# rzGate

z-axis rotation gate

Since R2023a

Installation Required: This functionality requires MATLAB Support Package for Quantum Computing.

## Syntax

``g = rzGate(targetQubit,theta)``

## Description

example

````g = rzGate(targetQubit,theta)` applies a z-axis rotation gate to a single target qubit and returns a `quantum.gate.SimpleGate` object. This gate rotates the qubit state around the z-axis by an angle of `theta`. If `targetQubit` and `theta` are vectors of the same length, `rzGate` returns a column vector of gates, where `g(i)` represents a z-axis rotation gate applied to a qubit with index `targetQubit(i)` with a rotation angle of `theta(i)`. If either `targetQubit` or `theta` is a scalar, and the other input is a vector, then MATLAB® expands the scalar to match the size of the vector input.```

## Examples

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Create a z-axis rotation gate that acts on a single qubit with rotation angle `pi/2`.

`g = rzGate(1,pi/2)`
```g = SimpleGate with properties: Type: "rz" ControlQubits: [1×0 double] TargetQubits: 1 Angles: 1.5708```

Get the matrix representation of the gate.

`M = getMatrix(g)`
```M = 0.7071 - 0.7071i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.7071 + 0.7071i```

Create an array of three z-axis rotation gates. The first gate acts on qubit 1 with rotation angle `pi/4`, the next gate acts on qubit 2 with rotation angle `pi/2`, and the final gate acts on qubit 3 with rotation angle `3*pi/4`.

`g = rzGate(1:3,pi/4*(1:3))`
```g = 3×1 SimpleGate array with gates: Id Gate Control Target Angle 1 rz 1 pi/4 2 rz 2 pi/2 3 rz 3 3pi/4```

## Input Arguments

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Target qubit of the gate, specified as a positive integer scalar index or vector of qubit indices.

Example: `1`

Example: `3:5`

Rotation angle, specified as a real scalar or vector.

Example: `pi`

Example: `(1:3)*pi/2`

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### Matrix Representation of z-Axis Rotation Gate

The matrix representation of a z-axis rotation gate applied to a target qubit with a rotation angle of $\theta$ is

`$\left[\begin{array}{cc}\mathrm{exp}\left(-i\text{\hspace{0.17em}}\frac{\theta }{2}\right)& 0\\ 0& \mathrm{exp}\left(i\text{\hspace{0.17em}}\frac{\theta }{2}\right)\end{array}\right].$`

Applying this gate with rotation angle $\theta =\pi$ is equivalent to applying a Pauli Z gate (`zGate`) up to a global phase factor.

This gate is also equivalent to the R1 gate (`r1Gate`) with a global phase difference.${R}_{1}\left(\theta \right)=\mathrm{exp}\left(i\text{\hspace{0.17em}}\frac{\theta }{2}\right)\left[\begin{array}{cc}\mathrm{exp}\left(-i\text{\hspace{0.17em}}\frac{\theta }{2}\right)& 0\\ 0& \mathrm{exp}\left(i\text{\hspace{0.17em}}\frac{\theta }{2}\right)\end{array}\right]=\mathrm{exp}\left(i\text{\hspace{0.17em}}\frac{\theta }{2}\right){R}_{z}\left(\theta \right)$

## Version History

Introduced in R2023a