A5: Generate state $\frac{1}{\sqrt{3}} (\ket{0} + \ket{1} + \ket{2}))$ II

Editorial

In problem A4, we prepared the superposition state $\frac{1}{\sqrt{2}} \ket{00} + \frac{1}{2}(\ket{10} + \ket{01})$, composed of the three computational basis states $\ket{00},\ \ket{10},\ \ket{01}$.

In this problem, we establish a uniform superposition state of these states, changing their complex amplitudes to $1/\sqrt{3}$.

This can be achieved by replacing the Hadamard gate in Equation $(1)$ of editorial A4 with a rotation gate. Choose a rotation that prepares $\frac{1}{\sqrt{3}} \ket{00} + \sqrt{\frac{2}{3}} \ket{10}$. The following gates then produce the desired uniform superposition state:

We will now find the axis of rotation and the angle to perform such a transformation.

An arbitrary state $\ket{\psi}$ of $1$ qubit can be expressed by two angle parameters, $\theta,\ \phi$, as follows:

The method of representing the quantum state of one qubit on the surface of a sphere with radius $1$ is called the Bloch sphere, and by using phase gates and rotation gates based on the angular parameters $\theta,\ \phi$, it is possible to transition any $1$-qubit state to any other arbitrary state.

Bloch sphere

Let us find the Bloch angular parameters $\theta,\ \phi$ for the state $\frac{1}{\sqrt{3}} \ket{00} + \sqrt{\frac{2}{3}} \ket{10}$.

Since the imaginary component is $0$, we can set $\phi = 0$, and establish the following system of equations:

By solving the above equation within the range $0 \leq \theta < 2 \pi$, we obtain the following solution:

Looking at the Bloch sphere, we see that the angle parameter $\theta$ represents the rotation angle about the $y$-axis.

Thus, using the $R_y$ gate, which is a rotation gate about the $y$-axis, we can implement the rotation by $\theta$ to achieve the transition $\ket{00} \xrightarrow{Ry(\theta, 0)} \frac{1}{\sqrt{3}} \ket{00} + \sqrt{\frac{2}{3}} \ket{10}$

As a result, the problem can be solved through the following transitions:

Sample Code

Below is a sample program:

from qiskit import QuantumCircuit
import math
 
def solve() -> QuantumCircuit:
    qc = QuantumCircuit(2)
 
    theta = 4 * math.atan(math.sqrt(6)/ (3 + math.sqrt(3)))
 
    qc.ry(theta, 0)
    qc.ch(0, 1)
    qc.cx(1, 0)
 
    return qc