# Defining standard qubit gates for qutrits

I am actually working on quantum computing with qutrits. I am trying to define standard qubit gates for qutrits. The CNOT gate for qubits is defined as follows: $$|x,y\rangle \to |x,y+x \bmod 2\rangle.$$

So the matrix is $$\mathrm{UCN} = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{pmatrix}.$$

For the definition of the same type of gate for qutrit I am left with two choices:

$$|x,y\rangle \to |x,y+x \bmod 3\rangle$$ $$|x,y\rangle \to |x,y+2x \bmod 3\rangle$$

Does anyone know if one of the two is more common or more usually defined?

PS: I know that applying one two times is equivalent to apply the other one time.

• Hi, Marco. Welcome to Quantum Computing SE! It is preferable that you use MathJax to typeset your posts. Review How to write a good question?. I've edited the question on your behalf. – Sanchayan Dutta May 7 '19 at 16:18
• Hi Sanchayan, thanks a lot for the edit and the advice. I will make sure that I write a good question next time. – Marco Falso May 7 '19 at 16:21

$$|x,y\rangle \to |x,y+x \bmod 3\rangle$$

is both more common and, IMO, easier to use. Especially as you point out that the second definition can be simulated by repeated application of the first. You can see the authors have used this version of the qutrit CNOT in Construction of two qutrit entanglement by using magnetic resonance selective pulse sequences (Çorbaci et al., 2016) and in Controlled Not (Cnot) Gate For Two Qutrit Systems, (KOCAKOÇ & TAPRAMAZ, 2018).

Additionally, you may want to consider the context of your qutrit gates. For example, section 2 of arXiv:1105.5485 first defines three versions of the qutrit X gate and then discusses what a controlled X would mean in these cases.

• Thanks a lot for the articles. None of them uses the Hadamard gate I am using but at least I have some articles to relate to about computation with qutrits. I am convinced that X gate and Hadamard gate must change all the states and not be only extensions of the two dimensional ones. – Marco Falso May 7 '19 at 17:40