chGate

Controlled Hadamard gate

Since R2023a

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

Syntax

g = chGate(controlQubit,targetQubit)

Description

g = chGate(controlQubit,targetQubit) applies a controlled Hadamard gate 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 applies a Hadamard gate (hGate) to the target qubit.

• If controlQubit and targetQubit are vectors of the same length, chGate returns a column vector of gates, where g(i) represents a controlled Hadamard gate applied to a control qubit with index controlQubit(i) and a target qubit with index targetQubit(i).

• If either controlQubit or targetQubit is a scalar, and the other input 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 Hadamard Gate and Its Matrix Representation

Create a controlled Hadamard gate that acts on a control qubit with index 1 and a target qubit with index 2.

g = chGate(1,2)

g = 

  SimpleGate with properties:

             Type: "ch"
    ControlQubits: 1
     TargetQubits: 2
           Angles: [1×0 double]

Get the matrix representation of the gate.

M = getMatrix(g)

M =

    1.0000         0         0         0
         0    1.0000         0         0
         0         0    0.7071    0.7071
         0         0    0.7071   -0.7071

Array of Controlled Hadamard Gates

Create an array of three controlled Hadamard gates. The first gate acts on control qubit 1 and target qubit 2, the next gate acts on control qubit 2 and target qubit 3, and the final gate acts on control qubit 3 and target qubit 4.

g = chGate(1:3,2:4)

g = 

  3×1 SimpleGate array with gates:

    Id   Gate   Control   Target
     1   ch     1         2     
     2   ch     2         3     
     3   ch     3         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

More About

Matrix Representation of Controlled Hadamard Gate

The matrix representation of a controlled Hadamard gate applied to control qubit 1 and target qubit 2 is

$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ 0& 0& \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right].$$

If the control qubit is in the $$|0\rangle$$ state, then this gate leaves the target qubit as is. If the control qubit is in the $$|1\rangle$$ state, then this gate maps the target qubit basis states from the Z basis to the X basis (and from the X basis to the Z basis)

Version History

Introduced in R2023a