Quantum Fourier transform (QFT) is an important part of many quantum algorithms. However, there are few reports on quantum fractional Fourier transform (QFRFT). The main reason is that the definitions of fractional Fourier transform (FRFT) are diverse, while some definitions do not include unitarity, which leads to some studies pointing out that there is no QFRFT. In this paper, we first present a reformulation of the weighted fractional Fourier transform (WFRFT) and prove its unitarity, thereby proposing a quantum weighted fractional Fourier transform (QWFRFT). The proposal of QWFRFT provides the possibility for many quantum implementations of signal processing.
Quantum Fourier transform (QFT) transformation plays a very important role in the design of many quantum algorithms. Fractional Fourier transform (FRFT), as an extension of the Fourier transform, is particularly important due to the design of its quantum algorithm. In this paper, a new reformulation of the weighted fractional Fourier transform (WFRFT) is proposed in order to realize quantum FRFT; however, we found that this reformulation can be applied to other transformations, and therefore, this paper presents the weighted fractional Hartley transform (WFRHT). For the universality of application, we further propose a general weighted fractional-order transform (WFRT). When designing the quantum circuits, we realized the quantum WFRFT via QFT and quantum phase estimation (QPE). Moreover, after extending our design to the WFRHT, we were able to formulate the quantum WFRHT. Finally, in accordance with the research results, we designed the quantum circuit of the general WFRT, and subsequently proposed the quantum WFRT. The research in this paper has great value as a reference for the design and application of quantum algorithms.
We present a new number theoretic definition of discrete fractional Fourier transform (DFrFT) . In this approach the DFrFT is defined as the
N×N dimensional unitary representation of the generator of the arithmetic rotational group
SO2[ZN], which is the finite set of
modN integer,
2×2 matrices acting on the points of the discrete toroidal phase space lattice
ZN×ZN, preserving the euclidean distance
modN. We construct explicitly, using techniques of the Finite Quantum Mechanics (FQM), the
pn dimensional unitary matrix representation of the group
SO2[Zpn] and especially we work out in detail the one which corresponds to the generator. This is our definition of the arithmetic fractional Fourier transform (AFrFT). Following this definition, we proceed to the construction of efficient quantum circuits for the AFrFT, on sets of
n p-dimensional qudits with
p a prime integer, by introducing novel quantum subcircuits for diagonal operators with quadratic phases as well as new quantum subcircuits for multipliers by a constant. The quantum subcircuits that we introduce provide a set capable to construct quantum circuits for any element of a more general group, the group of Linear Canonical Transformations (LCT),
SL2[ZN] of the toroidal phase space lattice. As a byproduct, extensions of the diagonal and multiplier quantum circuits for both the qudit and qubit case are given, which are useful alone in various applications. Also, we analyze the depth, width and gate complexity of the efficient AFrFT quantum circuit and we estimate its gate complexity which is of the order
O(n2), its depth which is of the order
O(n) with depth
n, while at the same time it has a structure permitting local interactions between the qudits.
The Quantum Fourier Transform (QFT)andQuantumPhaseEstimation (QPE)are twofundamental algorithmic frameworks in quantum computation, and they provide the mathematical basis of many exponential-speedup quan tum algorithms, such as Shor’s factoring algorithm and quantum simulation techniques. In this work we present a detailed theoretical and computational study of the Quantum Fourier Transform and its integration in the Quan tum Phase Estimation. The discussion begins with the linear algebraic structure of the Hilbert spaces, the unitary transformations, and the complex vector representations that enable quantum interference and superposition. The QFT is studied as the unitary transform of computational basis states in the amplitudes in the frequency domain by phase rotations and tensor-product decompositions. The efficient circuit implementation in terms of Hadamard and controlled phase gates of this transform is discussed. The advantage of the transform in terms of polynomial time complexity over the classical discrete Fourier transform in some quantum scenarios is discussed.The QFT is implemented and the Quantum Phase Estimation algorithm is proposed to obtain the eigenphase information of the unitary operators. In the report the mathematical representation of QPE is derived by using the controlled unitary evolution, the inverse Fourier transforms and the measurement theory in finite dimensional Hilbert spaces. We discuss the probabilistic interpretation of phase extraction in detail and the convergence properties of the bi nary phase approximation. We also discuss the relation of QPE to eigenvalue problems in quantum mechanics and its applications to Hamiltonian simulation, quantum chemistry, cryptography, and hidden subgroup prob lems.Implementation aspects of these algorithms are studied using quantum circuit representations and oracle based computation models. We also discuss error sensitivity, qubit precision requirements, decoherence limita tions and practical constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The work synthesizes the mathematical foundations, computational complexity analysis and physical realizations of the QFT and the QPE, demonstrating how the Fourier-based quantum transformations provide the essential mechanism underlying modern quantum algorithmic speedups and scalable quantum information processing.