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arxiv: 2604.09295 · v1 · submitted 2026-04-10 · 🪐 quant-ph · eess.SP

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

Matheus J. A. Oliveira , Israel F. Araujo , Jos\'e R. de Oliveira Neto , Juliano B. Lima This is my paper

Pith reviewed 2026-05-10 17:54 UTC · model grok-4.3

classification 🪐 quant-ph eess.SP
keywords dyadic-orderquantum fractional transformsquantum circuitsHartley transformcosine transformunitary operatorsquantum signal processingfractionalization
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The pith

Quantum circuits fractionalize dyadic-order unitary operators by superposing their integer powers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a circuit framework for fractionalizing unitary operators that have dyadic order, meaning raising them to a power of two yields the identity. It generalizes the quantum fractional Fourier transform by using an ancilla to create a superposition of the operator applied different numbers of times, with coefficients chosen to allow a continuous fractional parameter. This approach ensures the fractional operator satisfies the additive property, where applying it twice with parameters alpha and beta gives the operator with alpha plus beta. The method is then applied to create circuits for the fractional Hartley transform and fractional cosine transforms of certain types. A sympathetic reader would care because it provides a systematic way to implement tunable fractional transforms in quantum computing for signal processing tasks.

Core claim

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. The central construction is a coherent weighted superposition of integer powers, sum c_k(alpha) U^k, generated through an ancilla-domain quantum Fourier transform and a diagonal phase modulation.

What carries the argument

The Shih-type fractionalization circuit for dyadic-order unitaries, which uses an ancilla QFT to generate coefficients for the superposition of powers of U.

Load-bearing premise

The circuit works only if controlled implementations of the required powers of the unitary operator U are available.

What would settle it

Simulating the proposed circuit for the fractional Fourier transform case and checking whether it matches known quantum fractional Fourier transform behavior would confirm or refute the construction.

Figures

Figures reproduced from arXiv: 2604.09295 by Israel F. Araujo, Jos\'e R. de Oliveira Neto, Juliano B. Lima, Matheus J. A. Oliveira.

Figure 1
Figure 1. Figure 1: The controlled-NOT gate (CNOT) at the top and a controlled-Hadamard gate (CH) at the bottom. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuit for the QFrFT. The circuit uses two ancilla qubits, initialized in the state |00⟩, which control the application of different powers of the Fourier transform. Let |u⟩ denote the input quantum state upon which the fractional Fourier transform acts, consisting of q qubits. Consequently, the total system requires a register of size q + 2 qubits. The initial state is |ψ0⟩ = |00⟩|u⟩. The first o… view at source ↗
Figure 3
Figure 3. Figure 3: Diagram representing the quantum circuit for the generalized fractional transform. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagram representing the quantum circuit for the fractional transform of an involution (QFrIn). [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Diagram representing the quantum circuit for the QHT. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Diagram representing the quantum circuit for the QCT-I. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Diagram representing the quantum circuit for the Pn gate [9]. Since the implemented operator is the direct sum DCTI N+1 ⊕ DSTI N−1 , the cosine and sine components are not separated by a selector qubit. For this reason, this construction is more accurately described as a quantum cosine–sine transform of Type I (QCST-I). However, we will keep using its the usual name for compatibility with the literature. 3… view at source ↗
Figure 8
Figure 8. Figure 8: Diagram representing the quantum circuit for the QCT-IV. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

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, $\sum_k c_k(\alpha)U^k$, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript presents a generalized circuit framework for Shih-type fractionalizations of unitary operators U of dyadic order (U^{2^n}=I). Building on the quantum fractional Fourier transform architecture, it realizes the fractional operator as a coherent superposition sum_k c_k(alpha) U^k via an ancilla-register quantum Fourier transform followed by diagonal phase modulation. Under the explicit assumption that controlled implementations of the required powers U^k are available as black-box primitives, the construction interpolates the integer powers and satisfies the additive property of fractional transforms. Explicit quantum circuit realizations are derived for the quantum fractional Hartley transform and for the fractional cosine-transform families of Types I and IV.

Significance. If the constructions hold, the work supplies a systematic, assumption-conditioned template for fractionalizing a broad class of finite-order unitaries in quantum signal processing. The dyadic-order restriction ensures periodicity and well-defined interpolation on the circle, while the explicit gate decompositions for the Hartley and cosine cases offer immediately usable building blocks once the controlled-power primitives are supplied. This extends the QFrFT paradigm without introducing new free parameters or circular definitions.

major comments (1)
  1. [§3] §3, general construction: the claim that the additive property follows directly from the homomorphism of the phase factors is correct in outline, but the manuscript does not supply an explicit verification (e.g., the convolution identity for the coefficients c_k) for a small dyadic order such as n=2; this verification is load-bearing for the central claim that the family satisfies the fractional-transform axioms.
minor comments (3)
  1. [§2] Notation for the ancilla register size and the precise mapping of the phase gates to the fractional parameter alpha should be stated once in a single equation rather than repeated across sections.
  2. [§4] The resource counts (gate depth, ancilla qubits) for the QFrHT and cosine constructions are not tabulated; a comparison table with the standard QFrFT would clarify the overhead.
  3. [§5] A few sentences on how the controlled-U^k primitives might be compiled for the specific Hartley and cosine operators (e.g., via their known factorizations) would strengthen the practicality claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation for minor revision. We address the single major comment below and will incorporate the requested explicit verification into the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3, general construction: the claim that the additive property follows directly from the homomorphism of the phase factors is correct in outline, but the manuscript does not supply an explicit verification (e.g., the convolution identity for the coefficients c_k) for a small dyadic order such as n=2; this verification is load-bearing for the central claim that the family satisfies the fractional-transform axioms.

    Authors: We agree that an explicit verification for the smallest non-trivial case strengthens the exposition. Although the additive property is a direct consequence of the homomorphism property of the phase factors generated by the ancilla QFT (i.e., the eigenvalues multiply under composition), we will add a concise explicit check for n=2 in the revised §3. This will include the convolution identity for the coefficients c_k(α), confirming that the composition of two fractional operators with parameters α and β yields the operator with parameter α+β (modulo the dyadic order). The addition will be placed immediately after the general construction and before the applications to Hartley and cosine transforms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central construction is a direct circuit template: an ancilla-register QFT followed by diagonal phase gates realizing the superposition sum c_k(alpha) U^k, with the additive property following from the standard homomorphism of the phase coefficients under alpha addition. This holds under the explicit precondition that controlled-U^k primitives are available as black boxes. The Hartley and cosine applications are explicit gate decompositions once those primitives are granted. The reference to prior QFrFT architecture is background scaffolding, not a load-bearing self-citation or definitional reduction; no parameters are fitted, no uniqueness theorem is invoked, and no equation reduces to its own input by construction. The argument is therefore independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the domain assumption that controlled implementations of the integer powers of the base unitary U are already available; no free parameters or new invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Controlled implementations of the required powers of U are available
    Explicitly stated in the abstract as the precondition for the circuit to produce the fractional operator.

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Reference graph

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