David Evans CS150: Computer Science University of Virginia Computer Science Class 33: Computing with Photons From The.

Slides:



Advertisements
Similar presentations
University of Queensland
Advertisements

David Evans cs302: Theory of Computation University of Virginia Computer Science Lecture 13: Turing Machines.
CS 345: Chapter 9 Algorithmic Universality and Its Robustness
David Evans CS200: Computer Science University of Virginia Computer Science Class 30: Models of Computation.
Alice and Bob in the Quantum Wonderland. Two Easy Sums 7873 x 6761 = ? 7873 x 6761 = ? ? x ? = ? x ? =
Class 39: Universality cs1120 Fall 2009 David Evans University of Virginia.
David Evans CS150: Computer Science University of Virginia Computer Science Lecture 39: Lambda Calculus.
Recap CS605: The Mathematics and Theory of Computer Science.
Nathan Brunelle Department of Computer Science University of Virginia Theory of Computation CS3102 – Spring 2014 A tale.
Applied Computer Science II Chapter 3 : Turing Machines Prof. Dr. Luc De Raedt Institut für Informatik Albert-Ludwigs Universität Freiburg Germany.
1 Introduction to Computability Theory Lecture11: Variants of Turing Machines Prof. Amos Israeli.
David Evans CS150: Computer Science University of Virginia Computer Science Lecture 40: Computing with Glue and Photons.
March 11, 2015CS21 Lecture 271 CS21 Decidability and Tractability Lecture 27 March 11, 2015.
1 Polynomial Time Reductions Polynomial Computable function : For any computes in polynomial time.
University of Queensland
Quantum Computing Joseph Stelmach.
Turing Machines CS 105: Introduction to Computer Science.
David Evans cs302: Theory of Computation University of Virginia Computer Science Lecture 2: Modeling Computers.
Class 19: Undecidability in Theory and Practice David Evans cs302: Theory of Computation University of Virginia Computer.
Whatiscompn.ppt version: What Is Computation? William J. Rapaport Department of Computer Science & Engineering, Department of Philosophy, Department.
David Evans CS200: Computer Science University of Virginia Computer Science Class 31: Universal Turing Machines.
Lecture 14: Church-Turing Thesis Alonzo Church ( ) Alan Turing ( )
Dominique Unruh 3 September 2012 Quantum Cryptography Dominique Unruh.
More Theory of Computing
Lecture 20: λ Calculus λ Calculus & Computability Yan Huang Dept. of Comp. Sci. University of Virginia.
David Evans Turing Machines, Busy Beavers, and Big Questions about Computing.
Future Computers.
Class 37: Computability in Theory and Practice cs1120 Fall 2011 David Evans 21 November 2011 cs1120 Fall 2011 David Evans 21 November 2011.
Theory of Computing Lecture 15 MAS 714 Hartmut Klauck.
Limits and Horizon of Computing Post silicon computing.
Computational Complexity Theory Lecture 2: Reductions, NP-completeness, Cook-Levin theorem Indian Institute of Science.
Introduction to CS Theory Lecture 15 –Turing Machines Piotr Faliszewski
Cs3102: Theory of Computation Class 27: NP-Complete Desserts (DNA, RSA, BQP, NSA) Spring 2010 University of Virginia David Evans.
David Evans CS588: Security and Privacy University of Virginia Computer Science Lecture 15: Complexity Notes This is selected.
Cs3102: Theory of Computation Class 14: Turing Machines Spring 2010 University of Virginia David Evans.
Physics 2170 – Spring The Schrödinger equation Next homework assignment is available I will be giving a.
CSE 311 Foundations of Computing I Lecture 26 Computability: Turing machines, Undecidability of the Halting Problem Spring
Class 21: Introducing Complexity David Evans cs302: Theory of Computation University of Virginia Computer Science.
David Evans cs302: Theory of Computation University of Virginia Computer Science Lecture 16: Universality and Undecidability.
David Evans CS150: Computer Science University of Virginia Computer Science Lecture 36: Modeling Computing.
Strings Basic data type in computational biology A string is an ordered succession of characters or symbols from a finite set called an alphabet Sequence.
David Evans CS200: Computer Science University of Virginia Computer Science Lecture 15: Intractable Problems (Smiley.
Computer Theory Michael J. Watts
28 April 2005 CS588 Spring 2005 David Evans Phun with Photons.
David Evans CS200: Computer Science University of Virginia Computer Science Class 32: The Meaning of Truth.
CS216: Program and Data Representation University of Virginia Computer Science Spring 2006 David Evans Lecture 8: Crash Course in Computational Complexity.
Quantum Computation Stephen Jordan. Church-Turing Thesis ● Weak Form: Anything we would regard as “computable” can be computed by a Turing machine. ●
David Evans CS200: Computer Science University of Virginia Computer Science Lecture 14: P = NP?
David Evans CS200: Computer Science University of Virginia Computer Science Class 32: The Meaning of Truth.
David Evans CS200: Computer Science University of Virginia Computer Science Lecture 23: Intractable Problems (Smiley Puzzles.
Recall last lecture and Nondeterministic TMs Ola Svensson.
David Evans CS150: Computer Science University of Virginia Computer Science Class 32: Computability in Theory and Practice.
David Evans CS200: Computer Science University of Virginia Computer Science Class 26: Halting Problem It is plain at any.
Attendance Syllabus Textbook (hardcopy or electronics) Groups s First-time meeting.
Class 27: Universal Turing Machines CS150: Computer Science
Class 30: Models of Computation CS200: Computer Science
Lecture 37: A Universal Computer
Q Jeff Kinne.
Class 36: The Meaning of Truth CS200: Computer Science
Lecture 24: Metalinguistics CS200: Computer Science
OSU Quantum Information Seminar
Lecture 27: In Praise of Idleness CS200: Computer Science
Class 33: Making Recursion M.C. Escher, Ascending and Descending
Recall last lecture and Nondeterministic TMs
Class 31: Universal Turing Machines CS200: Computer Science
Class 34: Models of Computation CS200: Computer Science
Class 26: Modeling Computing CS150: Computer Science
Quantum Computing Joseph Stelmach.
The Church-Turing Thesis and Turing-completeness
Presentation transcript:

David Evans CS150: Computer Science University of Virginia Computer Science Class 33: Computing with Photons From The Tinkertoy Computer and Other Machinations by A. K. Dewdney

2 CS150 Fall 2005: Lecture 33: Alternate Computing Models Church-Turing Thesis Church’s original (1935) –Lambda calculus is equivalent to real world computers (can compute any computable function) Turing’s version –“Every function which would naturally be regarded as computable can be computed by a Turing machine.” Generalized version: –Any computation that can be done by an algorithm can be done by a mechanical computer –All “normal” computers are equivalent in computing power

3 CS150 Fall 2005: Lecture 33: Alternate Computing Models Turing Machines and Complexity Stronger version: –Complexity classes P, NP, and NP-complete are defined for Turing machine steps, but apply identically to all “normal” computers Today: “Abnormal” Computers –Might change what is computable (probably don’t) –Do change what a normal “step” is

4 CS150 Fall 2005: Lecture 33: Alternate Computing Models Normal Steps Turing machine: –Read one square on tape, follow one FSM transition rule, write one square on tape, move tape head one square Lambda calculus: –One beta reduction Your PC: –Execute one instruction (?) What one instruction does varies

5 CS150 Fall 2005: Lecture 33: Alternate Computing Models Generalized Normal Steps Require a constant amount of time Perform a fixed amount of work –Localized –Cannot scale (indefinitely) with input size

6 CS150 Fall 2005: Lecture 33: Alternate Computing Models Abnormal Imaginary Computer “Accelerating” TM –Like a regular TM, except the first step takes 1 second, second step takes ½ second, third step takes ¼ second,... n th step takes 1/2 n second Is our “Accelerating” TM more powerful than a regular TM?

7 CS150 Fall 2005: Lecture 33: Alternate Computing Models Quantum Physics for Dummies Light behaves like both a wave and a particle at the same time A single photon is in many states at once Can’t observe its state without forcing it into one state Schrödinger’s Cat –Put a live cat in a box with cyanide vial that opens depending on quantum state –Cat is both dead and alive at the same time until you open the box

8 CS150 Fall 2005: Lecture 33: Alternate Computing Models Quantum Computing Feynman, 1982 Quantum particles are in all possible states Can try lots of possible computations at once with the same particles In theory, can test all possible factorizations/keys/paths/etc. and get the right one! In practice, very hard to keep states entangled: once disturbed, must be in just one possible state

9 CS150 Fall 2005: Lecture 33: Alternate Computing Models Qubit Regular bit: either a 0 or a 1 Quantum bit: 0, 1 or in between –p% probability it is a 1 A single qubit is in 2 possible states at once If you have 7 bits, you can represent any one of 2 7 different states If you have 7 qubits, you have 2 7 different states (at once!)

10 CS150 Fall 2005: Lecture 33: Alternate Computing Models Quantum Computers Today Several quantum algorithms –Shor’s algorithm: factoring using a quantum computer Actual quantum computers –5-qubit computer built by IBM (2001) –Implemented Shor’s algorithm to factor: “World’s most complex quantum computation” –Los Alamos has built a 7-qubit computer To exceed practical normal computing need > 30 qubits –Adding another qubit is more than twice as hard 15 (= 5 * 3)

11 CS150 Fall 2005: Lecture 33: Alternate Computing Models Complexity for Quantum Computer Complexity classes are different than for regular computers, because a step is different Quantum computer: each step can take both possible decisions at once –Means a quantum computer is a nondeterministic computer! –It can solve problems in class NP in polynomial time! What matters? Number of qubits you need Number of (nondeterministic) steps

12 CS150 Fall 2005: Lecture 33: Alternate Computing Models Charge Exam 2 out Friday –Covers through Monday –All questions will assume only “normal” computers –Links to example exams on the web –Review session Wednesday, 7pm