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 $$|1\rangle$$ 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

Rotation angle, specified as a real scalar or vector.

Example: pi

Example: (1:3)*pi/2

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