Задача . C2. "Is the bit string periodic?" oracle

Implement a quantum oracle on \(N\) qubits which checks whether the bits in the input vector \(\vec{x}\) form a periodic bit string (i.e., implements the function \(f(\vec{x}) = 1\) if \(\vec{x}\) is periodic, and 0 otherwise).

A bit string of length \(N\) is considered periodic with period \(P\) (\(1 \le P \le N - 1\)) if for all \(i \in [0, N - P - 1]\) \(x_i = x_{i + P}\). Note that \(P\) does not have to divide \(N\) evenly; for example, bit string "01010" is periodic with period \(P = 2\).

You have to implement an operation which takes the following inputs:

• an array of \(N\) (\(2 \le N \le 7\)) qubits \(x\) in an arbitrary state (input register),
• a qubit \(y\) in an arbitrary state (output qubit),

and performs a transformation \(|x\rangle|y\rangle \rightarrow |x\rangle|y \oplus f(x)\rangle\). The operation doesn't have an output; its "output" is the state in which it leaves the qubits. Note that the input register \(x\) has to remain unchanged after applying the operation.

Your code should have the following signature:

namespace Solution {
    open Microsoft.Quantum.Primitive;
    open Microsoft.Quantum.Canon;

    operation Solve (x : Qubit[], y : Qubit) : Unit {
        body (...) {
            // your code here
        }
        adjoint auto;
    }
}

Note: the operation has to have an adjoint specified for it; adjoint auto means that the adjoint will be generated automatically. For details on adjoint, see Operation Definitions.

You are not allowed to use measurements in your operation.