5. Sequence of Pauli gates
Basic Quantum CircuitsSimplify the following sequence of Pauli gates $$ U = XYXXZXXXYXXXXZXXXXXY. $$
Hint
Think about what the result of n Pauli X gates in series is. Also what is the result of XY?
Solution
By inspection we have
$$
XX =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
= I
$$
Therefore, $XXX = X(XX) = X I = X$. So for n Pauli X gates where n is odd we end up with a single X gate and when n is even we end up with identity I. Same goes for Y gates and Z gates.
Moreover, we have $$ XY = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} = i Z $$
Therefore, $$ U = XYXXZXXXYXXXXZXXXXXY = (iZ)^5 = i Z $$