Description:
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.
Input Format:
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Output Format:
None
Note:
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