D4. TIE fighter
time limit per test
2 seconds
memory limit per test
1024 megabytes
input
standard input
output
standard output

Implement a unitary operation on $N$ qubits which is represented by a square matrix of size $2^N$ which has non-zero elements in the following positions:

• the central 2x2 sub-matrix,
• the diagonals of the top right and bottom left square sub-matrices of size $2^{N-1}-1$ that do not overlap with the central 2x2 sub-matrix,
• the anti-diagonals of the top left and bottom right square sub-matrices of size $2^{N-1}-1$ that do not overlap with the central 2x2 sub-matrix.

For example, for $N = 3$ the matrix of the operation should have the following shape:

..X..X...X....X.X......X...XX......XX...X......X.X....X...X..X..

Here X denotes a "non-zero" element of the matrix (a complex number which has the square of the absolute value greater than or equal to $10^{-5}$), and . denotes a "zero" element of the matrix (a complex number which has the square of the absolute value less than $10^{-5}$).

The row and column indices of the matrix follow little endian format: the least significant bit of the index is stored first in the qubit array. Thus, the first column of the matrix gives you the coefficients of the basis states you'll get if you apply the unitary to the $|00..0\rangle$ basis state, the second column - to the $|10..0\rangle$ basis state etc. You can use the DumpUnitary tool to get the coefficients of the matrix your unitary implements (up to relative phases between columns) and the corresponding pattern of Xs and .s.

You have to implement an operation which takes an array of $N$ ($2 \le N \le 5$) qubits as an input and applies the unitary transformation with the matrix of the described shape to it. If there are multiple unitaries which satisfy the requirements, you can implement any of them. The "output" of your operation is the pattern of the matrix coefficients implemented by it; you can see the testing harness in the UnitaryPatterns kata.

Your code should have the following signature:

namespace Solution {    open Microsoft.Quantum.Primitive;    open Microsoft.Quantum.Canon;    operation Solve (qs : Qubit[]) : Unit {        // your code here    }}

You are not allowed to use measurements in your operation.