Quantum Inf Process (2016) 15:1135–1188
Topological quantum computation within the anyonic
system the Kauffman–Jones version of SU(2)
Chern–Simons theory at level 4
Received: 3 June 2015 / Accepted: 12 January 2016 / Published online: 4 February 2016
© Springer Science+Business Media New York 2016
Abstract By braiding and measuring anyons, we realize irrational qubit and qutrit
phase gates within the anyonic system the Kauffman-Jones version of SU(2) Chern-
Simons theory at level 4. We obtain universality on 1-qubit and 1-qutrit gates. In the
qubit case, we also provide a protocol for realizing the controlled NOT gate, thus
leading to universality on n-qubit gates.
Keywords Quantum computation with anyons · Braids · Fusion measurements ·
Interferometric measurements · Ancilla preparation · Irrational qubit and qutrit phase
gates · Computational universality for n-qubit gates · Computational universality for
1-qutrit gates · Controlled NOT gate
We present protocols allowing to perform any unitary operation on n-qubits on a
topological quantum computer within the anyonic system the Kauffman–Jones version
. The minor modiﬁcations resulting from studying SU(2)
instead of the
Kauffman–Jones version of SU(2)
get explained in . The quantum gates are made
by braiding and measuring quasiparticles called anyons.
The ﬁrst two sections deal with 1-qudit gates for d = 2, 3, and the last section
deals with 2-qubit gates. In 2001, Ranee and Jean-Luc Brylinski showed as part of
their work that if we can approximate any 1-qubit gate and if we have only one 2-qubit
entangling gate, then we can approximate any n-qubit gate.
Claire Levaillant worked with the Microsoft Research Station Q team.