r1Gate

z-axis rotation gate with global phase

Since R2023a

Syntax

g = r1Gate(targetQubit,theta)

Description

Add-On Required: This feature requires the MATLAB Support Package for Quantum Computing add-on.

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 $$|1\rangle$$ 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.

Examples

R1 Gate and Its Matrix Representation

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

Array of R1 Gates

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

targetQubit — Target qubit of gate

positive integer scalar | positive integer vector

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

Example: 1

Example: 3:5

theta — Rotation angle

real scalar | real vector | string scalar | string vector | symbolic scalar | symbolic vector

Rotation angle, specified as a real scalar, real vector, string scalar, string vector, symbolic scalar, or symbolic vector. Specify the rotation angle as a vector to create an array of gates where each element of the vector corresponds to the rotation angle of one gate.

Use strings or symbolic values to specify variables for parameterized rotation angles. You can specify values for parameters when calling the simulate, getMatrix, and observe functions on a quantum circuit that contains parameterized gates. (since R2026a)

Example: pi

Example: (1:3)*pi/2

Example: ["x" "y" "z"]

More About

Matrix Representation of R1 Gate

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

$$\left[\begin{array}{cc}1& 0\\ 0& \exp \left(i\theta \right)\end{array}\right].$$

This gate changes the phase of the $$|1\rangle$$ state by angle of $$\theta$$ and leave the $$|0\rangle$$ 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.

$$R_1\left(\theta \right)=\exp \left(i\frac{\theta }{2}\right)\left[\begin{array}{cc}\exp \left(-i\frac{\theta }{2}\right)& 0\\ 0& \exp \left(i\frac{\theta }{2}\right)\end{array}\right]=\exp \left(i\frac{\theta }{2}\right)R_z\left(\theta \right)$$

Version History

Introduced in R2023a

R2026a: Create rotation gates with parameterized angles (July 2026)

Create a rotation gate with a parameterized angle by specifying the theta argument as a string scalar, string vector, symbolic scalar, or symbolic vector.

You can specify values for parameters when calling the simulate, getMatrix, and observe functions on a quantum circuit that contains parameterized gates.