Presentation at UIC

1. Topological Quantum
Computation
Joshua Jay Herman
Presentation is available under the Creative Commons Attribution-ShareAlike License;
Saturday, October 1, 11

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
computer
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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. What can they do
faster?
• Factoring (Shor’s algorithm)
• Approximating the Jones Polynominal
• Searching an unsorted database (Grover’s
Algorithm)
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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. 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. How do we do
Quantum Computation
• Qubits
• Entanglement
• Measurement
• Gates
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8. Qubits
• Short for quantum bit
• Can be |0> or |1>
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9. Entanglement
• Represented by the addition of two state
vectors
• Correlation of states between two vectors
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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=451883621
Saturday, October 1, 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=451883621
Saturday, October 1, 11

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=451883621
Saturday, October 1, 11

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=451883621
Saturday, October 1, 11

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=448532727
Saturday, October 1, 11

15. Measurement
• What happens when we observe a
quantum state
• What occurs is the quantum system
collapses
• What you get back is one state
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16. Topology
• Reidmeister Moves
• Anyons
• Braid Group
• Yang Baxter Equation
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17. Reidemeister Moves I,II
Kurt Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg 5 (1926), 24-32 Diagram from
Reidemeister 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=374283067
Saturday, October 1, 11

18. Reidemeister Move III
■ Kurt Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg 5 (1926), 24-32
Saturday, October 1, 11

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=448305722
Saturday, October 1, 11

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=450094400
Saturday, October 1, 11

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=450094400
Saturday, October 1, 11

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=450094400
Saturday, October 1, 11

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. 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. 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. Future Work
• What new quantum algorithms are faster
than classical algorithms
• Hidden subgroup problem
Wikipedia contributors, "Jones polynomial," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Jones_polynomial&oldid=413672903 (accessed
September 2, 2011).
Saturday, October 1, 11

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=445380938
Saturday, October 1, 11

28. Future Work
Saturday, October 1, 11
• How large is BQP?

29. Questions?
?
Saturday, October 1, 11