cr1Gate

Controlled z-axis rotation gate with global phase

Since R2023a

Syntax

g = cr1Gate(controlQubit,targetQubit,theta)

Description

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

g = cr1Gate(controlQubit,targetQubit,theta) applies a controlled R1 gate (controlled z-axis rotation gate with global phase) to a target qubit based on the state of a control qubit and returns a quantum.gate.SimpleGate object. If the control qubit is in the $$|0\rangle$$ state, then this gate does nothing. If the control qubit is in the $$|1\rangle$$ state, then this gate changes the phase of the target qubit $$|1\rangle$$ state by an angle of theta.

• If controlQubit, targetQubit, and theta are vectors of the same length, cr1Gate returns a column vector of gates, where g(i) represents a controlled R1 gate applied to the ith element triplet of controlQubit, targetQubit, and theta.

• If any of controlQubit, targetQubit, or theta is a scalar, and at least one of the inputs is a vector, then MATLAB^® expands the scalar to match the size of the vector input.

• The qubit indices in each element pair of controlQubit and targetQubit must not be the same.

Examples

Controlled R1 Gate and Its Matrix Representation

Create a controlled R1 gate that acts on a control qubit with index 1 and a target qubit with index 2 using a rotation angle of pi/2.

g = cr1Gate(1,2,pi/2)

g = 

  SimpleGate with properties:

             Type: "cr1"
    ControlQubits: 1
     TargetQubits: 2
           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 + 0.0000i
   0.0000 + 0.0000i   1.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   1.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 1.0000i

Array of Controlled R1 Gates

Create an array of three controlled R1 gates. The first gate acts on control qubit 1 and target qubit 2 with rotation angle pi/4, the next gate acts on control qubit 2 and target qubit 3 with rotation angle pi/2, and the final gate acts on control qubit 3 and target qubit 4 with rotation angle 3*pi/4.

g = cr1Gate(1:3,2:4,pi/4*(1:3))

g = 

  3×1 SimpleGate array with gates:

    Id   Gate   Control   Target   Angle
     1   cr1    1         2        pi/4 
     2   cr1    2         3        pi/2 
     3   cr1    3         4        3pi/4

Input Arguments

controlQubit — Control qubit of gate

positive integer scalar | positive integer vector

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

Example: 2

Example: 6:8

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 Controlled R1 Gate

The matrix representation of a controlled R1 gate applied to control qubit 1 and target qubit 2 with a rotation angle of $$\theta$$ is

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

Because this gate affects only the $$|11\rangle$$ state, swapping the control and target qubits does not change the gate operation.

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.