Dyadic-Order Quantum Fractional Transforms: Circuit Constructions and Applications to Hartley and Cosine Transform Families
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arXiv
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| Format: | Preprint |
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2026
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| author | Oliveira, Matheus J. A. Araujo, Israel F. Neto, José R. de Oliveira Lima, Juliano B. |
| author_facet | Oliveira, Matheus J. A. Araujo, Israel F. Neto, José R. de Oliveira Lima, Juliano B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09295 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dyadic-Order Quantum Fractional Transforms: Circuit Constructions and Applications to Hartley and Cosine Transform Families Oliveira, Matheus J. A. Araujo, Israel F. Neto, José R. de Oliveira Lima, Juliano B. Quantum Physics Signal Processing 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. |
| title | Dyadic-Order Quantum Fractional Transforms: Circuit Constructions and Applications to Hartley and Cosine Transform Families |
| topic | Quantum Physics Signal Processing |
| url | https://arxiv.org/abs/2604.09295 |