱丁堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一族弱测量循环量子振幅放大算法，通过可调弱测量与Grover旋转交错，在保持$\Theta(1/\sqrt{p})$搜索尺度的同时，实现预言机节省和自适应搜索，优于标准Grover常数。
AI 中文摘要
我们提出一族量子振幅放大算法，这些算法在循环中将普通的Grover旋转与可调弱测量交错进行。一次成功的测量会得到目标态并终止算法，而失败后，循环从测量后的状态继续，无需重新启动。我们将期望Grover迭代次数中的主要成本表示为当 $p \to 0$ 时的渐近形式。对于已知的目标概率 $p$，在接近角度 $\pi/4$ 的状态下，精确的弱测量条件循环使用 $(\pi/8+1/4+o(1))/\sqrt{p}$ 次迭代，这优于标准Grover常数 $\pi/4$ 以及优化后的重启Grover，并且与我们在连续角度分析中的相应下确界相匹配。当仅知道下界 $0 < p_0 \leq p < 1/2$ 时，一个离散Lyapunov方程给出了对于每个固定测量强度的精确期望预言机成本，以及一个闭式的最优强度。使用从 $p_0$ 选择的强度，期望的Grover迭代次数至多为 $\frac{1}{2\sqrt{2}}(\frac{1}{\sqrt{p}}+\frac{1}{\sqrt{p_0}})$，其中期望成本随实际 $p$ 的增大而降低，并且在承诺边界 $p = p_0$ 处给出前导常数 $1/\sqrt{2}$。对于完全未知的 $p$，我们确定测量强度 $\kappa(t)=\Theta(1/t)$ 是此处考虑的规则调度中的临界尺度，并分析 $\kappa_b(t)=\min\{1/2,b/t\}$。对于每个固定的 $b > 2$，我们严格推导出一个闭式的Gamma函数表达式 $C(b)$，给出 $(C(b)/2+o(1))/\sqrt{p}$ 的期望迭代次数。对该显式表达式的数值最小化得到 $b \approx 5.2$ 和 $C(b)/2 \approx 1.01$。结果表明，弱测量保持了 $\Theta(1/\sqrt{p})$ 的搜索尺度，同时增加了一个显式的定点控制机制和可证明的预言机节省。
英文摘要
We present a family of quantum amplitude amplification algorithms that interleave ordinary Grover rotations with tunable weak measurements in loops. A successful measurement yields the target state and stops the algorithm, while after a failure, the loop resumes from the post-measurement state without restarting. We express the leading costs in expected Grover iterations as $p \to 0$. For a known target probability $p$, an exact weak measurement-conditioned loop for states near the angle $π/4$ uses $(π/8+1/4+o(1))/\sqrt{p}$ iterations, improving on the standard Grover constant $π/4$ and on optimised restart Grover, and matching the corresponding infimum in our continuous-angle analysis. When only a lower bound $0 < p_0 \leq p < 1/2$ is available, a discrete Lyapunov equation gives the exact expected oracle cost for every fixed measurement strength and a closed-form optimal strength. Using the strength selected from $p_0$, the expected number of Grover iterations is at most $\frac{1}{2\sqrt{2}}(\frac{1}{\sqrt{p}}+\frac{1}{\sqrt{p_0}})$, where the expected cost decreases as the actual $p$ increases, and gives the leading constant $1/\sqrt{2}$ at the promise boundary $p = p_0$. For completely unknown $p$, we identify measurement strength $κ(t)=Θ(1/t)$ as the critical scale within the regular schedules considered here and analyse $κ_b(t)=\min\{1/2,b/t\}$. For each fixed $b > 2$, we rigorously derive a closed-form Gamma-function expression $C(b)$ giving $(C(b)/2+o(1))/\sqrt{p}$ expected iterations. Numerical minimisation of this explicit expression gives $b \approx 5.2$ and $C(b)/2 \approx 1.01$. The results show that weak measurements preserve the $Θ(1/\sqrt{p})$ search scale while adding an explicit fixed-point control mechanism and provable oracle savings.
Comments41 pages, 3 figures
