Quantum Computing History Future and Algorithms John Lavigne

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

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

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!

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

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

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”

Schrödinger's Cat

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

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

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

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

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

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

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

Shor's Algorithm Part 2 - cont.

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

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

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”

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

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)

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

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

References, cont. Jonietz, Erika. "Quantum Calculation. " Technology Review, July 2005. http: //www. technologyreview. com/Infotech/14591 Aaronson, Scott, The Limits of Quantum, Scientific American, Mar 2008, Vol. 298 Issue 3, p 62 -69, 8 p Castelvecchi, Davide, 15 = 3 x 5, Science News, 12/8/2007, Vol. 172 Issue 23, p 356 -358, 3 p 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/vol 4/spb 3/ "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/0301079 v 1. pdf

Any Questions?

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