Topological Quantum                               Computation                                                             ...
How is quantum                          computation topological?                     • Particles that interact in two dime...
Why Quantum                                         Computation?            Quantum          computers are             fas...
What can they do                               faster?                     • Factoring (Shor’s algorithm)                 ...
Why Topological                              Quantum Computers?Topological quantum computers are faster AND have error    ...
Why Topological                              Quantum Computers?   Also, the approximation of the Jones Polynominal was firs...
How do we do                  Quantum Computation                     • Qubits                     • Entanglement         ...
Qubits                     • Short for quantum bit                     • Can be |0> or |1>Saturday, October 1, 11
Entanglement                     • Represented by the addition of two state                          vectors              ...
Quantum Gates                     • Hadamard                     • Phase shift gate                     • Toftoli Gate    ...
Hadamard         Representation of a rotation by             Pi on the x and z axes         Important in the Hadamard Test...
Phase Shift                       Gate         Rotates the input vector(s) by a               specific phase pi/2 Quantum g...
2                                                3                                                                        ...
2                                                                 3                                               1       ...
Measurement                     • What happens when we observe a                          quantum state                   ...
Topology                     • Reidmeister Moves                     • Anyons                     • Braid Group           ...
Reidemeister Moves I,IIKurt Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg 5 (1926),...
Reidemeister Move III                           ■   Kurt Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math....
Braid Group                      •      Closed under                             concatenation                      •     ...
Anyons                     • In 3-D we encounter Bosons and Fermions                     • In 2-D we encounter Anyons (Due...
Anyons Continued                     • Due to the properties of the particles being                            in 2-D we c...
Anyons Continued                     • The anyonic wavefunctions are simply 1                            dimentional repre...
Yang Baxter Equation            quant-ph/0401090 Braiding Operators are Universal Quantum Gates. Louis H. Kauffman, Samuel...
Yang Baxter & Braiding            quant-ph/0401090 Braiding Operators are Universal Quantum Gates. Louis H. Kauffman, Samu...
The R Gate          •      A universal quantum                 gate.          •      Shown to be equivalent               ...
Future Work                     • What new quantum algorithms are faster                            than classical algorit...
Hidden Subgroup                                               Problem                        • Given a group G and a finite...
Future WorkSaturday, October 1, 11                           • How large is BQP?
Questions?                             ?Saturday, October 1, 11
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Topological quantumcomputation

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Topological quantumcomputation

  1. 1. Topological Quantum Computation Joshua Jay HermanPresentation is available under the Creative Commons Attribution-ShareAlike License;Saturday, October 1, 11
  2. 2. How is quantum computation topological? • Particles that interact in two dimentions and braid according to paths in space and time can create a topological quantum computerSaturday, October 1, 11
  3. 3. Why Quantum Computation? Quantum computers are faster. Citation: Wikipedia contributors, "BQP," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=BQP&oldid=432055421 (accessed September 2, 2011).Saturday, October 1, 11
  4. 4. What can they do faster? • Factoring (Shor’s algorithm) • Approximating the Jones Polynominal • Searching an unsorted database (Grover’s Algorithm)Saturday, October 1, 11
  5. 5. Why Topological Quantum Computers?Topological quantum computers are faster AND have error correcting properties.[1] P. W. Shor, Fault-tolerant quantum computation, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science, edited by R. S. Sip- ple, IEEE (IEEE Press, Los Alamitos, CA, 14–16 Oct. 1996, Burlington, VT, USA, 1996), pp. 56–65, ISBN 0-8186-7594-2, doi:10.1137/S0097539795293172, arXiv:quant-ph/9605011.Saturday, October 1, 11
  6. 6. Why Topological Quantum Computers? Also, the approximation of the Jones Polynominal was first done on a Topological Quantum Computer[1] P. W. Shor, Fault-tolerant quantum computation, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science, edited by R. S. Sip- ple, IEEE (IEEE Press, Los Alamitos, CA, 14–16 Oct. 1996, Burlington, VT, USA, 1996), pp. 56–65, ISBN 0-8186-7594-2, doi:10.1137/S0097539795293172, arXiv:quant-ph/9605011.Saturday, October 1, 11
  7. 7. How do we do Quantum Computation • Qubits • Entanglement • Measurement • GatesSaturday, October 1, 11
  8. 8. Qubits • Short for quantum bit • Can be |0> or |1>Saturday, October 1, 11
  9. 9. Entanglement • Represented by the addition of two state vectors • Correlation of states between two vectorsSaturday, October 1, 11
  10. 10. Quantum Gates • Hadamard • Phase shift gate • Toftoli Gate • CNOT Quantum gate. (2011, September 22). In Wikipedia, The Free Encyclopedia. Retrieved 16:05, October 1, 2011, from //en.wikipedia.org/w/index.php? title=Quantum_gate&oldid=451883621Saturday, October 1, 11
  11. 11. Hadamard Representation of a rotation by Pi on the x and z axes Important in the Hadamard Test Quantum gate. (2011, September 22). In Wikipedia, The Free Encyclopedia. Retrieved 16:05, October 1, 2011, from //en.wikipedia.org/w/index.php? title=Quantum_gate&oldid=451883621Saturday, October 1, 11
  12. 12. Phase Shift Gate Rotates the input vector(s) by a specific phase pi/2 Quantum gate. (2011, September 22). In Wikipedia, The Free Encyclopedia. Retrieved 16:05, October 1, 2011, from //en.wikipedia.org/w/index.php? title=Quantum_gate&oldid=451883621Saturday, October 1, 11
  13. 13. 2 3 1 0 0 0 CNOT Gate 60 6 1 0 077 Basically a not gate which can be 40 0 0 15 switched on and off given another input. 0 0 1 0 Quantum gate. (2011, September 22). In Wikipedia, The Free Encyclopedia. Retrieved 16:05, October 1, 2011, from //en.wikipedia.org/w/index.php? title=Quantum_gate&oldid=451883621Saturday, October 1, 11
  14. 14. 2 3 1 0 0 0 0 0 0 0 60 1 0 0 0 0 0 07 6 7 60 07 Toftoli Gate 6 60 6 0 0 1 0 0 1 0 0 0 0 0 0 07 7 7 60 Also a reversible classical gate. 6 0 0 0 1 0 0 077 Also called a CCNOT gate. 60 6 0 0 0 0 1 0 077 40 0 0 0 0 0 0 15 0 0 0 0 0 0 1 0 Toffoli gate. (2011, September 5). In Wikipedia, The Free Encyclopedia. Retrieved 16:26, October 1, 2011, from //en.wikipedia.org/w/index.php? title=Toffoli_gate&oldid=448532727Saturday, October 1, 11
  15. 15. Measurement • What happens when we observe a quantum state • What occurs is the quantum system collapses • What you get back is one stateSaturday, October 1, 11
  16. 16. Topology • Reidmeister Moves • Anyons • Braid Group • Yang Baxter EquationSaturday, October 1, 11
  17. 17. Reidemeister Moves I,IIKurt Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg 5 (1926), 24-32 Diagram fromReidemeister move. (2010, July 19). In Wikipedia, The Free Encyclopedia. Retrieved 02:13, October 1, 2011, from//en.wikipedia.org/w/index.php?title=Reidemeister_move&oldid=374283067Saturday, October 1, 11
  18. 18. Reidemeister Move III ■ Kurt Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg 5 (1926), 24-32Saturday, October 1, 11
  19. 19. Braid Group • Closed under concatenation • Can represent any knot Braid group. (2011, September 4). In Wikipedia, The Free Encyclopedia. Retrieved 14:31, October 1, 2011, from //en.wikipedia.org/w/index.php?title=Braid_group&oldid=448305722Saturday, October 1, 11
  20. 20. Anyons • In 3-D we encounter Bosons and Fermions • In 2-D we encounter Anyons (Due to the Fractional Quantum Hall Effect) Anyon. (2011, September 12). In Wikipedia, The Free Encyclopedia. Retrieved 16:32, October 1, 2011, from //en.wikipedia.org/w/index.php? title=Anyon&oldid=450094400Saturday, October 1, 11
  21. 21. Anyons Continued • Due to the properties of the particles being in 2-D we can have crossing and knotted structures • Anyons braid due to their worldlines or paths through time and space. Anyon. (2011, September 12). In Wikipedia, The Free Encyclopedia. Retrieved 16:32, October 1, 2011, from //en.wikipedia.org/w/index.php? title=Anyon&oldid=450094400Saturday, October 1, 11
  22. 22. Anyons Continued • The anyonic wavefunctions are simply 1 dimentional representations of the braid group Anyon. (2011, September 12). In Wikipedia, The Free Encyclopedia. Retrieved 16:32, October 1, 2011, from //en.wikipedia.org/w/index.php? title=Anyon&oldid=450094400Saturday, October 1, 11
  23. 23. Yang Baxter Equation quant-ph/0401090 Braiding Operators are Universal Quantum Gates. Louis H. Kauffman, Samuel J. Lomonaco Jr. physics.quant-ph.Saturday, October 1, 11
  24. 24. Yang Baxter & Braiding quant-ph/0401090 Braiding Operators are Universal Quantum Gates. Louis H. Kauffman, Samuel J. Lomonaco Jr. physics.quant-ph.Saturday, October 1, 11
  25. 25. The R Gate • A universal quantum gate. • Shown to be equivalent to a CNOT gate. quant-ph/0401090 Braiding Operators are Universal Quantum Gates. Louis H. Kauffman, Samuel J. Lomonaco Jr. physics.quant-ph.Saturday, October 1, 11
  26. 26. Future Work • What new quantum algorithms are faster than classical algorithms • Hidden subgroup problemWikipedia contributors, "Jones polynomial," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Jones_polynomial&oldid=413672903 (accessedSeptember 2, 2011).Saturday, October 1, 11
  27. 27. Hidden Subgroup Problem • Given a group G and a finite set X. Let there be a function from G to X which hides the group. • The function is given by a oracle. • The problem is to determine the subgroup Hidden subgroup problem. (2011, August 17). In Wikipedia, The Free Encyclopedia. Retrieved 16:22, October 1, 2011, from //en.wikipedia.org/w/index.php?title=Hidden_subgroup_problem&oldid=445380938Saturday, October 1, 11
  28. 28. Future WorkSaturday, October 1, 11 • How large is BQP?
  29. 29. Questions? ?Saturday, October 1, 11

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