Quantum Gates as Rotations: The Hadamard

In the last post, we built the Bloch sphere: every qubit state is a point on its surface. The north pole is $|0\rangle$, the south pole is $|1\rangle$, the equator is superposition. We have the map. Now we need operations: ways to move that point. But not every map from sphere to sphere is a valid quantum gate. The constraints are tight, and they have a name: unitarity. Understanding them is the same as understanding why the Hadamard works.

Gates as rotations

In the last post we established that a qubit’s state is a point on the surface of the Bloch sphere. The north pole is $|0\rangle$, the south pole is $|1\rangle$, and every other point is a superposition parametrized by $\theta$ and $\varphi$.

A quantum gate is an operation that moves that point. The question is: which moves are allowed?

Not all moves preserve what needs to be preserved. A quantum state is a unit vector; its amplitudes satisfy $|\alpha|^2 + |\beta|^2 = 1$. That constraint is the requirement that probabilities sum to 1. Any gate that stretched or compressed the sphere would violate it: after the gate, the probabilities would no longer add up.

The operations that preserve the sphere are exactly the rigid rotations. A rotation moves the point without changing the fact that it’s on the sphere.

There is a second requirement: quantum mechanics is reversible. Every valid gate must be undoable. Rotations are reversible; we can always rotate back. A projection or a map that collapses two states to one would be irreversible and therefore not a valid gate.

Unitarity

The formal name for the class of operations that are rigid rotations on the Bloch sphere is unitary transformations. A matrix $U$ is unitary if:

$U^\dagger U = UU^\dagger = I$

where $U^\dagger$ is the conjugate transpose of $U$. This is the matrix statement of “rigid rotation,” and it implies two things directly:

Norm preservation: for any state $|\psi\rangle$, applying $U$ gives a state with the same norm. Probabilities still sum to 1.

Invertibility: $U^{-1} = U^\dagger$. Every gate has an inverse, and that inverse is the conjugate transpose. To undo a gate, apply its conjugate transpose.

Every valid single-qubit gate is a $2 \times 2$ unitary matrix. No more, no less. [1, §4.2]

A classical NOT gate is a special case: it swaps $|0\rangle \leftrightarrow |1\rangle$, which is a 180° rotation around the x-axis of the Bloch sphere. Quantum gates generalize this to rotations around any axis, by any angle.

The Hadamard geometrically

The gate is named after Jacques Hadamard (1865–1963), a French mathematician whose work touched prime number theory, partial differential equations, and the geometry of determinants. In an 1893 paper [3], Hadamard studied a question about how large the determinant of a real matrix can be relative to its entries, and showed that matrices with entries in $\{+1, -1\}$ achieve the maximum. These are now called Hadamard matrices. The $2 \times 2$ case (up to the normalizing $1/\sqrt{2}$ factor) is the gate we’re studying. The name carried into quantum computing through the canonical textbook by Nielsen and Chuang [1], and has since become universal.

The Hadamard gate, written $H$, is a 180° rotation around the axis pointing in the $(x+z)/\sqrt{2}$ direction, the diagonal halfway between the x and z axes of the Bloch sphere.

Try it

The matrix

With the geometry in place, the matrix is just a coordinate description of the same rotation. In the standard $|0\rangle$/$|1\rangle$ basis:

$H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$

Check unitarity: $H$ is real and symmetric, so $H^\dagger = H$. And:

$H^2 = \frac{1}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\begin{pmatrix}1&1\\1&-1\end{pmatrix} = \frac{1}{2}\begin{pmatrix}2&0\\0&2\end{pmatrix} = I$

Both properties confirm what the geometry told us: $H$ is its own inverse.

Reading a quantum circuit

The widget uses the standard notation for quantum circuits. Here is what we need.

A wire represents a qubit over time, read left to right. The qubit starts in a known state at the left end and evolves as gates act on it.

A box on the wire is a gate. The label names the gate: $H$ for Hadamard, $X$ for NOT, $Z$ for the phase flip. Read a wire from left to right to see the sequence of operations.

A meter symbol at the end of a wire is a measurement. It collapses the qubit to a classical bit, 0 or 1, based on the outcome probabilities. The double wire after the meter carries that classical bit.

Gates in sequence compose: the output of one gate is the input to the next. The widget shows $|0\rangle \to H \to H \to$ measure. The first H creates superposition; the second H undoes it. Stepping through makes H·H = I visible.

Order matters. Matrix multiplication is not commutative in general, and neither is gate composition. Applying $X$ then $H$ is not the same as $H$ then $X$. As a worked example: $X|0\rangle = |1\rangle$, then $H|1\rangle = |{-}\rangle$, so $X$ followed by $H$ takes $|0\rangle$ to $|{-}\rangle$. Compare that with $H|0\rangle = |{+}\rangle$, then $X|{+}\rangle = |{+}\rangle$ (since $|{+}\rangle$ is a $+1$ eigenstate of $X$), so $H$ followed by $X$ takes $|0\rangle$ to $|{+}\rangle$. The two results are different states with different phases. Order matters.

$H$ creates superposition, but superposition alone does not explain why quantum algorithms work. What matters is what happens to the phase, the longitude $\varphi$ on the Bloch sphere, before measurement. $|{+}\rangle$ and $|{-}\rangle$ have the same measurement probabilities but different phases. That difference becomes visible when we apply another gate before measuring. Two paths can reinforce each other or cancel out, depending on their relative phase.

That mechanism is quantum interference.