# Identifying a matrix with the fundamental quantum logic gates [closed]

I have the following matrix

$$\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0\\ 0 & \frac{-i}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix}$$ I want to understand the operation by this matrix in terms of the familiar quantum logic gates. More precisely, how can one write the above matrix in terms of the quantum logic gates?

• Are you restricting to any particular set of quantum logic gates? This actually is a gate that is included in some sets. Have you tried squaring the matrix? Related (up to some phase gates): quantumcomputing.stackexchange.com/a/2468/1837 Apr 17 '19 at 9:15
• Thanks, @DaftWullie. I am not restricting to any specific gate. SWAP involves the unwanted 1/2 factor. Can one get rid of that? Apr 17 '19 at 9:31
• 1/2 factor? My point is that, apart from some factors of $e^{i\pi/4}$, your matrix is square root of swap. You can fix that easily with phase gates. Apr 17 '19 at 9:34
• My matrix resembles to XX gate given here: en.wikipedia.org/wiki/Quantum_logic_gate, apart from the two $e^{i(\phi - \pi/2)}$. But not sure how to get rid of this. Apr 17 '19 at 10:26
• I'm voting to close this question as off-topic because it's a homework-type question showing insufficient effort. Please check our policy on homework and exercise questions. However, since it has already received a good answer, I believe this thread is worth preserving. In any case, please avoid deleting questions when others have put in the effort to answer. Instead, try to edit and improve them. May 9 '19 at 17:45

This gate, which I'll denote as $$U$$ is essentially the square root of not gate, whose decomposition has already been discussed elsewhere. Hence, we only need the conversion:
Specifically, if we perform the corresponding matrix multiplications, we have $$\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & i \\ \end{array} \right)\cdot\left(\left( \begin{array}{cc} 1 & 0 \\ 0 & e^{-\frac{i \pi }{4}} \\ \end{array} \right)\otimes \left( \begin{array}{cc} 1 & 0 \\ 0 & e^{-\frac{i \pi }{4}} \\ \end{array} \right)\right)\cdot\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \frac{e^{\frac{i \pi }{4}}}{\sqrt{2}} & \frac{e^{-\frac{i \pi }{4}}}{\sqrt{2}} & 0 \\ 0 & \frac{e^{-\frac{i \pi }{4}}}{\sqrt{2}} & \frac{e^{\frac{i \pi }{4}}}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)=\\ \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & -\frac{i}{\sqrt{2}} & 0 \\ 0 & -\frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)$$ as required.