Rz gate

As explained in Building Quantum Software with Python

The RZ gate is a single-qubit quantum gate that rotates the zero and one amplitudes in opposite directions by half of a specified angle.

The RZ gate rotates the amplitudes of a single-qubit state in opposite directions. For angle θ, the |0⟩ amplitude turns clockwise by θ/2, while the |1⟩ amplitude turns counterclockwise by θ/2. Their magnitudes, and therefore the outcome probabilities, remain unchanged. section 3.2.2

Represent a state as a pair of amplitudes and update both directions:

• Multiply the |0⟩ amplitude by cis(-θ/2). section 3.2.2
• Multiply the |1⟩ amplitude by cis(θ/2). section 3.2.2
• Keep both amplitude magnitudes, and thus both measurement probabilities, unchanged. section 3.2.2

Applying RZ(π/3)

theta = pi/3 state = [cis(-theta/2)*state[0], cis(theta/2)*state[1]] section 3.2.2

Figure 3.22 The general form of the effect of the R_Z gate on a pair of amplitudes

Visualizes the opposite rotations applied to the two amplitudes. section 3.2.2

How Building Quantum Software with Python covers it

How deep it goes

Explains amplitude directions, probability preservation, and a direct state-update implementation, without deriving matrix internals.

Examples

Demonstrates RZ(π/3) through a two-element state-update expression and amplitude-direction diagrams.

What it assumes

Assumes single-qubit amplitudes and probabilities, while using complex-number multiplication to express rotations.

What you can do after

Apply it by multiplying each amplitude by an opposite half-angle complex rotation.

How it connects

Connects single-qubit states, complex amplitudes, phase, and measurement probabilities with the Z, P, H, and Y gates.

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Related concepts

• rotational gatekind of · A rotational gate is a single-qubit quantum gate that rotates a qubit’s state by a specified angle about the Y axis of the Bloch sphere.
• phase gatecontrasts with · The Z gate is a single-qubit Pauli quantum operation that leaves |0⟩ unchanged and multiplies |1⟩ by −1.

This explainer was generated by Manning's knowledge graph from the text of Building Quantum Software with Python. The section links point to the parts of the book it draws on.