The Exercise

The exercise is about a single qubit and two of its Pauli operators, $X$ and $Y$. It asks for the product $XYX$, the operator $Y$ applied between two bit flips, and the answer is $-Y$.^✦

The Pauli Matrices

The library states the claim for the matrices of $X$ and $Y$. The Pauli matrix $X$ is the complex $2 \times 2$ matrix $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$$(1)^✦ The Pauli matrix $Y$ is the complex $2 \times 2$ matrix $$Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},$$(2)^✦ with $i$ the imaginary unit.

The Claim

What Is Checked

The equality (3) is one of matrices: all four entries agree, with no global phase left free.