B7: Quantum Arithmetic III

Problem Statement

You are given integers $n,\ m,\ S_0,\ S_1,\ \cdots,\ S_{n-1}$.

For an integer $x = x_0 + 2^1 x_1 + \cdots 2^{n-1}x_{n-1}$ $\left(x_i \in \{0, 1\}\right)$ where $0 \leq x < 2^n$, define the function $f(x)$ by

Implement the $(n + m)$-qubit oracle $O$ acting on computational basis states as

for any pair of integers $(x,\ y)$ such that $0 \leq x < 2^n$ and $0 \leq y < 2^m$.

Constraints

• $1 \leq n, m \leq 9$
• $n + m \leq 10$
• $0 \leq S_k < 2^m$
• The circuit depth must not exceed $50$.
• Integers are encoded by little-endian notation, i.e., $\ket{100} = 1 \neq \ket{001}$.
• Changes in the global phase are ignored.
  
• The submitted code must follow the specified format:

from qiskit import QuantumCircuit, QuantumRegister
 
 
def solve(n: int, m: int, S: list[int]) -> QuantumCircuit:
    x, y = QuantumRegister(n), QuantumRegister(m)
    qc = QuantumCircuit(x, y)
    # Write your code here:
 
    return qc