Pauli group

In physics and mathematics, the Pauli group G 1 {displaystyle G_{1}} on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I {displaystyle I} and all of the Pauli matrices In physics and mathematics, the Pauli group G 1 {displaystyle G_{1}} on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I {displaystyle I} and all of the Pauli matrices together with the products of these matrices with the factors − 1 {displaystyle -1} and ± i {displaystyle pm i} : The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli. The Pauli group on n qubits, G n {displaystyle G_{n}} , is the group generated by the operators described above applied to each of n {displaystyle n} qubits in the tensor product Hilbert space ( C 2 ) ⊗ n {displaystyle (mathbb {C} ^{2})^{otimes n}} . As an abstract group, G 1 ≅ C 4 ∘ D 4 {displaystyle G_{1}cong C_{4}circ D_{4}} is the central product of a cyclic group of order 4 and the dihedral group of order 8.