Rotation Matrices

Understand how single-qubit gates rotate states on the Bloch sphere.

Rotation Matrices on the Bloch Sphere

Every single-qubit gate is a rotation on the Bloch sphere. Understanding this gives you the ability to construct any gate from first principles.

The Three Rotation Axes

Rotation about X-axis by angle θ:

R X (θ) = (cos (θ /2) - i sin (θ /2) - i sin (θ /2) cos (θ /2))

qc.rx(theta, qubit)   # theta in radians

Rotation about Y-axis by angle θ:

R Y (θ) = (cos (θ /2) sin (θ /2) - sin (θ /2) cos (θ /2))

qc.ry(theta, qubit)

Rotation about Z-axis by angle θ:

R Z (θ) = (e^{- i θ /2} 0 0 e^{i θ /2})

qc.rz(theta, qubit)

Familiar Gates as Rotations

Gate	Rotation
X	RX(π)
Y	RY(π)
Z	RZ(π)
H	RZ(π) then RY(π/2)
S	RZ(π/2)
T	RZ(π/4)

The H row is a circuit-order sequence: apply RZ(π) first, then RY(π/2). As a matrix product that is RY(π/2)·RZ(π), which equals H up to a global phase: $H = e^{iπ /2} R_{Y} (π /2) \cdot R_{Z} (π)$ . The Pauli, S, and T rows also hold up to a global phase, since RZ uses the symmetric convention.

The Y gate in that table deserves a word, since this track has so far leaned on X and Z. Y is the third Pauli gate: a combined bit flip and phase flip (Y = iXZ), and it rotates about the Y axis exactly as X and Z rotate about theirs.

Key Insight

Any single-qubit unitary can be decomposed as:

U = e^{i δ} R Z (α) \cdot R Y (β) \cdot R Z (γ)

for some angles α, β, γ and a global phase δ. This ZYZ decomposition is how quantum compilers translate arbitrary unitary matrices into physical gates.

Commutation and Non-Commutation

•RX and RY do not commute: RX(θ)RY(φ) ≠ RY(φ)RX(θ) in general
•RZ and RZ do commute: applying two Z-rotations is equivalent to one combined rotation
•This is why gate ordering matters in quantum circuits!

How this lesson works

A guided reading lesson with interactive knowledge checks. Concepts are explained step by step with circuit diagrams and runnable examples, and you confirm understanding before moving on.

Part of: Quantum Foundations

Learn the basics: qubits, gates, superposition, and measurement.

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