Expressing Power of Elementary Quantum Recursion Schemes for Quantum Logarithmic-Time Computability

Abstract

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 schematic definitions, which constitute a few initial quantum functions and a few construction schemes, including composition, branching, and multi-qubit quantum recursion. By taking a similar step, 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, which helps us capture quantum logarithmic-time computability.