# How to implement the 4 Bell states on the IBM Q (composer)?

I would like to simulate the 4 "Bell States" on the IBM composer?

How can I best implement those 4 Bell states using the existing set of gates ?

Here below you see the definition of the 4 Bell states.

The first bell state can be easily implemented on the composer through a Hadamard gate and a CNOT gate as follows:

but how can I implement the other 3 bell states ?

Remember that the Hadamard (H) gate maps $|0\rangle$ to $\frac{|0\rangle+|1\rangle}{\sqrt{2}}$ and $|1\rangle$ to $\frac{|0\rangle-|1\rangle}{\sqrt{2}}$, while the CNOT gate has the following conversion table:

So, you can use the same circuit:

but begin in the states $|01\rangle$, $|10\rangle$ and $|11\rangle$ to get the other three Bell states (you can easily convert to these states from $|00\rangle$ using the Pauli-X gate).

One of the possible solutions:

$$|\Phi^+\rangle = \textrm{CNOT} \cdot H_1 |00 \rangle$$
$$|\Phi^-\rangle = Z_1 |\Phi^+\rangle = Z_1 \cdot \textrm{CNOT} \cdot H_1 |00 \rangle$$
$$|\Psi^+\rangle = X_2 |\Phi^+\rangle = X_2 \cdot \textrm{CNOT} \cdot H_1 |00 \rangle$$
$$|\Psi^-\rangle = Z_1 |\Psi^+\rangle = Z_1 \cdot X_2 \cdot \textrm{CNOT} \cdot H_1 |00 \rangle$$

Where $$Z_i$$ and $$X_i$$ act on the $$i^\textrm{th}$$ qubit.

So here is $$|\Phi^-\rangle$$:

Here is $$|\Psi^+\rangle$$:

And here is $$|\Psi^-\rangle$$:

Another solution is:

$$|\Phi^+\rangle = \textrm{CNOT} \cdot H_1 |00 \rangle$$
$$|\Phi^-\rangle = X_1 \cdot \textrm{CNOT} \cdot H_1 |00 \rangle$$
$$|\Psi^+\rangle = X_2 \cdot \textrm{CNOT} \cdot H_1 |00 \rangle$$
$$|\Psi^-\rangle = Z_1 X_2\cdot \textrm{CNOT} \cdot H_1 |00 \rangle$$

• Are you assuming that we can only begin in the state $|00\rangle$? Is that a necessity for the IBM Q composer? Asking because I don't have experience with the IBM Q composer. Jun 7, 2018 at 15:45
• @Blue, yes IBM Q composer seems always to begin in state $|00\rangle$ but it can be easily converted to e.g. state $|01\rangle$ by using an $X$ gate. Jun 7, 2018 at 16:01
• I've included the solution that starts with states different from $|00\rangle$, but do I have to put the diagrams in there too or is what I have good enough for you? Jun 7, 2018 at 16:11