我們給出了一個深度有限的量子近似計數演算法。該演算法接收輸入 $\epsilon, \delta, T_{max}$，並輸出 $\Tilde{K}$，即標記項數量的估計值，滿足 $\epsilon K \geq |\Tilde{K} - K|$，其中 $K$ 是標記項的數量，失敗概率小於 $\delta$，且電路深度小於 $T_{max}$。當 $T_{max} \leq \frac{1}{\epsilon}\sqrt{\frac{N}{K}}$ 時，我們的演算法使用 $O\left( \frac{1}{T_{max} \epsilon^2} \frac{N}{K}\right)$ 查詢；當 $T_{max} \geq \frac{1}{\epsilon}\sqrt{\frac{N}{K}}$ 時，使用 $O\left(\frac{1}{\epsilon}\sqrt{\frac{N}{K}}\right)$ 查詢。

我們還給出了深度限制的近似計數的匹配查詢下界 $ \Omega \left( \frac{N}{T_{max} \epsilon^2 K} \right)$，在以下限制條件下：演算法僅運行到最大深度 $T_{max}$ 的 Grover 算子，且 $T_{max}$ 滿足 $(2T_{max}+1) \theta \leq
i/2$。

We give an depth-limited quantum approximate counting algorithm. The algorithm takes input $\epsilon,\delta$, $T_{max}$ and outputs $\Tilde{K}$, an estimate of the number of marked items, satisfying $\epsilon K \geq |\Tilde{K} - K|$, where $K$ is the number of marked items, with failure probability less than $\delta$ and circuit depth less than $T_{max}$. Our algorithm uses $O\left( \frac{1}{T_{max} \epsilon^2} \frac{N}{K}\right)$ queries when $T_{max} \leq \frac{1}{\epsilon}\sqrt{\frac{N}{K}}$ and $O\left(\frac{1}{\epsilon}\sqrt{\frac{N}{K}}\right)$ queries when $T_{max} \geq \frac{1}{\epsilon}\sqrt{\frac{N}{K}}$.
We also give a matching query lower bound of depth-limited approximate counting of $ \Omega \left( \frac{N}{T_{max} \epsilon^2 K} \right)$ under the restrictions that the algorithm only runs Grover iterations to the maximum depth $T_{max}$, and $T_{max}$ satisfies $(2T_{max}+1) \theta \leq
i/2$.

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