r1Gate

z-axis rotation gate with global phase

Since R2023a

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

Description

example

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.

Examples

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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

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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 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