Quantum computing has been studied over the past four decades based on two computational models of quantum circuits and quantum Turing machines. To capture quantum polynomial-time computability, a new recursion-theoretic approach was taken lately by Yamakami [J. Symb. Logic 80, pp. 1546--1587, 2020] by way of recursion schematic definitions, which constitute six initial quantum functions and three construction schemes of composition, branching, and multi-qubit quantum recursion. By taking a similar approach, we look into quantum logarithmic-time computability and further explore the expressing power of elementary schemes designed for such quantum computation. In particular, we introduce an elementary form of the quantum recursion, called the fast quantum recursion and formulate EQS (elementary quantum schemes) of "elementary" quantum functions. This class EQS captures exactly quantum logarithmic-time computability, represented by BQPOLYLOGTIME. We also demonstrate the separation of BQPOLYLOGTIME from NLOGTIME and PPOLYLOGTIME. As a natural extension of EQS, we further consider an algorithmic procedural scheme that implements the well-known divide-and-conquer strategy. This divide-and-conquer scheme helps compute the parity function but the scheme cannot be realized within our system EQS.

翻译：量子计算在过去四十年中基于量子电路和量子图灵机两种计算模型进行了研究。为了刻画量子多项式时间可计算性，Yamakami [J. Symb. Logic 80, pp. 1546--1587, 2020] 近期采用了一种新的递归理论方法，通过递归方案定义（包括六种初始量子函数以及复合、分支和多量子比特递归三种构造方案）来实现。采用类似方法，我们探讨了量子对数时间可计算性，并进一步研究了为此类量子计算设计的初等方案的表达能力。特别地，我们引入了量子递归的一种初等形式，称为快速量子递归，并构建了“初等”量子函数的EQS（初等量子方案）类。该类EQS恰好刻画了由BQPOLYLOGTIME表示的量子对数时间可计算性。我们还证明了BQPOLYLOGTIME与NLOGTIME及PPOLYLOGTIME的可分离性。作为EQS的自然扩展，我们进一步考虑了一种实现经典分治策略的算法过程方案。该分治方案有助于计算奇偶函数，但无法在我们的EQS系统中实现。