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Quantum Computing: Fourier Transform Applications & Quantum Algorithms

This chapter delves into the applications and benefits of Quantum Fourier Transform in quantum computing, showcasing its advantages over classical computation. The content highlights Quantum Phase Estimation algorithm, its significance, and its role in solving complex problems efficiently. The Quantum Fourier Transform's involvement in various quantum algorithms, such as period finding and discrete logarithms, is emphasized. Additionally, the chapter explores the military applications and general advantages of quantum computing compared to classical computers. Ultimately, it provides insights into why investing in quantum computing is crucial for advancing technology.

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Quantum Computing: Fourier Transform Applications & Quantum Algorithms

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  1. Chap 5 Q Fourier Transform: p 216-247 Avery Leider The information presented here, although greatly condensed, comes almost entirely from the course textbook: Quantum Computation and Quantum Information by Nielsen & Chuang and the tutorials written by Prof Ronald Frank

  2. 5.1 Why the investment in Quantum Computing? • Prime factorization of an n-bit integer on a classical computer using the number field sieve: • Same task with a quantum algorithm: Quantum is faster!

  3. Military Applications Quantum Computing

  4. 5.1 Quantum Fourier Transform • Quantum Fourier Transform is an algorithm used in • quantum factoring of primes, • and other quantum algorithms. • Useful for warfare. Quantum Computing

  5. Fourier seriesillustration by Pierre Guilleminot • https://bl.ocks.org/jinroh/7524988 Research Seminar 1 Review

  6. 5.1 Quantum Fourier Transform From Box 5.1 on page 220 of Nielsen, M.A. and Chuang, I., 2002. Quantum computation and quantum information. Quantum Computing

  7. 5.2 Quantum Phase Estimation algorithm • Phase Estimation uses the inverse Quantum Fourier transform • Phase Estimation makes it possible to estimate the phase that a unitary transformation adds to one of its eigenvectors.

  8. Quantum Phase Estimation • Phase Estimation is a subroutine that prepares an eigenstate of the Hermitian operator in one register • It stores the corresponding eigenvalue in a 2nd register • It uses momentum and phase shifts • & inverse Quantum Fourier transform https://quantumexperience.ng.bluemix.net/proxy/tutorial/full-user-guide/004-Quantum_Algorithms/100-Quantum_Phase_Estimation.html Quantum Computing

  9. Phase Estimation Quantum Computing

  10. Phase Estimation Research Seminar 1 Review

  11. Phase Estimation • Phase Estimation, combined with quantum search algorithm, can solve the problem of counting solutions to a search problem. • Phase Estimation supports solutions to the order-finding problem and the factoring problem. Research Seminar 1 Review

  12. 5.4 General Applications of the quantum Fourier transform • Period finding • Discrete logarithms • Hidden subgroup problem • Other quantum algorithms? Quantum Computing

  13. 5.4 General Applications of the quantum Fourier transform • Quantum Computing can solve some problems exponentially better than classical computers. • Other problems it is polynominally better. • Quantum Fourier Transform is involved in most of these advantages • But in some cases, it is not any better. Quantum Computing

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