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arXiv 0910.4145quant-ph
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量子模拟的随机化算法及其下界

Randomized Algorithms and Lower Bounds for Quantum Simulation

Chi Zhang

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6 pages
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摘要：

我们考虑确定性和{ \em随机化}量子算法模拟$e^{-iHt}$的幺正算子$e^{-iA_jt_j}$，$j=1,...,N$的乘积，其中$A_j\in\{H_1,...,H_m\}$，$H=\sum_{i=1}^m H_i$和$t_j > 0$为每个$j$。随机算法是通过混合量子态近似系统最终状态的算法。首先，我们提供了一个方案来限制随机化算法的最终量子态的迹距离。然后，我们展示了一些随机算法，它们具有与某些确定性算法相同的效率，但比它们的同类算法复杂。此外，我们证明了模拟$e^{-iHt}$与错误$\e$的确定性和随机算法至少有$Ω(t^{3/2}\e^{-1/2})$指数。

英文摘要：

We consider deterministic and {\em randomized} quantum algorithms simulating $e^{-iHt}$ by a product of unitary operators $e^{-iA_jt_j}$, $j=1,...,N$, where $A_j\in\{H_1,...,H_m\}$, $H=\sum_{i=1}^m H_i$ and $t_j > 0$ for every $j$. Randomized algorithms are algorithms approximating the final state of the system by a mixed quantum state. First, we provide a scheme to bound the trace distance of the final quantum states of randomized algorithms. Then, we show some randomized algorithms, which have the same efficiency as certain deterministic algorithms, but are less complicated than their opponentes. Moreover, we prove that both deterministic and randomized algorithms simulating $e^{-iHt}$ with error $\e$ at least have $Ω(t^{3/2}\e^{-1/2})$ exponentials.

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