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

Editorial

The objective is to prepare a superposition of the three computational basis states $\ket{00}$, $\ket{10}$, and $\ket{11}$. Note we are treating the leftmost qubit as the least significant.

A superposition 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 first qubit effects the transformation

Subsequently applying a controlled-Hadamard gate which targets the right qubit and which is controlled by the left qubit effects

Finally, applying a controlled-$X$ gate which targets the left qubit and is controlled by the right qubit effects

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

As we will see in A5, this is not quite the state we wish to prepare.

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