Introduction to quantum gates

An operation, not an assignment

An ideal quantum gate applies a unitary matrix to a state. X exchanges amplitudes. Z changes the sign of the 1 component. H mixes components so they can reinforce or cancel.

If H is applied first and Z second, the resulting state is $ZH\lvert\psi\rangle$. The rightmost matrix acts first. Circuit diagrams here read left to right, so follow the physical order when translating to algebra.

A small gate dictionary

Gate	Action on a one-qubit state $(\alpha,\beta)^T$
X	$(\beta,\alpha)^T$
Z	$(\alpha,-\beta)^T$
H	$(\alpha+\beta,\alpha-\beta)^T/\sqrt2$
S	$(\alpha,i\beta)^T$
T	$(\alpha,e^{i\pi/4}\beta)^T$

Predict, then build

Apply H to q0 and CX from q0 to q1. Predict which two basis outcomes survive. Use Undo to check the circuit one step at a time.

L02 · INTERACTIVE LABRuns on your device

Quantum gate playground

Gates act on amplitudes. Build a small circuit one operation at a time and compare its exact predictions with repeated measurements.

Make a prediction first

Apply H to q0, then CX from q0 to q1. Which two outcomes should disappear?

Experiment controls

Add a gate · left to right

H q0X q0Z q0H q1X q1CX 0→1

Load Bell-state exampleUndo last gate

|00⟩ → I

Maximum 20 gates. CX has q0 as control, q1 as target.Random seed

Run measurements →Reset

Exact probability

|00⟩

100.0%

|01⟩

0.0%

|10⟩

0.0%

|11⟩

0.0%

Exact ideal-model predictions · run a measurement experiment to compare.

MODEL · Ideal pure-state vector, complex double precisionORDER · q0 is the leftmost bit

Take it further

Create the Bell state, then undo the gates in reverse order. Where did the randomness go?

Prediction: Apply H to q0, then CX from q0 to q1. Which two outcomes should disappear? Challenge: Create the Bell state, then undo the gates in reverse order. Where did the randomness go?

Order can matter

On $\lvert0\rangle$, H followed by Z gives $\lvert-\rangle$. Z followed by H gives $\lvert+\rangle$. Their Z probabilities match, so add a final H to reveal the difference.

Do not infer that two circuits are equivalent merely because they give the same counts for one input in one basis. A gate is a transformation with a contract over its allowed inputs.

Challenge: find two gates in the table that square to I. Then explain why S is not one of them.

Reference: IBM Quantum — Bits, gates, and circuits.