Main Content

# r1Gate

z-axis rotation gate with global phase

Since R2023a

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

## Syntax

``g = r1Gate(targetQubit,theta)``

## Description

````g = r1Gate(targetQubit,theta)` applies an R1 gate (z-axis rotation gate with global phase) to a single target qubit and returns a `quantum.gate.SimpleGate` object. This gate changes the phase of the $|1〉$ state by an angle of `theta`. If `targetQubit` and `theta` are vectors of qubit indices and angles with the same length, `r1Gate` returns a column vector of gates, where `g(i)` represents a z-axis rotation gate with global phase 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.```

example

## Examples

collapse all

Create an R1 gate that acts on a single qubit with rotation angle `pi/2`.

`g = r1Gate(1,pi/2)`
```g = SimpleGate with properties: Type: "r1" ControlQubits: [1×0 double] TargetQubits: 1 Angles: 1.5708```

Get the matrix representation of the gate.

`M = getMatrix(g)`
```M = 1.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 1.0000i```

Create an array of three R1 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 = r1Gate(1:3,pi/4*(1:3))`
```g = 3×1 SimpleGate array with gates: Id Gate Control Target Angle 1 r1 1 pi/4 2 r1 2 pi/2 3 r1 3 3pi/4```

## Input Arguments

collapse all

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`

## More About

collapse all

### Matrix Representation of R1 Gate

The matrix representation of an R1 gate applied to a target qubit with a rotation angle of $\theta$ is

`$\left[\begin{array}{cc}1& 0\\ 0& \mathrm{exp}\left(i\text{\hspace{0.17em}}\theta \right)\end{array}\right].$`

This gate changes the phase of the $|1〉$ state by angle of $\theta$ and leave the $|0〉$ state as is. Applying this gate with rotation angle $\theta =\pi$ is equivalent to applying a Pauli Z gate (`zGate`).

This gate is also equivalent to the z-axis rotation gate (`rzGate`) 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