Exploiting recursive structures for the design of novel quantum primitives

The advent of fault-tolerant quantum computers marks a significant milestone, yet the development of practical quantum algorithms remains a critical challenge. Effective quantum algorithms are essential for leveraging the power of quantum computers, and their design is often non-intuitive. This paper addresses the issue of generating novel quantum primitives by focusing on recursive circuits. We explore the recursive circuit structures prevalent in existing quantum algorithms and demonstrate how these structures can be exploited to design new, potentially advantageous quantum algorithms. We base our discussion on the quantum Fourier transform (QFT), which is a primitive that is widely used in quantum algorithms. We show that the recursive structure in well-established fast classical transforms forms a fruitful bridge with quantum algorithms, enabling the design of novel quantum primitives and the discovery of new discrete numerical transforms. The discussion is split into two complementary parts, the forward and the reverse direction, in which existing classical transforms are implemented using polynomial-time quantum circuits and recursive circuits are used to find novel non-sparse classical transforms with guaranteed quantum speedup, respectively. We comment on the potential impact on quantum algorithms, numerical analysis, and signal processing.