Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Quantum ComputationThe Mathematics of Information<br />J. Caleb Wherry<br />Austin Peay State University<br />Departments ...
Outline<br />Classical Computation<br />History<br />Babbage, ENIAC, Vacuum Tubes, & the Transistor<br />Moore’s Law<br />...
3<br />Classical Computation<br />
History<br />4<br />Difference Engine - 1823<br />Pascaline - 1623<br />Step Reckoner - 1673<br />
History<br />5<br />ENIAC - 1946<br />Vacuum Tubes<br />
History<br />6<br />Texas Instruments 1954 Transistor<br />
History<br />7<br />Moore’s Law<br />
Computation & Complexity Theory<br />8<br />What is computation?<br />
Computation & Complexity Theory<br />9<br />Computation<br />	A process following a well-defined model that is understood ...
Computation & Complexity Theory<br />10<br />
Computation & Complexity Theory<br />11<br />
Cbits, Logic Gates, & the Circuit Model<br />12<br />Classical Bits<br /><ul><li> 2-state system (Boolean Algebra)
 Possible states: 0 or 1 (Off or On)
 0 -> No voltage
 1 -> 0.5 voltage</li></ul>If we have n classical bits, how much information do we have?<br />
Cbits, Logic Gates, & the Circuit Model<br />13<br />Basic Classical Logic Gates<br />Logic Gates<br /><ul><li>{One,Two}-a...
 Universal set of gates: (AND, NOT, & FANOUT)
 What does universal mean?
 Are they reversible?
What does reversible mean?</li></li></ul><li>Cbits, Logic Gates, & the Circuit Model<br />14<br />
Moore’s Law Revisited<br />15<br />Moore’s Law<br />
16<br />Quantum Computation<br />
Mathematical Formalisms<br />17<br />Qubit – Quantum Bit<br />Orthonormal Basis Set<br />Superposition of 0 & 1<br />|0 ...
Neat Mathematics<br />18<br />Where do qubits live?<br />|<br /><br />lives in a Hilbert Space H .<br />H   is a complete...
Mathematical Formalisms<br />19<br />Quantum Logic Gates = Linear Transformations<br />Pauli Matrices<br />Hadamard Gate<b...
Mathematical Formalisms<br />20<br />Quantum Weirdness<br />Superposition<br />Entanglement<br />Teleportation<br />
Mathematical Formalisms<br />21<br />Quantum Weirdness I<br />Superposition & Interference<br />
Mathematical Formalisms<br />22<br />Quantum Weirdness I<br />Superposition & Interference<br />
Mathematical Formalisms<br />23<br />Quantum Weirdness II<br />Entanglement – EPR Paradox<br />“Spookiness at a distance” ...
Mathematical Formalisms<br />24<br />Quantum Weirdness III<br />Teleportation<br />
BQP & the Power of Q.C.<br />17<br />If we have nqubits, how much information do we have?<br />
Quantum Implementations<br />26<br />NMR<br />Ion Trap<br />Superconducting Qubits<br />Topological Q.C.<br />
Quantum Algorithms<br />27<br />Grover’s Search<br />Normal amount of time a database search takes?<br />N items takes O(n...
Quantum Algorithms<br />28<br />Shor’s Factoring<br />Fastest Classical Factoring Algorithm:<br />General Number Field Sie...
29<br />Other Computational Paradigms<br />
Upcoming SlideShare
Loading in …5
×

Quantum Computation: The Mathematics of Information

753 views

Published on

Published in: Technology
  • Be the first to comment

Quantum Computation: The Mathematics of Information

  1. 1. Quantum ComputationThe Mathematics of Information<br />J. Caleb Wherry<br />Austin Peay State University<br />Departments of Computer Science, Mathematics, & Physics<br />
  2. 2. Outline<br />Classical Computation<br />History<br />Babbage, ENIAC, Vacuum Tubes, & the Transistor<br />Moore’s Law<br />Computation & Complexity Theory<br />Cbits, Logic Gates, & the Circuit Model<br />Moore’s Law Revisited<br />Quantum Computation<br />Mathematical Formalisms (Linear Algebra & Quantum Mechanics)<br />Qubits, Quantum Gates, & the Quantum Circuit Model<br />BQP & the Power of Q.C.<br />Quantum Q.C. Implementations<br />NMR, Iron Trap, Superconducting Qubits, & Topological Q.C.<br />Quantum Algorithms<br />Grover’s Search & Shor’s Factoring Algorithms<br />Other Computational Paradigms<br />Zeno’s Computer<br />Relativity Computer<br />Closed Timelike Curve Computation<br />DNA Computing<br />2<br />
  3. 3. 3<br />Classical Computation<br />
  4. 4. History<br />4<br />Difference Engine - 1823<br />Pascaline - 1623<br />Step Reckoner - 1673<br />
  5. 5. History<br />5<br />ENIAC - 1946<br />Vacuum Tubes<br />
  6. 6. History<br />6<br />Texas Instruments 1954 Transistor<br />
  7. 7. History<br />7<br />Moore’s Law<br />
  8. 8. Computation & Complexity Theory<br />8<br />What is computation?<br />
  9. 9. Computation & Complexity Theory<br />9<br />Computation<br /> A process following a well-defined model that is understood and can be expressed in an algorithm, protocol, network topology, etc.<br />Computational Complexity<br /> The measure of the resources (e.g. time, space, basic operations, energy) used by a computation. Measured as a function of the input size.<br />Turing Machine<br /> A very simplistic computer in which computations can be executed on. <br />Tape – Infinitely Long. Finite Alphabet.<br />Head – Reads/Writes, Moves Tape 1 Cell L/R.<br />Table – Finite Set of Instructions.<br />State Register – Current Finite State of TM.<br />Strong Church-Turing Thesis <br />A probabilistic Turing machine (e.g. a classical computer that can make fair coin flips) can efficiently simulate any realistic model of computing.<br />
  10. 10. Computation & Complexity Theory<br />10<br />
  11. 11. Computation & Complexity Theory<br />11<br />
  12. 12. Cbits, Logic Gates, & the Circuit Model<br />12<br />Classical Bits<br /><ul><li> 2-state system (Boolean Algebra)
  13. 13. Possible states: 0 or 1 (Off or On)
  14. 14. 0 -> No voltage
  15. 15. 1 -> 0.5 voltage</li></ul>If we have n classical bits, how much information do we have?<br />
  16. 16. Cbits, Logic Gates, & the Circuit Model<br />13<br />Basic Classical Logic Gates<br />Logic Gates<br /><ul><li>{One,Two}-ary Operations on our Boolean Algebra
  17. 17. Universal set of gates: (AND, NOT, & FANOUT)
  18. 18. What does universal mean?
  19. 19. Are they reversible?
  20. 20. What does reversible mean?</li></li></ul><li>Cbits, Logic Gates, & the Circuit Model<br />14<br />
  21. 21. Moore’s Law Revisited<br />15<br />Moore’s Law<br />
  22. 22. 16<br />Quantum Computation<br />
  23. 23. Mathematical Formalisms<br />17<br />Qubit – Quantum Bit<br />Orthonormal Basis Set<br />Superposition of 0 & 1<br />|0 + |1<br />|0<br />|1<br />|0<br /><br />|<br />|1<br />E.g.<br /><br />=<br />|<br />Qubits: Photons, Electrons, Ions, etc.<br />*Spin of above particles.<br />Bloch Sphere<br />
  24. 24. Neat Mathematics<br />18<br />Where do qubits live?<br />|<br /><br />lives in a Hilbert Space H .<br />H is a complete Vector Space with a defined inner product. <br />What does complete mean?<br />Formal definition: a space is complete if every Cauchy Sequence converges to a point within the set.<br />But what does that mean?<br />Fields: N, Q, R, C, H<br />
  25. 25. Mathematical Formalisms<br />19<br />Quantum Logic Gates = Linear Transformations<br />Pauli Matrices<br />Hadamard Gate<br />Pauli-X <br />Pauli-Y <br />Hadamard<br />Pauli-Z <br />
  26. 26. Mathematical Formalisms<br />20<br />Quantum Weirdness<br />Superposition<br />Entanglement<br />Teleportation<br />
  27. 27. Mathematical Formalisms<br />21<br />Quantum Weirdness I<br />Superposition & Interference<br />
  28. 28. Mathematical Formalisms<br />22<br />Quantum Weirdness I<br />Superposition & Interference<br />
  29. 29. Mathematical Formalisms<br />23<br />Quantum Weirdness II<br />Entanglement – EPR Paradox<br />“Spookiness at a distance” - Einstein <br />
  30. 30. Mathematical Formalisms<br />24<br />Quantum Weirdness III<br />Teleportation<br />
  31. 31. BQP & the Power of Q.C.<br />17<br />If we have nqubits, how much information do we have?<br />
  32. 32. Quantum Implementations<br />26<br />NMR<br />Ion Trap<br />Superconducting Qubits<br />Topological Q.C.<br />
  33. 33. Quantum Algorithms<br />27<br />Grover’s Search<br />Normal amount of time a database search takes?<br />N items takes O(n) searches.<br />Grover’s Search takes O( SQRT(N) ) searches for N items.<br />
  34. 34. Quantum Algorithms<br />28<br />Shor’s Factoring<br />Fastest Classical Factoring Algorithm:<br />General Number Field Sieve<br />O(e^((log N)^1/3 (log log N)^2/3))<br />Shor’s Algorithm Factors in:<br /> O(log(N)^3)<br />Exponential Speedup!<br />
  35. 35. 29<br />Other Computational Paradigms<br />
  36. 36. Other Computational Paradigms<br />30<br />Zeno’s Computer<br />STEP 1<br />STEP 2<br />Time (seconds)<br />STEP 3<br />STEP 4<br />STEP 5<br />
  37. 37. 31<br />Other Computational Paradigms<br />Relativity Computer<br />DONE<br />
  38. 38. Other Computational Paradigms<br />32<br />Closed Timelike Curve Computation<br />S. Aaronson and J. Watrous. Closed Timelike Curves Make Quantum and Classical Computing Equivalent, Proceedings of the Royal Society A 465:631-647, 2009. arXiv:0808.2669. <br />
  39. 39. Other Computational Paradigms<br />33<br />DNA Computing<br />
  40. 40. References<br />34<br />[1] Arora, S., Barak, B., “Computational Complexity: A Modern Approach.”<br />[2] Bernstein, E., Vazirani, U., “Quantum Complexity Theory.”<br />[3] Chuang, I., “Quantum Algorithms and their Implementations: QuISU – An Introduction for Undergraduates.”<br />[4] Lloyd, S., “Quantum Information Science.”<br />[5] Nielson, M., Chuang, I., “Quantum Computation and Quantum Information.”<br />[6] Images Courtesy of Wikipedia.<br />[7] Thanks to Scott Aaronson & Michele Mosca for Slide Inspirations & Figures. <br />
  41. 41. Questions & Comments<br />35<br />Questions?<br />Comments?<br />

×