Quantum Computing

1. Quantum Computing History, Future, and Algorithms John Lavigne COT 4810 April 15, 2008

2. Summary • Intro to Quantum Physics • Issues with Practical Application • QC History • QC Limits • Deutsch Algorithm • Shor's Algorithm • Grover's Algorithm

3. Quantum Physics 101 • To understand how Quantum Computing works, we need to understand some things about atoms • Don't worry, Quantum Physics isn't as hard as you think!

4. Quantum Physics 101 • Quantum Mechanics - a theory of the mechanics of atoms, molecules, and other physical systems that are subject to the uncertainty principle

5. Quantum Physics 101 • Bohr Model of the Atom • Electrons spinning around a Proton and Neutron Core

6. Quantum Physics 101 • Spinning of atoms can represent bits • Spinning down represents 0 • Spinning up represents 1 • Spinning both ways is called a “super-position” • These bits is called “qubits”

7. Schrödinger's Cat

8. Quantum Physics 101 • These bits is called “qubits” • Using several of these qubits together is the basis of a quantum computer • Ket Notation

9. Immediate Problems • Heisenburg Uncertainty Principle • Any Interference will change the value of the qubit. • This causes the information to become lost. • This known as decoherence • To solve this, we use a property known as “quantum entanglement” • Two entangled particles will always have one particle spinning up and the other down • By taking advantage of this, it is possible to predict the value of a given atom

10. What these Algorithms can do • No Magic Solutions • NP-Complete problems are still difficult. • Don't try to find a single solution • Make assumptions about the entire set of solutions

11. The Deutsch Algorithm • Given a function f: {0,1} -> {0,1} • Can be either constant or balanced • Can determine which in only one step • Further iterations developed • Computationally Insignificant • Proved the advantages to QC techniques in certain situations

12. Shor's Algorithm • 1994 – Peter Shor, Mathematician at MIT • Used to factorize large numbers • 2001 – Algorithm “proved” on a 7-qubit machine • 15 factored in 5 and 3 • 15 = 1111 => 4 qubits required

13. Shor's Algorithm • Part 1 • set all Qubits to their superposition

14. Shor's Algorithm • Part 2 • N is the number we want to factorize • X is a random number, 1 < X < N-1 • X is raised to the power of Register A and divided by N • The remainder is placed into Register B

15. Shor's Algorithm • Part 2 - cont.

16. Shor's Algorithm • Part 3 • Take the frequency of repitition, f, and use it in this equation: • P = X^(f/2) -1 • The answer is not guaranteed to be correct, but it is easy to check the answer • multiply it out again

17. Grover's Algorithm • Used to search databases • Many practical uses • Knowing a phone number and no name, find the person in a phone book • Assumes a lack of knowledge about the order of items

18. Grover's Algorithm • “similar to dropping multiple pebbles in a pond so that the waves cross and interact in a particular way” (Maney)‏ • “undesired answers cancel out”

19. Grover's Algorithm • Classical Computer: O(N)‏ • Average: N/2 • Quantum Computer: • Average: sqrt(N)‏ • Averages for a DB of N=1,000,000 • Classical computer: 500,000 searches • Quantum computer: 1000 searches

20. Grover's Algorithm • Set all qubits to their super position • Checks all possible answers at once • Made possible through Quantum Parallelism • Works by “increasing the amplitude of the states that carry the desired result” (Lavor)‏

21. Other Approaches • Qutrits? • |0>, |1>, and |2> • Takes advantage of a base e system • Using Photons instead of Atoms

22. History • 1982 – First time Quantum Theory applied to computers • 1989 – First Quantum Algorithm (Deutsch)‏ • 1994 – Shor's Algorithm developed • 1996 – Grover's Algorithm developed • 1998 – 2-Qubit register developed • 2001 – Shor's Algorithm ran on 7 bit QC at Los Alamos labs • May 2006 – Experimental 12-bit QC built by

23. References, cont. • Jonietz, Erika. "Quantum Calculation." Technology Review, July 2005. http://www.technologyreview.com/Infotech/14591 • Aaronson, Scott, The Limits of Quantum, Scientific American, Mar2008, Vol. 298 Issue 3, p62-69, 8p • Castelvecchi, Davide, 15 = 3 x 5, Science News, 12/8/2007, Vol. 172 Issue 23, p356-358, 3p • Bone, Simone and Matias Castro. "A Brief History of Quantum Computing." Imperial College, London, Department of Computing. 1997. http://www.doc.ic.ac.uk/~nd/surprise_97/journal/vol4/spb3/ • "12-qubits Reached In Quantum Information Quest." Science Daily, May 2006. http://www.sciencedaily.com/releases/2006/05/060508164700.htm • Hagar, Amit “Quantum Computing” Stanford Encyclopedia of Philosophy, Feb 2007 • http://plato.stanford.edu/entries/qt-quantcomp/ • Lavor, C., Manssur, L.R.U, “Grover's Algorithm: Quantum Database Search*”, Universidade do Estado de Rio de Janeiro, Feb 2008, http://arxiv.org/PS_cache/quant-ph/0301/0301079v1.pdf

24. Any Questions?

25. Homework 1. What are some possible applications of Quantum Computers? 2. What are the names of the two Algorithms presented here? 3. What are the three possible states of an atom?