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:

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.

见题面 Input 部分

见题面 Output 部分