Checkpoint: Linear Algebra for Qubits

This lesson brings together every linear-algebra tool from Module 2 and shows how each one has a direct role in describing single-qubit operations.

The qubit state space

A single qubit lives in $\mathbb{C}^2$. Any pure state is a normalised column vector

The computational basis vectors $|0\rangle = \begin{pmatrix}1\\0\end{pmatrix}$ and $|1\rangle = \begin{pmatrix}0\\1\end{pmatrix}$ are orthonormal: their inner products satisfy $\langle 0|0\rangle = \langle 1|1\rangle = 1$ and $\langle 0|1\rangle = 0$. The norm condition $\|\,|\psi\rangle\,\| = 1$ is therefore equivalent to the Born-rule requirement that probabilities sum to 1.

Single-qubit gates as $2 \times 2$ unitary matrices

Every physically allowed operation on a qubit is a unitary linear map: a $2 \times 2$ matrix $U$ satisfying $U^\dagger U = I$. Two gates that appear constantly are the Pauli-X and the Hadamard.

The Pauli-X gate flips the two basis states. Its matrix is

One can verify $X^\dagger X = I$ directly: $X$ is real and symmetric, so $X^\dagger = X$, and $X^2 = I$.

The Hadamard gate creates an equal superposition:

A short calculation confirms it is unitary:

Composing gates: matrix multiplication

Applying $H$ and then $X$ to $|0\rangle$ means computing the matrix product $X \cdot (H|0\rangle)$, or equivalently $(XH)|0\rangle$. Working step by step:

Step 1 — apply $H$ to $|0\rangle$:

This is the $|{+}\rangle$ eigenstate of $X$, with both amplitudes equal to $\tfrac{1}{\sqrt{2}}$.

Step 2 — apply $X$ to $|{+}\rangle$:

The state is unchanged: $|{+}\rangle$ is an eigenvector of $X$ with eigenvalue $+1$. This is a direct consequence of the eigenvalue equation $X|{+}\rangle = (+1)|{+}\rangle$, which can be verified by the characteristic polynomial of $X$:

The eigenvalue $+1$ eigenvector satisfies $(X - I)\mathbf{v} = 0$, giving $v_1 = v_2$, i.e. the normalised form $\tfrac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix} = |{+}\rangle$.

Eigenvalues and measurement outcomes

The eigenvalue equation connects linear algebra directly to physics: measuring a qubit in the eigenbasis of the observable $X$ (the Pauli-X operator) gives outcome $+1$ with certainty when the state is $|{+}\rangle$, because $|{+}\rangle$ is the $+1$ eigenstate of $X$. More generally, if the state is $|\psi\rangle$ and we measure observable $A$ with eigenvalues $\lambda_j$ and normalised eigenvectors $|j\rangle$, the probability of outcome $\lambda_j$ is $|\langle j|\psi\rangle|^2$ (the Born rule) and the expected value is

This expectation value formula is a direct application of matrix multiplication and inner products — the same operations practised throughout this module.

Try it

Build a one-qubit circuit that applies $H$ and then $X$ to qubit 0. After pressing Run you should see both amplitudes equal $\tfrac{1}{\sqrt{2}} \approx 0.707$ — confirming that $XH|0\rangle = |{+}\rangle$ exactly as computed above.

Sign in on the full site to ask questions and join the discussion.