A4: Generate state $\frac{1}{\sqrt{3}} (\ket{0} + \ket{1} + \ket{2}))$ I

Editorial

The objective is to prepare a superposition state of the three computational basis states $\ket{00},\ \ket{10},\ \ket{01}$. Note we are treating the leftmost qubit as the least significant.

A superposition state involving a non-power-of-two number of computational basis states can be prepared by using controlled-Hadamard ($CH$) gates.

First, applying a Hadamard gate to the $1$st qubit effects the transformation

Subsequently applying a controlled-Hadamard gate with the $1$st qubit as the control qubit and the $2$nd as the target qubit effects

Finally, applying a controlled-$X$ gate with the $1$st qubit as the target qubit and the $2$nd as the control qubit effects

In this way, a superposition state of the three computational basis states $\ket{00},\ \ket{10},\ \ket{01}$ is prepared.

Sample Code

Below is a sample program:

from qiskit import QuantumCircuit
 
def solve() -> QuantumCircuit:
    qc = QuantumCircuit(2)
 
    qc.h(0)
    qc.ch(0, 1)
    qc.cx(1, 0)
 
    return qc