- In this paper we present a method to compute the coefficients of the fractional Fourier transform (FrFT) on a quantum computer using quantum gates of polynomial complexity of the order O(n^3). The FrFt, a generalization of the DFT, has wide applications in signal processing and is particularly useful to implement the Pseudopolar and Radon transforms. Even though the FrFT is a non-unitary operation, to develop its quantum counterpart, we develop a unitary operator called the quantum Pseudo-fraction Fourier Transform (QPFrFT) in a higher-dimensional Hilbert space, in order to computer the coefficients of the FrFT. In this process we develop a unitary operator denoted U by which is an essential step to implement the QPFrFT. We then show the application of the operator U in the problem of quantum phase estimation.
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.
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.
Quantum phase estimation (QPE) is one of the most important quantum algorithms which is used as a subroutine for other important quantum algorithms like Shor's factoring algorithm, simulation of quantum systems, quantum counting and QFT on arbitrary Zp. In this paper we develop the theoretical framework for the multivalued quantum logic version of the QPE algorithm using d valued qudits and show a quantum circuit to implement QPE with a complexity of O(nlogn) single qudit operations. The multivalued QPE algorithm, when compared to the binary quantum logic version, turns out to be more robust and leads to a significant decrease in the number of qudits required along with drastic improvement in the precision and success probability. We derive the requirements to amplify the probability of success to a value very close to 1 (for a given precision), thereby generalizing the previously obtained result in the binary case. Also, we note that the failure probability of QPE algorithm decreases exponentially as d increases.
In this paper, we first use the quantum Fourier transform (QFT) and quantum phase estimation (QPE) to realize the quantum fractional Fourier transform (QFrFT). As diverse definitions of the discrete fractional Fourier transform (DFrFT) exist, the relationship between the QFrFT and a classical algorithm is then established; that is, we determine the classical algorithm corresponding to the QFrFT. Second, we observe that many definitions of the multi-fractional Fourier transform (mFrFT) are flawed: when we attempt to propose a design scheme for the quantum mFrFT, we find that there are many invalid weighting terms in the definition of the mFrFT. This flaw may have very significant impacts on relevant algorithms for signal processing and image encryption. Finally, we analyze the circuit of the QFrFT and the reasons for the observed defects.
Quantum Fourier transform (QFT) plays a principal role in the development of efficient quantum algorithms. Since the number of quantum bits that can be built is limited, while many quantum technologies are inherently three (or more) valued, we consider extending the reach of the realistic quantum systems by building a QFT over ternary quantum digits. Compared to traditional binary QFT, the q-valued transform improves approximation properties and increases the state space by a factor of (q/2)n. Further, we use nonbinary QFT derivation to generalize and improve the approximation bounds for QFT