



from __future__ import annotations

UpperCAmelCase__     = list[list[int]]

# assigning initial values to the grid
UpperCAmelCase__     = [
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
UpperCAmelCase__     = [
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]





def 							A			(  _UpperCAmelCase       :     Matrix							,     _UpperCAmelCase       :     int							,     _UpperCAmelCase       :     int							,     _UpperCAmelCase       :     int					)       ->				bool:







			'''simple docstring'''
			for i in range(9					):
						if grid[row][i] == n or grid[i][column] == n:
									return False

			for i in range(3					):
						for j in range(3					):
									if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
												return False

			return True





def 							A			(  _UpperCAmelCase       :     Matrix					)       ->				tuple[int, int] | None:







			'''simple docstring'''
			for i in range(9					):
						for j in range(9					):
									if grid[i][j] == 0:
												return i, j
			return None





def 							A			(  _UpperCAmelCase       :     Matrix					)       ->				Matrix | None:







			'''simple docstring'''
			if location := find_empty_location(_UpperCAmelCase					):
						_UpperCAmelCase ,  _UpperCAmelCase	 =     location
			else:
						# If the location is ``None``, then the grid is solved.
						return grid

			for digit in range(1							,     10					):
						if is_safe(_UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase					):
									_UpperCAmelCase	 =     digit

									if sudoku(_UpperCAmelCase					) is not None:
												return grid

									_UpperCAmelCase	 =     0

			return None





def 							A			(  _UpperCAmelCase       :     Matrix					)       ->				None:







			'''simple docstring'''
			for row in grid:
						for cell in row:
									print(_UpperCAmelCase							,     end=' '					)
						print()


if __name__ == "__main__":
							# make a copy of grid so that you can compare with the unmodified grid
							for example_grid in (initial_grid, no_solution):
														print("\nExample grid:\n" + "=" * 20)
														print_solution(example_grid)
														print("\nExample grid solution:")
														UpperCAmelCase__     = sudoku(example_grid)
														if solution is not None:
																					print_solution(solution)
														else:
																					print("Cannot find a solution.")





import qiskit





def 							A			(  _UpperCAmelCase       :     int							,     _UpperCAmelCase       :     int					)       ->				qiskit.result.counts.Counts:







			'''simple docstring'''
			_UpperCAmelCase	 =     qiskit.Aer.get_backend('aer_simulator'					)

			# Create a Quantum Circuit acting on the q register
			_UpperCAmelCase	 =     qiskit.QuantumCircuit(_UpperCAmelCase							,     _UpperCAmelCase					)

			# Apply X (NOT) Gate to Qubits 0 & 1
			circuit.x(0					)
			circuit.x(1					)

			# Map the quantum measurement to the classical bits
			circuit.measure([0, 1]							,     [0, 1]					)

			# Execute the circuit on the qasm simulator
			_UpperCAmelCase	 =     qiskit.execute(_UpperCAmelCase							,     _UpperCAmelCase							,     shots=1_000					)

			# Return the histogram data of the results of the experiment.
			return job.result().get_counts(_UpperCAmelCase					)


if __name__ == "__main__":
							UpperCAmelCase__     = single_qubit_measure(2, 2)
							print(f"""Total count for various states are: {counts}""")
