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 series illustration by Pierre Guilleminot
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
Quantum Computing

9. Phase Estimation
Quantum Computing

10. Research Seminar 1 Review
Phase Estimation
Research Seminar 1 Review

11. Research Seminar 1 Review
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