Lecture 11 - Multiple Qubit Gates

There are an infinite number of multi-qubit gates. We can see a list via:

But for our studies we can just study the two-qubit gates and see how it extends to $n$ states or whatever.

Some highlights are:

• CNOT: Controlled NOT, if the top control bit is $|1⟩$ then the operation bit is on the other gate. Otherwise, nothing happens.
  
• SWAP: swaps the states
• $\sqrt{\text{SWAP}}$: Similar to SWAP except adds some phase differences on the way.
• CCNOT or Toffoli gate: is a NOT gate on an arbitrary $n_{sub}$ control bits (2, 3, ...)

The thing is that a lot of these are just online when you need them.

Controlled Gates

When the control qubit is in $|1⟩$, the gate will be applied to the target qubit. When the control qubit is in $|0⟩$ then nothing will be applied to the target qubit.

This brings up the question if you have something in between like $|+⟩$ or something weird. You have to look at the matrix itself to get this information.

Single qubit gates in a two-qubit world (Kronecker product or the Tensor Product).

Practice

Let's try $\text{cPHASE}(\frac{1}{\sqrt{2}}|00⟩+\frac{1}{\sqrt{2}}|01⟩)$, where here the phase is $\pi$, reflected by the following matrix:

The result will be:

\boxed{\frac{1}{2}(\ket{00} + \ket{01} + \ket{10} - \ket{11})}

\boxed{\frac{1}{2}(\ket{00} - i\ket{01} + i\ket{10} + \ket{11})}