Ex: Can you prepare $\ket{\omega}?$

Problem Statement

You are given an integer $n$, a real number $\theta$, an $n$-qubit quantum gate $U$ and an $n$-qubit parametric quantum gate $R(\cdot)$ of unknown structures.

There exists a vector $\ket{\omega}$ satisfying the following two conditions:

• $P \leq \left|\braket{\omega \mid U \mid 0}\right|^2 \leq 4P$
• $R(\theta) = I - (1 - e^{i\theta})\ket{\omega}\bra{\omega}$

Using the quantum gates $U$ and $R(\cdot)$, implement an operation $O$ that prepares the quantum state $\ket{\omega}$ on a quantum circuit $\mathrm{qc}$ with $n$ qubits.

The operation $O$ must satisfy $\left|\braket{\omega \mid O \mid 0}\right|^2 \geq 0.99$ within an error of $0.01$.

Constraints

• $1 \leq n \leq 5$
• $0.02 \leq P \leq 1$
• The number of applications of $R(\cdot)$ must not exceed $100$. (If exceeded, a DLE (depth limit exceeded) error will be displayed)
• Changes in the global phase are ignored.
  
• The submitted code must follow the specified format:

from qiskit import QuantumCircuit
 
"""
You can apply U and R as follows:
qc.compose(U(), inplace=True)
qc.compose(R(theta), inplace=True)
"""
 
 
def solve(n: int, P: float, U, R) -> QuantumCircuit:
    qc = QuantumCircuit(n)
    # Write your code here:
 
    return qc

Hints

qc.compose(U().inverse(), inplace=True)
qc.compose(R(theta).inverse(), inplace=True)