``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.