rxxGate

Ising XX coupling gate

Since R2023a

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

Syntax

g = rxxGate(targetQubit1,targetQubit2,theta)

Description

g = rxxGate(targetQubit1,targetQubit2,theta) applies an Ising XX coupling gate to two target qubits with a phase parameter theta and returns a quantum.gate.SimpleGate object.

• If targetQubit1, targetQubit2, and theta are vectors of the same length, rxxGate returns a column vector of gates, where g(i) represents an Ising XX coupling gate applied to the ith element triplet of targetQubit1, targetQubit2, and theta.

• If any of targetQubit1, targetQubit2, 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 targetQubit1 and targetQubit2 must not be the same.

Examples

Ising XX Coupling Gate and Its Matrix Representation

Create an Ising XX coupling gate that acts on two target qubits with indices 1 and 2 using a phase parameter of pi/2.

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

g = 

  SimpleGate with properties:

             Type: "rxx"
    ControlQubits: [1×0 double]
     TargetQubits: [1 2]
           Angles: 1.5708

Get the matrix representation of the gate.

M = getMatrix(g)

M =

   0.7071 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 - 0.7071i
   0.0000 + 0.0000i   0.7071 + 0.0000i   0.0000 - 0.7071i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 - 0.7071i   0.7071 + 0.0000i   0.0000 + 0.0000i
   0.0000 - 0.7071i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.7071 + 0.0000i

Array of Ising XX Coupling Gates

Create an array of three Ising XX coupling gates. The first gate acts on target qubits 1 and 2 with phase parameter pi/4, the next gate acts on target qubits 2 and 3 with phase parameter pi/2, and the final gate acts on target qubits 3 and 4 with phase parameter 3*pi/4.

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

g = 

  3×1 SimpleGate array with gates:

    Id   Gate   Control   Target   Angle
     1   rxx              [1,2]    pi/4 
     2   rxx              [2,3]    pi/2 
     3   rxx              [3,4]    3pi/4

Input Arguments

targetQubit1 — First target qubit of gate

positive integer scalar | positive integer vector

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

Example: 1

Example: 3:5

targetQubit2 — Second target qubit of gate

positive integer scalar | positive integer vector

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

Example: 2

Example: 6:8

theta — Phase parameter

real scalar | real vector

Phase parameter, specified as a real scalar or vector.

Example: pi

Example: (1:3)*pi/2

More About

Matrix Representation of Ising XX Coupling Gate

The matrix representation of an Ising XX coupling gate applied to two target qubits with a phase parameter of $$\theta$$ is

$$\exp \left(-i\frac{\theta }{2}X\otimes X\right)=\exp \left(-i\frac{\theta }{2}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\otimes \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\right)=\left[\begin{array}{cccc}\cos \left(\frac{\theta }{2}\right)& 0& 0& -i\sin \left(\frac{\theta }{2}\right)\\ 0& \cos \left(\frac{\theta }{2}\right)& -i\sin \left(\frac{\theta }{2}\right)& 0\\ 0& -i\sin \left(\frac{\theta }{2}\right)& \cos \left(\frac{\theta }{2}\right)& 0\\ -i\sin \left(\frac{\theta }{2}\right)& 0& 0& \cos \left(\frac{\theta }{2}\right)\end{array}\right].$$

Version History

Introduced in R2023a