  In this paper we construct a new $8\times8$ $\mathbb{M}$ matrix from the
$4\times4$ $M$ matrix, where $\mathbb{M}$ / $M$ is the image of the braid group
representation. The $ 8\times8 $ $\mathbb{M}$ matrix and the $4\times4$ $M$
matrix both satisfy extraspecial 2-groups algebra relations. By
Yang-Baxteration approach, we derive a unitary $ \breve{R}(\theta, \phi)$
matrix from the $\mathbb{M}$ matrix with parameters $\phi$ and $\theta$.
Three-qubit entangled states can be generated by using the
$\breve{R}(\theta,\phi)$ matrix. A Hamiltonian for 3 qubits is constructed from
the unitary $\breve{R}(\theta,\phi)$ matrix. We then study the entanglement and
Berry phase of the Yang-Baxter system.

Quantum Physics