Dyadic-Order Quantum Fractional Transforms: Circuit Constructions and Applications to Hartley and Cosine Transform Families

This paper presents a generalized circuit framework for constructing Shih-type fractionalizations of unitary operators of dyadic order, i.e., operators $U$ satisfying $U^{2^n}=I$. Building upon the architecture of the quantum fractional Fourier transform (QFrFT), we show that fractionalization can be implemented coherently as a weighted superposition of integer powers, $\sum_k c_k(α)U^k$, where the coefficients are generated through an ancilla-domain quantum Fourier transform and a diagonal phase modulation. Under the assumption that controlled implementations of the required powers of $U$ are available, the resulting circuit yields a parameterized family of operators that interpolates the integer powers of $U$ and satisfies the additive property of fractional transforms. As concrete applications, we derive explicit quantum circuit realizations of the quantum fractional Hartley transform (QFrHT) and of the fractional cosine-transform families associated with Types~I and~IV. These constructions demonstrate the versatility of the proposed dyadic-order fractionalization framework for structured operators arising in quantum signal processing.