Quantum Fourier Transform
In quantum computing, the quantum Fourier transform is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform. The quantum Fourier transform is a part of many quantum algorithms, notably Shor's algorithm for factoring and computing the discrete logarithm, the quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator, and algorithms for the hidden subgroup problem.
The quantum Fourier transform can be performed efficiently on a quantum computer, with a particular decomposition into a product of simpler unitary matrices. Using a simple decomposition, the discrete Fourier transform can be implemented as a quantum circuit consisting of only O(n^2)Hadamard gates and controlled phase shift gates, where n is the number of qubits. This can be compared with the classical discrete Fourier transform, which takes O(n2^n) gates (where n is the number of bits), which is exponentially more than O(n^2). However, the quantum Fourier transform acts on a quantum state, whereas the classical Fourier transform acts on a vector, so not every task that uses the classical Fourier transform can take advantage of this exponential speedup.
The best quantum Fourier transform algorithms known today require only O(n \log n) gates to achieve an efficient approximation.

This is an excerpt from the article Quantum Fourier Transform from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.
The article Quantum Fourier Transform at en.wikipedia.org was accessed 1,453 times in the last 30 days. (as of: 08/24/2013)
Images on Quantum Fourier Transform
Preview image:
Original:
Search results from Google and Bing
1
>30
1
Quantum Fourier transform - Wikipedia, the free encyclopedia
In quantum computing, the Quantum Fourier Transform is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform.
en.wikipedia.org/wiki/Quantum_Fourier_transform
2
>30
2
Quantum Fourier Transform - Usc
This lecture will concentrate almost entirely upon a single unitary transformation: the Quantum Fourier Transform. This is a discrete Fourier transform, not upon.
www-bcf.usc.edu/~tbrun/Course/lecture13.pdf
3
>30
3
The Quantum Fourier Transform and Jordan's Algorithm
This algorithm makes us of the Quantum Fourier Transform. ... deviate to discuss the (quantum) discrete Fourier transform and see an application of this transform.
courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes9.pdf
4
>30
4
Lecture 7: Quantum Fourier transform - Helsinki.fi
Let us instead focus on the Quantum Fourier Transform... Department of Physical ... Quantum Fourier Transform is the same transformation with somewhat different  ...
theory.physics.helsinki.fi/~kvanttilaskenta/Lecture7.pdf
5
>30
5
Lecture 6: The Quantum Fourier Transform - Computer Science
show how the Quantum Fourier Transform (QFT) emerges from the DFT. • construct a quantum circuit that performs the QFT. Lecture 6: The Quantum Fourier ...
www.cs.bham.ac.uk/internal/courses/intro-mqc/current/lecture06_handout.pdf
6
>30
6
pdf - www-inst.eecs.berkeley.edu
Quantum Fourier Transform. 10/28/03. Fall 2003. Lecture 19. 1 Fourier Transform . Definition 19.1 (FTN): The Fourier transform mod N is the N ×N matrix given by.
www-inst.eecs.berkeley.edu/~cs191/sp05/lectures/lecture19.pdf
7
>30
7
Integer Factoring and Fourier Transform
Jesko Hüttenhain. Lars Wallenborn. May 6th, 2011. Contents. 1 Abstract. 1. 2 The Quantum Fourier Transform. 2. 3 Quantum Integer Factoring. 5. 4 The Hidden ...
page.math.tu-berlin.de/~jesko/written/quantum.fourier.pdf
8
>30
8
A quantum Fourier transform algorithm
Apr 9, 2004 ... Abstract: Algorithms to compute the Quantum Fourier Transform over a cyclic group are fundamental to many quantum algorithms. This paper ...
arxiv.org/abs/quant-ph/0404060
9
>30
9
How does Quantum Fourier Transform work? - Quora
The Quantum Fourier Transform works just like the Classical Fourier Transform however it has very deep and elegant applications in exponentially ...
www.quora.com/Quantum-Algorithms/How-does-Quantum-Fourier-Transform-work
>30
1
10
Fourier Transform
A thorough tutorial of the Fourier Transform, for both the laymen and the practicing scientist. This site is designed to present a comprehensive overview of the ...
www.thefouriertransform.com/
Search results for "Quantum Fourier Transform"
Google: approx. 3.770.000
Quantum Fourier Transform in science
[PDF]Lecture 7: Quantum Fourier transform - Helsinki.fi
Let us instead focus on the Quantum Fourier Transform... Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/˜quantumgas/ – p.
[PDF]Lecture 8: Applications of Quantum Fourier transform - Helsinki.fi
Department of Physical Sciences, University of Helsinki ... problems. Search for crypto keys. Fourier transform. Quantum. Discrete log. Factoring. Order−finding.
Generic Quantum Fourier Transforms
Apr 9, 2003 ... Generic Quantum Fourier Transforms. Cristopher Moore (University of New Mexico), Daniel Rockmore (Dartmouth), Alexander Russell ...
[PDF]CS 294-6, Quantum Computing (Umesh Vazirani) Fall 2004 ...
Fall 2004, University of California, Berkeley. Lecture #8 (Quantum Fourier Transform). DRAFT Notes by Boris Bukh. 30 September 2004. Fall 2004. Lecture 8.
Fourier transform - Wikipedia, the free encyclopedia
The Fourier transform, named after Joseph Fourier, is a mathematical ... 8.2 Fourier transform spectroscopy; 8.3 Quantum mechanics and signal processing ...... Chandrasekharan K. (1949), Fourier Transforms, Princeton University Press ...
[PDF]GENERIC QUANTUM FOURIER TRANSFORMS - Department of ...
University of Connecticut acr@cse.uconn.edu. Abstract. The Quantum Fourier Transform (QFT) is the principal ingredient of most efficient quantum algorithms. We.
About things like the quantum Fourier transform,.. – Introduction
The lyric "About things like the Quantum Fourier Transform, and a" from " Introduction" by Umesh Vazirani (Ft. Berkeley & University of California) has not yet been ...
[PDF]Shor's Algorithm and the Quantum Fourier ... - McGill University
1 COMPUTATION AND COMPLEXITY CLASSES. Shor's Algorithm and the Quantum Fourier Transform. Fang Xi Lin. McGill University fangxi.lin@mail.mcgill. ca.
[PDF]De-quantisation of the Quantum Fourier Transform - Department of ...
Department of Computer Science, University of Auckland, Private Bag 92109, Auckland, New Zealand. Abstract. The Quantum Fourier Transform (QFT) plays an  ...
[PDF]Quantum Phase Estimation and Arbitrary Size Quantum Fourier
Department of Computer Science & Engineering, University of Washington. What use is the Quantum Fourier Transform? Well we've already seen Jordan's ...
Books on the term Quantum Fourier Transform
Principles of Quantum Computation and Information: Basic ...
Principles of Quantum Computation and Information: Basic ...
Giuliano Benenti, Giulio Casati, Giuliano Strini, 2007
So far, no other implementation of a quantum processor has been able to produce similar results. In this section, we briefly describe three relevant experiments. Quantum Fourier Transform. This has been implemented (Weinstein et al., 2001) ...
Quantum Computation and Quantum Information: 10th Anniversary Edition
Quantum Computation and Quantum Information: 10th Anniversary Edition
Michael A. Nielsen, 2011
"In conclusion, "Mike and Ike" is still the authoritative reference point for anybody studying or researching in the field of quantum computing." Alessandro Berni, Computing Reviews
Quantum Computation and Quantum Information: 10th ...
Quantum Computation and Quantum Information: 10th ...
Michael A. Nielsen, Isaac L. Chuang, 2010
Therefore, this circuit provides a Θ(n2) algorithm for performing the Quantum Fourier Transform. In contrast, the best classical algorithms for computing the discrete Fourier transform on 2n elements are algorithms such as the Fast Fourier  ...
Quantum Computing: A Gentle Introduction (Scientific and Engineering Computation)
Quantum Computing: A Gentle Introduction (Scientific and Engineering Computation)
2011
The collection of exercises is a treasure… I could open any chapter and follow its content without having to turn to previous chapters for notions and notation… precious for the beginner... [a] masterpiece. I need not say more. (Valerio Scarani, Physics Today Physics Today)Rieffel and Polak have produced a pedagogical triumph. While reviewing this ...
Quantum Information: An Overview
Quantum Information: An Overview
Gregg Jaeger, 2007
noting that the quantum version of the fast Fourier transform can be carried out efficiently, since the number of quantum-parallel computations increases exponentially with number with the number of bits involved. Integer prime factoring is the ...
Quantum Computing for Computer Scientists
Quantum Computing for Computer Scientists
Noson S. Yanofsky, 2008
The multidisciplinary field of quantum computing strives to exploit some of the uncanny aspects of quantum mechanics to expand our computational horizons. Quantum Computing for Computer Scientists takes readers on a tour of this fascinating area of cutting-edge research. Written in an accessible yet rigorous fashion, this book employs ideas and tec...
Quantum Computing: A Gentle Introduction
Quantum Computing: A Gentle Introduction
Eleanor Rieffel, Wolfgang Polak, 2011
Thus, any computation in BQP can be simulated classically in polynomial space. 7.8 Quantum Fourier Transformations The Quantum Fourier Transformation (QFT) is the single most important quantum subroutine. It and its generalizations are ...
Quantum Optics
Quantum Optics
2012
"The reader will find this book to be an excellent, modern review of the field of quantum optics and its applications. It is written for graduate students with a strong background in quantum mechanics and classical electrodynamics. If you studied quantum optics more than a decade ago, it is a must-read to refresh your knowledge of this rapidly...
Foundations of Quantum Mechanics: From Photons to Quantum ...
Foundations of Quantum Mechanics: From Photons to Quantum ...
Charlotte Augusta Ayres Professor of Physics Wesleyan University Connecticut Reinhold Blumel, Reinhold Blumel, 2011
The crucial element of Shor's quantum algorithm for the factorization of large semi-primes is the application of a Quantum Fourier Transform. The Quantum Fourier Transform is itself one of the most important quantum algorithms and is frequently ...
Quantum Computer Science: An Introduction
Quantum Computer Science: An Introduction
N. David Mermin, 2007
In the 1990's it was realized that quantum physics has some spectacular applications in computer science. This book is a concise introduction to quantum computation, developing the basic elements of this new branch of computational theory without assuming any background in physics. It begins with an introduction to the quantum theory from a co...
Development of Google searches


Blog posts on the term
Quantum Fourier Transform
Quantum Fourier Transform - The Student Room
www.thestudentroom.co.uk/showthread.php?t=2482133
viXra.org e-Print archive, viXra:1311.0058, Non-Local Quantum Theory is Fourier-Transformed Classical Mechanics. Planck’s Constant as Adiabatic Invariant Characterized by Hubble's and Cosmological Constants
vixra.org/abs/1311.0058
Phys. Rev. A 87, 032333 (2013): Scaling laws for Shor's algorithm with a banded quantum Fourier transform
pra.aps.org/abstract/PRA/v87/i3/e032333
[1210.1550] Quantum Fourier Transforms and the Complexity of Link Invariants for Quantum Doubles of Finite Groups
arxiv.org/abs/1210.1550
Into the Continuum: The Quantum Fourier Transform
In the previously introduced algorithms, the Hadamard transform played an essential role in being able to created a certain superposition of states that was exploited in the various ways by the algorithms. Recall the action of the Hadamard transform \(H^{\otimes n}\) on an arbitrary basis state \(\left|\mathbf{x}\right>\in\mathcal{H}^{2^n}\):\[H^{\otimes n}\left|\mathbf{x}\right>=\displaystyle \frac{1}{\sqrt{2^{n}}}\displaystyle\sum\limits_{\mathbf{y}\in\{0,1\}^n}(-1)^{\mathbf{x}\cdot\mathbf{y}}\left|\mathbf{y}\right>,\]which can be thought of as effectively encoding the string \(\mathbf{x}\) into the relative paste factors present in the amplitudes of the states in the superposition.
intothecontinuum.blogspot.com/2013/09/the-quantum-fourier-transform.html
Ali Abbas » Shor’s algorithm and Quantum Fourier Transform
alouche.net/2012/10/28/shor-algorithm-quantum-fourier-transform/
Quantum logic gates | Complex Projective 4-Space
You are probably aware of classical logic gates, such as the Boolean AND and OR gates. The input(s) and output(s) of a logic gate take values in the set {0, 1}, usually identified with 'false' and 'true', respectively. One example with multiple inputs and outputs is the full adder, so called because of its ability to perform…
cp4space.wordpress.com/2013/11/09/quantum-logic-gates/
Math is Awesome: The Importance of Fourier Transforms - Tested
www.tested.com/science/458960-fourier-transforms-behind-everything/
Nuit Blanche: Phase Retrieval from masked Fourier transforms - implementation -
In the following paper, I note four items of note beyond the very interesting results of the paper: an implementation of the paper is made available with the paper the possibility of a sharp phase transition in phase retrieval (see figure below and mentioned in 2. 3) Terry Tao mentioned it in the selected paper network, and you may recall the last words in this week's Sunday Morning Insight: The Map Makers that directed to Gilles Pisier's Grothendieck's theorem past and present In addition, GT [Grothendieck's theorem] independently surfaced in several quite unrelated fi elds: .
nuit-blanche.blogspot.com/2013/11/phase-retrieval-from-masked-fourier.html
functions - Fourier transformations in Simon's quantum paper - Mathematics Stack Exchange
math.stackexchange.com/questions/450322/fourier-transformations-in-simons-quantum-paper
123