Sturm-Liouville特征值问题的经典和量子复杂性
Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem
摘要：
研究了Sturm-Liouville问题在经典和量子情形下最小本征值的近似。我们考虑一个单变量Sturm-Liouville特征值问题的非负函数$q$类$C^2([0,1])$和研究的最小数量$n(\e)$的功能评估或查询是必要的计算一个$\e$ -近似的最小特征值。我们证明了$n(\e)=Θ(\e^{-1/2})$在（确定性）最差情况设置中，以及$n(\e)=Θ(\e^{-2/5})$在随机设置中。量子设置提供了{ \it bit}查询的多项式加速和{ \it power}查询的指数加速。位查询类似于Grover算法中使用的Oracle调用，适当地扩展到实值函数。功率查询用于包括相位估计在内的许多问题。它们是通过考虑离散系统在不同时刻的传播子得到的。它们允许我们使用酉矩阵$\exp(\tfrac12 {\rm i}M)$的幂，其中$M$是从Sturm-Liouville微分算子的标准离散化获得的$n\times n$矩阵。在$n$中，利用多个基本量子门（polylog）实现幂查询是一个悬而未决的问题。
英文摘要：
We study the approximation of the smallest eigenvalue of a Sturm-Liouville problem in the classical and quantum settings. We consider a univariate Sturm-Liouville eigenvalue problem with a nonnegative function $q$ from the class $C^2([0,1])$ and study the minimal number $n(\e)$ of function evaluations or queries that are necessary to compute an $\e$-approximation of the smallest eigenvalue. We prove that $n(\e)=Θ(\e^{-1/2})$ in the (deterministic) worst case setting, and $n(\e)=Θ(\e^{-2/5})$ in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it power} queries. Bit queries are similar to the oracle calls used in Grover's algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix $\exp(\tfrac12 {\rm i}M)$, where $M$ is an $n\times n$ matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in $n$ is an open issue.
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