Ex1: Convert Bit-Flip into Phase-Flip II

Problem Statement

You are given an integer $n$ and an oracle $O$. For any pair of integers $(x,y)$ satisfying $0\leq x\lt 2^n$ and $0\leq y\lt 2$, the oracle $O$ satisfies

where $\oplus$ denotes the XOR operator, and $f(x)$ is a function that returns either $0$ or $1$ for any integer $x$ with $0\leq x\lt 2^n$.

Implement an operation on a quantum circuit $\mathrm{qc}$ that satisfies

for any integer $x$ satisfying $0\leq x\lt 2^n$.

Constraints

• $1\leq n\leq 10$
• The number of applied oracle must not exceed $1$. (If exceeded, a DLE (depth limit exceeded) error will be displayed)
• Integers must be encoded by little-endian notation, i.e., $\ket{100} = 1 \neq \ket{001}$.
• Global phase is ignored in judge.
• The submitted code must follow the specified format:

from qiskit import QuantumCircuit, QuantumRegister
 
"""
You can apply oracle as follows:
qc.compose(o, inplace=True)
"""
 
 
def solve(n: int, o: QuantumCircuit) -> QuantumCircuit:
    x, y = QuantumRegister(n), QuantumRegister(1)
    qc = QuantumCircuit(x, y)
    # Write your code here:
 
    return qc

Sample Input

• $n = 2,\ (f(00), f(10), f(01), f(11)) = (0, 1, 0, 1)$: Implemented circuit $\mathrm{qc}$ should perform the following transformation. $$\frac{1}{\sqrt{4}} ( \ket{00}+\ket{10}+\ket{01}+\ket{11} )\ket{0} \xrightarrow{\mathrm{qc}} \frac{1}{\sqrt{4}} (\ket{00} - \ket{10} + \ket{01} - \ket{11})\ket{0}$$

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