Ex: Reverse and Shift

Problem Statement

Let $n$ be a positive integer. Consider a list $[p_0, p_1, ... , p_{2^n-1}]$ of length $2^n$.

We define the following $2$ operations:

• Operation 1: Move the first element to the end

• Operation 2: Reverse the entire list

Let $g(m, k)$ be the concatenation of the following actions:

• First, apply Operation 1 for $k$ times.
• Then, apply Operation 2 for $m$ times.

Given integers $n$, $m_{\text{before}}$, $k_{\text{before}}$, $m_{\text{after}}$ and $k_{\text{after}}$, implement an $n+1$-qubit oracle $O$ on a quantum circuit $\mathrm{qc}$.

For any pair of integers $(m,k)$ satisfying $0 \leq m < 2$ and $0 \leq k < 2^{n}$, the oracle must satisfy

Here, $0 \leq m' < 2$ and $0 \leq k' < 2^{n}$ must satisfy the following condition:

• The operation $g(m', k')$ is equivalent to applying $g(m_{\text{before}}, k_{\text{before}}) \rightarrow g(m, k) \rightarrow g(m_{\text{after}}, k_{\text{after}})$ in order.

Constraints

• $2 \leq n \leq 10$
• $0 \leq m_{\text{before}} < 2$
• $0 \leq k_{\text{before}} < 2^n$
• $0 \leq m_{\text{after}} < 2$
• $0 \leq k_{\text{after}} < 2^n$
• The quantum circuit depth must not exceed $300$.
• Integers must be encoded by little-endian.
• Global phase is ignored in judge.
• The submitted code must follow the specified format:

from qiskit import QuantumCircuit
 
 
def solve(n: int, m_before: int, k_before: int, m_after: int, k_after: int) -> QuantumCircuit:
    qc = QuantumCircuit(n + 1)
    # Write your code here:
 
    return qc