Content deleted Content added
Citation bot (talk | contribs) Add: bibcode, authors 1-1. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Dominic3203 | Category:Fourier analysis | #UCB_Category 98/126 |
Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.9.5 |
||
Line 138:
{{see also|Hadamard transform#Quantum computing applications}}
Using the generalized [[Fourier transform on finite groups|Fourier transform on finite (abelian) groups]], there are actually two natural ways to define a quantum Fourier transform on an ''n''-qubit [[quantum register]]. The QFT as defined above is equivalent to the DFT, which considers these n qubits as indexed by the cyclic group <math>\Z / 2^n \Z</math>. However, it also makes sense to consider the qubits as indexed by the [[Boolean group]] <math>(\Z / 2 \Z)^n</math>, and in this case the Fourier transform is the [[Hadamard transform]]. This is achieved by applying a [[Hadamard gate]] to each of the n qubits in parallel.<ref>[https://www.staff.uni-mainz.de/pommeren/Kryptologie/Bitblock/A_Nonlin/Fourier.pdf Fourier Analysis of Boolean Maps– A Tutorial –, pp. 12-13] {{Webarchive|url=https://web.archive.org/web/20210501173437/https://www.staff.uni-mainz.de/pommeren/Kryptologie/Bitblock/A_Nonlin/Fourier.pdf |date=2021-05-01 }}{{full|date=June 2024}}</ref><ref>[https://web.archive.org/web/20210713134541/https://www.cse.iitk.ac.in/users/rmittal/prev_course/s19/reports/5_algo.pdf Lecture 5: Basic quantum algorithms, Rajat Mittal, pp. 4-5]</ref> [[Shor's algorithm]] uses both types of Fourier transforms, an initial Hadamard transform as well as a QFT.
== For other groups ==
| |||