# Matrices size of n-qubit controlled gates and n+2 states qubit?

Operations on many qubits at the same time is the same as operation of qubits with many states.

For example the CNOT gate matrix with 1 control qubit will be the same of a "gate" acting on a qubit with 4 possible states , no?

• what does "4 possible states" mean in this sentence? I don't know if that's what you're trying to ask, but an $n$-qubit state is essentially a single $2^n$-dimensional qudit, yes. With some underlying assumption on the type of operations and questions you're going to ask it.
– glS
Dec 1, 2023 at 11:54

It is true that 2 qubits can be seen as one system with 4 basis states $$\left|00\right>$$, $$\left|01\right>$$, $$\left|10\right>$$, $$\left|11\right>$$. In general it is not $$n+2$$, but $$2n$$ — the system containing a qubit and a $$n$$-dimensional qudit has $$2n$$ basis states (of the form $$\left|0k\right>$$ and $$\left|1k\right>$$).
• you probably meant $2^n$ basis states, not $2n$
• For $n$ qubits yes, the dimension would be $2^n$. I meant 1 qubit and 1 $n$-dimensional qudit, then it is $2n$. Dec 1, 2023 at 14:36