Rxx Gate in Quantum Computing

What is it?

The gate is a quantum gate used in quantum computing to implement a common rotation around the X axis for two qubits, which can generate entanglement between them.

The Definition of Rxx Gate

The operation of this gate is defined by the following exponential operator:

R_{xx}(\theta) = \exp\left(-i \frac{\theta}{2} X \otimes X\right)

The expansion of the exponential operator often utilizes Taylor series expansion or Euler’s formula. According to Euler’s formula, for any real number x, the following equation holds

e^{ix} = \cos(x) + i \sin(x)

Applying this to our matrix exponential gives:

\begin{align} R_{xx}(\theta) & = \cos\left(\frac{\theta}{2}\right)(I \otimes I) -i \sin\left(\frac{\theta}{2}\right)(X \otimes X) \\ & = \cos\left(\frac{\theta}{2}\right)\left(\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix} \otimes \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\right) -i \sin\left(\frac{\theta}{2}\right)\left(\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix} \otimes \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\right) \\ & = \cos\left(\frac{\theta}{2}\right)\left(\begin{pmatrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{pmatrix}\right) -i \sin\left(\frac{\theta}{2}\right)\left(\begin{pmatrix}0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{pmatrix}\right) \\ & = \begin{pmatrix} \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{pmatrix}\ \end{align}

These terms describe how the gate interacts with a two-qubit system, inducing changes in the quantum state that can include the generation of entanglement.

The behavior of this gate

The behavior of the RXX gate changes significantly depending on the value of the angle θ. This angle determines the degree of entanglement generated between two qubits and the overall effect of the rotation on the system. Below, I detail how the effects of the Rxx gate vary with different values of θ.

θ=0

Identity Matrix: When θ=0, the RXX gate is equivalent to the identity matrix I⊗I and does not affect the quantum state of the qubits. In this setting, the gate effectively does “nothing.”

θ=pi

Bit-Swap-Like Operation: At θ=π, the Rxx can strongly entangle two qubits. This setting allows the gate to have an effect equivalent to an X⊗X operation, maximizing entanglement. This implies flipping each qubit into the opposite state, thereby creating entanglement across the entire system.

Caution

The operation above is Bit-Swap-“Like Operation“, not Swap Gate itself. The Swap Gate never cause entanglement, but the Bit-Swap-“Like Operation” is going to cause entangle between qubits.