带查询的量子环境中的Sturm-Liouville特征值问题和NP-完全问题
The Sturm-Liouville eigenvalue problem and NP-complete problems in the quantum setting with queries
摘要：
我们展示了一些NP-完全以及NP-困难的问题可以减少到斯特姆-刘维特征值问题的量子设置与查询。我们认为权力的查询，这是来自系统的传播演变与汉密尔顿从Sturm-Liouville算子的离散化。我们证明了解决本文所研究的问题所需的幂查询数以及量子比特数是一个低次多项式。利用多项式数量的基本量子门实现幂次查询是一个悬而未决的问题。如果这个问题对于用于Sturm-Liouville特征值问题的幂查询是正解的，那么量子计算机将是一个非常强大的计算设备，允许我们在多项式时间内解决NP完全问题。
英文摘要：
We show how a number of NP-complete as well as NP-hard problems can be reduced to the Sturm-Liouville eigenvalue problem in the quantum setting with queries. We consider power queries which are derived from the propagator of a system evolving with a Hamiltonian obtained from the discretization of the Sturm-Liouville operator. We show that the number of power queries as well the number of qubits needed to solve the problems studied in this paper is a low degree polynomial. The implementation of power queries by a polynomial number of elementary quantum gates is an open issue. If this problem is solved positively for the power queries used for the Sturm-Liouville eigenvalue problem then a quantum computer would be a very powerful computation device allowing us to solve NP-complete problems in polynomial time.
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