Translating Circuits to Math

Every quantum circuit has an equivalent mathematical description: a unitary matrix $U$ that maps input states to output states. Being able to derive that matrix from the diagram is one of the most useful skills in quantum computing — it lets you predict outputs without running a simulator, spot gate identities, and debug circuits by inspection.

Each gate is a matrix; composition is multiplication

A single-qubit gate is a $2 \times 2$ unitary matrix. The familiar Pauli and Hadamard matrices are:

When a circuit applies gate $A$ first and then gate $B$, the combined operation is $B \cdot A$ — matrix multiplication with the later gate on the left. For three gates applied in order $G_1$, $G_2$, $G_3$, the total unitary is $U = G_3 G_2 G_1$.

A worked example: $H \cdot X \cdot H$

Let us derive the unitary for the three-gate sequence $H$, $X$, $H$. First compute $X \cdot H$:

Now multiply on the left by $H$:

The circuit $H \to X \to H$ is exactly the $Z$ gate. This makes geometric sense on the Bloch sphere: $H$ swaps the $z$- and $x$-axes, so a flip about the $x$-axis (which is what $X$ does) becomes, after swapping axes back with $H$, a flip about the $z$-axis — and that is precisely $Z$.

Why unitarity matters

Every physically valid quantum operation must be unitary: $U^\dagger U = I$. Unitarity preserves the total probability (the norm of the state vector) and guarantees reversibility — you can always undo a unitary by applying $U^\dagger$. For the example above, $Z^\dagger = Z$ (it is its own inverse), so $H X H H X H = Z Z = I$, confirming reversibility.

Try it

Implement the circuit $H \to X \to H$, whose unitary is $Z$. The grader compares the full $2 \times 2$ unitary of your circuit, so returning an empty circuit will not pass.

After running, open the Unitary tab to see the resulting matrix: it should be diagonal with entries $1$ and $-1$, exactly matching $Z$.

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