Quantum entanglement properties of geometrical and topological quantum gates

Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.

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Extraspecial two-Groups, generalized Yang-Baxter equations and braiding quantum gates

Quantum Information and Computation ◽

10.26421/qic10.7-8-8 ◽

2010 ◽

Vol 10 (7&8) ◽

pp. 685-702

Author(s):

E.C. Rowell ◽

Y. Zhang ◽

Y.-S. Wu ◽

M.-L. Ge

Keyword(s):

Quantum Computing ◽

Braid Group ◽

Quantum Error Correction ◽

Unitary Representations ◽

Quantum Gates ◽

Group Representations ◽

Unitary Evolution ◽

Ghz States ◽

Topological Quantum ◽

Quantum Error

In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates generate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly an \emph{odd}) number of qubits, from the product basis. We also discuss the Yang-Baxterization of the new braid group representations, which describes unitary evolution of the GHZ states. Our study suggests that through their connection with braiding gates, extraspecial 2-groups and the GHZ states may play an important role in quantum error correction and topological quantum computing.

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GEOMETRIC PHASE AND TOPOLOGY QUANTUM GATES USING PARALLEL TRANSPORTATION

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887808003041 ◽

2008 ◽

Vol 05 (05) ◽

pp. 789-798

Author(s):

YU-MEI GAO ◽

XIN-DING ZHANG ◽

LIAN HU

Keyword(s):

Josephson Junction ◽

Geometric Phase ◽

Fault Tolerant ◽

Quantum Gates ◽

Initial State ◽

Geometric Evolution ◽

And Topology ◽

Topological Quantum ◽

State Method ◽

Parallel Transportation

We present a novel evolution method to obtain pure geometric phase with orthogonal superposed initial state method (OGSM). As examples we illustrate in detail the geometric evolution both in NMR and superconducting Josephson junction systems, which may be further designed to construct fault-tolerant geometric quantum gates. In the end we also propose a simple way to construct topological quantum gates based on OGSM.

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