M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Hadamard Eigen-Basis


The Hadamard gate \(H\) is given by

\begin{equation*} H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \end{equation*}

Since \(H^{2} = I\), the eigenvalues of the Hadamard matrix are \(\pm 1\). Let \(\lvert H_{\pm} \rangle\) be the eigenvector with eigenvalue \(\pm1\). Then

\begin{align*} \lvert H_{+} \rangle &= \frac{1}{\sqrt{2}} \sqrt{1 + \frac{1}{\sqrt{2}}} \lvert {0} \rangle + \frac{1}{\sqrt{2}} \sqrt{1 - \frac{1}{\sqrt{2}}} \lvert {1} \rangle \\ | H_{-} \rangle &= -\frac{1}{\sqrt{2}} \sqrt{1 - \frac{1}{\sqrt{2}}} \lvert {0} \rangle + \frac{1}{\sqrt{2}} \sqrt{1 + \frac{1}{\sqrt{2}}} \lvert {1} \rangle \end{align*}

Note that

\begin{align*} \frac{1}{\sqrt{2}} \sqrt{1 + \frac{1}{\sqrt{2}}} &= \frac{\sqrt{2 + \sqrt{2}}}{2} = \cos\left( \frac{\pi}{8} \right) = 0.923879532511... \\ \frac{1}{\sqrt{2}} \sqrt{1 - \frac{1}{\sqrt{2}}} &= \frac{\sqrt{2 - \sqrt{2}}}{2} = \sin\left( \frac{\pi}{8} \right) = 0.382683432365... \end{align*}

Thus, the eigenvectors can also be written as

\begin{align*} \lvert H_{+} \rangle &= \cos\left( \frac{\pi}{8} \right) \lvert {0} \rangle + \sin\left( \frac{\pi}{8} \right) \lvert {1} \rangle & \lvert H_{-} \rangle &= -\sin\left( \frac{\pi}{8} \right) \lvert {0} \rangle + \cos\left( \frac{\pi}{8} \right) \lvert {1} \rangle \end{align*}

or, equivalently,

\begin{align*} \lvert H_{+} \rangle &= \cos\left( -\frac{\pi}{8} \right) \lvert {0} \rangle - \sin\left( -\frac{\pi}{8} \right) \lvert {1} \rangle & \lvert H_{-} \rangle &= \sin\left( -\frac{\pi}{8} \right) \lvert {0} \rangle + \cos\left( -\frac{\pi}{8} \right) \lvert {1} \rangle \end{align*}

The eigen-basis for \(X\) and \(Z\) can also be written in this form:

\begin{align*} \lvert Z_{+} \rangle &= \lvert {0} \rangle = \cos(0) \lvert {0} \rangle - \sin(0) \lvert {1} \rangle & \lvert Z_{-} \rangle &= \lvert {1} \rangle = \sin(0) \lvert {0} \rangle + \cos(0) \lvert {1} \rangle \\ \lvert X_{-} \rangle &= \lvert {-} \rangle = \cos\left( \frac{\pi}{4} \right) \lvert {0} \rangle - \sin\left( \frac{\pi}{4} \right) \lvert {1} \rangle & \lvert X_{+} \rangle &= \lvert {+} \rangle = \sin\left( \frac{\pi}{4} \right) \lvert {0} \rangle + \cos\left( \frac{\pi}{4} \right) \lvert {1} \rangle \end{align*}

Now you know.