Abstract: This talk will begin with an introduction to the general idea of quantum computing and quantum algorithms. Then we will discuss how there is the possibility of using topology to both investigate topology problems through quantum formulations and to use physical situations that involve topology to create quantum computers. The latter is an active area of research, due to surprising and sometimes simple relationships between topology and physics. For example, the algebra of fermion creation and annihilation operators is generated by a Clifford algebra of Majorana fermion operators. These operators (call them a, b and c))
satisfy a^2 = b^2 = c^2 = 1 and ab = -ba, ac = -ca, bc = - cb.
Then the quaternions arise via I = ba, J = cb, K = ac with
I^2 = J^2 = K^2 = IJK = -1 and if we define R = (1 + I)/sqrt(2),
S = (1 + J)/sart(2) and T = (1 + K)/sqrt(2), then RSR = SRS, STS = TST,
RTR = TRT giving unitary braid group representations associated with fermions.
Topology occurs naturally in basic quantum physics.
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