topological quantumcomputation

Topological Quantum Computation

Joshua Jay Herman

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Saturday, October 1, 11

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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


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).


http://en.wikipedia.org/w/index.php?title=BQP&oldid=432055421

http://en.wikipedia.org/w/index.php?title=BQP&oldid=432055421

What can they do faster?

• Factoring (Shor’s algorithm)

• Approximating the Jones Polynominal

• Searching an unsorted database (Grover’s Algorithm)


Why Topological Quantum Computers?

[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.

Topological quantum computers are faster AND have error correcting properties.


Why Topological Quantum Computers?

[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.

Also, the approximation of the Jones Polynominal was first done on a Topological Quantum Computer


How do we do Quantum Computation

• Qubits

• Entanglement

• Measurement

• Gates


Qubits

• Short for quantum bit

• Can be |0> or |1>


Entanglement

• Represented by the addition of two state vectors

• Correlation of states between two vectors


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


HadamardRepresentation of a rotation by

Pi on the x and z axesImportant in the Hadamard Test



Phase Shift Gate

Rotates the input vector(s) by a specific phase pi/2



2

664

1 0 0 00 1 0 00 0 0 10 0 1 0

3

775CNOT Gate

Basically a not gate which can be switched on and off given

another input.



2

66666666664

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 0

3

77777777775

Toftoli GateAlso a reversible classical gate.

Also called a CCNOT gate.

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


Measurement

• What happens when we observe a quantum state

• What occurs is the quantum system collapses

• What you get back is one state


Topology

• Reidmeister Moves

• Anyons

• Braid Group

• Yang Baxter Equation


Reidemeister Moves I,IIKurt 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


http://en.wikipedia.org/wiki/Kurt_Reidemeister


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




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


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


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.



Anyons Continued

• The anyonic wavefunctions are simply 1 dimentional representations of the braid group



Yang Baxter Equationquant-ph/0401090 Braiding Operators are Universal Quantum Gates. Louis H. Kauffman, Samuel J. Lomonaco Jr. physics.quant-ph.


http://front.math.ucdavis.edu/0401.4090


http://front.math.ucdavis.edu/author/L.Kauffman


http://front.math.ucdavis.edu/author/S.Lomonaco-Jr


http://front.math.ucdavis.edu/physics.quant-ph


Yang Baxter & Braidingquant-ph/0401090 Braiding Operators are Universal Quantum Gates. Louis H. Kauffman, Samuel J. Lomonaco Jr. physics.quant-ph.










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.










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).


http://en.wikipedia.org/w/index.php?title=Jones_polynomial&oldid=413672903

http://en.wikipedia.org/w/index.php?title=Jones_polynomial&oldid=413672903

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


Future Work

• How large is BQP?Saturday, October 1, 11

Questions?

?Saturday, October 1, 11

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