Preparing Bell state $(1/\sqrt{2}) (|01\rangle + |10\rangle)$ in Qiskit

Question

I'm working through the Qiskit textbook right now, and wanted to complete part 1 of exercise 3.4, which asks me to use qiskit to produce the Bell state:

$$\frac{|01\rangle + |10\rangle}{\sqrt{2}}$$

which is equivalent to

$$\begin{bmatrix} 0 \\ \frac{1}{\sqrt2} \\ \frac{1}{\sqrt2}\\ 0 \end{bmatrix}. $$

When you apply a CNOT to this, we get:

$$\begin{bmatrix} 0 \\ 0 \\ \frac{1}{\sqrt2}\\ \frac{1}{\sqrt2} \end{bmatrix}$$

Which is just $|1\rangle\otimes |+\rangle$. However, I can't figure out how to set my qubits up in the right way.

Any tips on how to use qiskit to get that last matrix?

Answer 1

The circuit to prepare the state $|\psi \rangle = \dfrac{|01\rangle + |10\rangle}{\sqrt{2}} $ is as follows:

     ┌───┐          
q_0: ┤ H ├──■───────
     └───┘┌─┴─┐┌───┐
q_1: ─────┤ X ├┤ X ├
          └───┘└───┘

This can be written in matrix notation as:

\begin{align} U &= (I \otimes X)\cdot CNOT \cdot (H\otimes I) \\ &= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{pmatrix} \cdot \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ &= \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{pmatrix} \cdot \dfrac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\end{pmatrix} \\ &= \dfrac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & -1 & 0\\ 0 & 1 & 0 & -1 \end{pmatrix} \end{align}

And $$ U |00\rangle =\dfrac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & -1 & 0\\ 0 & 1 & 0 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} = \dfrac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix} = \dfrac{|01\rangle + |10\rangle}{\sqrt{2}} $$

If you want to generate this circuit with qiskit you can do it as follows:

from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit
qreg_q = QuantumRegister(2, 'q')
creg_c = ClassicalRegister(2, 'c')
circuit = QuantumCircuit(qreg_q, creg_c)
circuit.h(qreg_q[0])
circuit.cx(qreg_q[0], qreg_q[1])
circuit.x(qreg_q[1])

Answer 2

To provide general answer, Bell states are prepared with circuit described by matrix $CNOT(H \otimes I)$. Which Bell state is returned depends on input to the circuit:

• input $|00\rangle$ returns $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$
• input $|01\rangle$ returns $\frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$
• input $|10\rangle$ returns $\frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$
• input $|11\rangle$ returns $\frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$

Clearly, value of the second input qubit specifies whether both output qubits will be same (the second input qubit is $|0\rangle$) or opposite (the second input qubit is $|1\rangle$). The first qubit specifies a phase - when the qubit is $|0\rangle$, the phase is 0, when it is $|1\rangle$, the phase is $\pi$ ($\mathrm{e^{i\pi}}=-1$).

Concerning the input, to obtain for example $|01\rangle$, put $X$ gate on second qubit and then apply the circuit for Bell state. Similarly for other inputs.