Quantum Fourier Transforms

2021 ◽  
pp. 157-192
Keyword(s):  
Fourier Transform ◽  
Information Processing ◽  
Fourier Transforms ◽  
Quantum Algorithms ◽  
Experimental Results ◽  
Quantum Fourier Transform ◽  
Formula Derivation ◽  
Quantum Fourier Transforms ◽  
Proof Of Correctness ◽  
Complete Framework

Quantum Fourier transform (QFT) plays a key role in many quantum algorithms, but the existing circuits of QFT are incomplete and lacking the proof of correctness. Furthermore, it is difficult to apply QFT to the concrete field of information processing. Thus, this chapter firstly investigates quantum vision representation (QVR) and develops a model of QVR (MQVR). Then, four complete circuits of QFT and inverse QFT (IQFT) are designed. Meanwhile, this chapter proves the correctness of the four complete circuits using formula derivation. Next, 2D QFT and 3D QFT based on QVR are proposed. Experimental results with simulation show the proposed QFTs are valid and useful in processing quantum images and videos. In conclusion, this chapter develops a complete framework of QFT based on QVR and provides a feasible scheme for QFT to be applied in quantum vision information processing.

scholarly journals EXACT QUANTUM FOURIER TRANSFORMS AND DISCRETE LOGARITHM ALGORITHMS

2004 ◽  
Vol 02 (01) ◽  
pp. 91-100 ◽  
Author(s):  
MICHELE MOSCA ◽  
CHRISTOF ZALKA
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Quantum Algorithms ◽  
Discrete Logarithm ◽  
Quantum Algorithm ◽  
Quantum Fourier Transform ◽  
The Family ◽  
The Fourier Transform ◽  
Quantum Fourier Transforms ◽  
Amplitude Amplification

We show how the Quantum Fast Fourier Transform (QFFT) can be made exact for arbitrary orders (first showing it for large primes). Most quantum algorithms only need a good approximation of the quantum Fourier transform of order 2n to succeed with high probability, and this QFFT can in fact be done exactly. Kitaev1 showed how to approximate the Fourier transform for any order. Here we show how his construction can be made exact by using the technique known as "amplitude amplification". Although unlikely to be of any practical use, this construction allows one to make Shor's discrete logarithm quantum algorithm exact. Thus we have the first example of an exact non black box fast quantum algorithm, thereby giving more evidence that "quantum" need not be probabilistic. We also show that in a certain sense the family of circuits for the exact QFFT is uniform. Namely, the parameters of the gates can be approximated efficiently.

scholarly journals Implementation of the quantum Fourier transform on a hybrid qubit–qutrit NMR quantum emulator

2015 ◽  
Vol 13 (07) ◽  
pp. 1550059 ◽  
Author(s):  
Shruti Dogra ◽  
Arvind Dorai ◽  
Kavita Dorai
Keyword(s):  
Fourier Transform ◽  
Quantum Algorithms ◽  
Quantum Computers ◽  
Quantum Fourier Transform ◽  
Specific Implementation ◽  
Circuit Decomposition

The quantum Fourier transform (QFT) is a key ingredient of several quantum algorithms and a qudit-specific implementation of the QFT is hence an important step toward the realization of qudit-based quantum computers. This work develops a circuit decomposition of the QFT for hybrid qudits based on generalized Hadamard and generalized controlled-phase gates, which can be implemented using selective rotations in NMR. We experimentally implement the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits.

scholarly journals Human-Competitive Evolution of Quantum Computing Artefacts by Genetic Programming

2006 ◽  
Vol 14 (1) ◽  
pp. 21-40 ◽  
Author(s):  
Paul Massey ◽  
John A. Clark ◽  
Susan Stepney
Keyword(s):  
Fourier Transform ◽  
Quantum Computing ◽  
Genetic Programming ◽  
Quantum Algorithms ◽  
Quantum Circuits ◽  
Quantum Fourier Transform ◽  
Competitive Evolution ◽  
Specific Quantum

We show how Genetic Programming (GP) can be used to evolve useful quantum computing artefacts of increasing sophistication and usefulness: firstly specific quantum circuits, then quantum programs, and finally system-independent quantum algorithms. We conclude the paper by presenting a human-competitive Quantum Fourier Transform (QFT) algorithm evolved by GP.

Estimation of the heating rate of ions due to laser fluctuations when implementing quantum algorithms

2007 ◽  
Vol 7 (7) ◽  
pp. 573-583
Author(s):  
S. Fujiwara ◽  
S. Hasegawa
Keyword(s):  
Fourier Transform ◽  
White Noise ◽  
Heating Rate ◽  
Laser Intensity ◽  
Quantum Algorithms ◽  
Trapped Ions ◽  
Quantum Fourier Transform ◽  
Phase Fluctuations ◽  
Order Of Magnitude ◽  
Magnitude Difference

We analyze numerically the heating of trapped ions due to laser intensity and phase fluctuations when implementing Grover's algorithm and the Quantum Fourier Transform. For a simpler analysis we assume that the stochastic processes are white noise processes and average over each noise as in [Phys. Rev. A. \textbf{57}, 3748, (1998)]. We investigate the fidelity and the heating rate for these algorithms using parameters estimated from experiments, and we can see the order of magnitude difference in the heating rate depending on the quantum algorithms.

scholarly journals Photonic scheme of discrete quantum Fourier transform for quantum algorithms via quantum dots

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Jino Heo ◽  
Kitak Won ◽  
Hyung-Jin Yang ◽  
Jong-Phil Hong ◽  
Seong-Gon Choi
Keyword(s):  
Quantum Dots ◽  
Fourier Transform ◽  
Quantum Algorithms ◽  
Quantum Fourier Transform

Efficient classical simulations of quantum Fourier transforms and Normalizer circuits over Abelian groups

2013 ◽  
Vol 13 (11&12) ◽  
pp. 1007-1037
Author(s):  
Maarten Van den Nest
Keyword(s):  
Fourier Transforms ◽  
Finite Abelian Group ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Quadratic Functions ◽  
Matrix Formalism ◽  
Hidden Subgroup Problem ◽  
Linear Diophantine Equation ◽  
Subgroup Problem ◽  
Quantum Fourier Transforms

The quantum Fourier transform (QFT) is an important ingredient in various quantum algorithms which achieve superpolynomial speed-ups over classical computers. In this paper we study under which conditions the QFT can be simulated efficiently classically. We introduce a class of quantum circuits, called \emph{normalizer circuits}: a normalizer circuit over a finite Abelian group is any quantum circuit comprising the QFT over the group, gates which compute automorphisms and gates which realize quadratic functions on the group. In our main result we prove that all normalizer circuits have polynomial-time classical simulations. The proof uses algorithms for linear diophantine equation solving and the monomial matrix formalism introduced in our earlier work. Our result generalizes the Gottesman-Knill theorem: in particular, Clifford circuits for $d$-level qudits arise as normalizer circuits over the group ${\mathbf Z}_d^m$. We also highlight connections between normalizer circuits and Shor's factoring algorithm, and the Abelian hidden subgroup problem in general. Finally we prove that quantum factoring cannot be realized as a normalizer circuit owing to its modular exponentiation subroutine.

scholarly journals Quantum Computing Concepts with Deutsch Jozsa Algorithm

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Poornima Aradyamath ◽  
Naghabhushana N M ◽  
Rohitha Ujjinimatad
Keyword(s):  
Fourier Transform ◽  
Quantum Computing ◽  
Quantum Computation ◽  
Mathematical Description ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Computers ◽  
Quantum Fourier Transform ◽  
Basic Concepts ◽  
Classical Computer

In this paper, we briefly review the basic concepts of quantum computation,  entanglement,  quantum cryptography and quantum fourier  transform.   Quantum algorithms like Deutsch Jozsa, Shor’s   factorization and Grover’s data search are developed using fourier  transform  and quantum computation concepts to build quantum computers.  Researchers are finding a way to build quantum computer that works more efficiently than classical computer.  Among the  standard well known  algorithms  in the field of quantum computation  and communication we  describe  mathematically Deutsch Jozsa algorithm  in detail for  2  and 3 qubits.  Calculation of balanced and unbalanced states is shown in the mathematical description of the algorithm.

scholarly journals Quantum Fourier Operators and Their Application

2021 ◽  
Author(s):  
Eric Sakk
Keyword(s):  
Fourier Transform ◽  
Quantum Computation ◽  
Quantum Algorithms ◽  
Quantum Fourier Transform ◽  
Discrete Logarithms ◽  
Universal Computation ◽  
Shor's Algorithm ◽  
Proper Perspective ◽  
Classical Computer ◽  
Novel Applications

The application of the quantum Fourier transform (QFT) within the field of quantum computation has been manifold. Shor’s algorithm, phase estimation and computing discrete logarithms are but a few classic examples of its use. These initial blueprints for quantum algorithms have sparked a cascade of tantalizing solutions to problems considered to be intractable on a classical computer. Therefore, two main threads of research have unfolded. First, novel applications and algorithms involving the QFT are continually being developed. Second, improvements in the algorithmic complexity of the QFT are also a sought after commodity. In this work, we review the structure of the QFT and its implementation. In order to put these concepts in their proper perspective, we provide a brief overview of quantum computation. Finally, we provide a permutation structure for putting the QFT within the context of universal computation.

scholarly journals A Comparison of Quantum and Traditional Fourier Transform Computations

2020 ◽  
Author(s):  
Damian Musk
Keyword(s):  
Fourier Transform ◽  
Computational Complexity ◽  
Time Complexity ◽  
Quantum Algorithms ◽  
Quantum Fourier Transform ◽  
Initial State ◽  
Output Vector ◽  
Output State ◽  
The Fourier Transform

The quantum Fourier transform (QFT) can calculate the Fourier transform of a vector of size N with time complexity 𝒪(log2N) as compared to the classical complexity of 𝒪(NlogN). However, if one wanted to measure the full output state, then the QFT complexity becomes 𝒪(Nlog2N), thus losing its apparent advantage, indicating that the advantage is fully exploited for algorithms when only a limited number of samples is required from the output vector, as is the case in many quantum algorithms. Moreover, the computational complexity worsens if one considers the complexity of constructing the initial state. In this paper, this issue is better illustrated by providing a concrete implementation of these algorithms and discussing their complexities as well as the complexity of the simulation of the QFT in MATLAB.

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