





import qiskit

def   a_   (			lowerCAmelCase_				:      int,     lowerCAmelCase_				:      int     ):
   __lowerCAmelCase							     =     qiskit.Aer.get_backend('aer_simulator'     )

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

   # 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
   __lowerCAmelCase							     =     qiskit.execute(lowerCAmelCase_,     lowerCAmelCase_,     shots=1000     )

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


if __name__ == "__main__":
     _snake_case  :   Optional[int]						   =  single_qubit_measure(2, 2)
     print(F"""Total count for various states are: {counts}""")








from math import factorial



class     _UpperCAmelCase   :

  """simple docstring"""


  def __init__(    self			:		Dict ,  lowerCAmelCase_			:		Optional[int] ,  lowerCAmelCase_			:		str					)							->						Union[str, Any]:
     __lowerCAmelCase							     =     real
     if isinstance(lowerCAmelCase_ ,  lowerCAmelCase_					):
        __lowerCAmelCase							     =     [1] * rank
     else:
        __lowerCAmelCase							     =     rank






  def __repr__(    self			:		Optional[Any]					)							->						Tuple:
     return (
         f"""{self.real}+"""
         f"""{"+".join(str(lowerCAmelCase_					)+"E"+str(n+1					)for n,dual in enumerate(self.duals					)					)}"""
     )






  def     lowercase		(    self			:		str					)							->						Dict:
     __lowerCAmelCase							     =     self.duals.copy()
     while cur[-1] == 0:
        cur.pop(-1					)
     return Dual(self.real ,  lowerCAmelCase_					)






  def __add__(    self			:		Union[str, Any] ,  lowerCAmelCase_			:		Tuple					)							->						Optional[Any]:
     if not isinstance(lowerCAmelCase_ ,  lowerCAmelCase_					):
        return Dual(self.real + other ,  self.duals					)
     __lowerCAmelCase							     =     self.duals.copy()
     __lowerCAmelCase							     =     other.duals.copy()
     if len(lowerCAmelCase_					) > len(lowerCAmelCase_					):
        o_dual.extend([1] * (len(lowerCAmelCase_					) - len(lowerCAmelCase_					))					)
     elif len(lowerCAmelCase_					) < len(lowerCAmelCase_					):
        s_dual.extend([1] * (len(lowerCAmelCase_					) - len(lowerCAmelCase_					))					)
     __lowerCAmelCase							     =     []
     for i in range(len(lowerCAmelCase_					)					):
        new_duals.append(s_dual[i] + o_dual[i]					)
     return Dual(self.real + other.real ,  lowerCAmelCase_					)

  a_      =			__add__






  def __sub__(    self			:		int ,  lowerCAmelCase_			:		Dict					)							->						Optional[Any]:
     return self + other * -1






  def __mul__(    self			:		int ,  lowerCAmelCase_			:		Optional[int]					)							->						Dict:
     if not isinstance(lowerCAmelCase_ ,  lowerCAmelCase_					):
        __lowerCAmelCase							     =     []
        for i in self.duals:
           new_duals.append(i * other					)
        return Dual(self.real * other ,  lowerCAmelCase_					)
     __lowerCAmelCase							     =     [0] * (len(self.duals					) + len(other.duals					) + 1)
     for i, item in enumerate(self.duals					):
        for j, jtem in enumerate(other.duals					):
           new_duals[i + j + 1] += item * jtem
     for k in range(len(self.duals					)					):
        new_duals[k] += self.duals[k] * other.real
     for index in range(len(other.duals					)					):
        new_duals[index] += other.duals[index] * self.real
     return Dual(self.real * other.real ,  lowerCAmelCase_					)

  a_      =			__mul__






  def __truediv__(    self			:		Union[str, Any] ,  lowerCAmelCase_			:		str					)							->						Dict:
     if not isinstance(lowerCAmelCase_ ,  lowerCAmelCase_					):
        __lowerCAmelCase							     =     []
        for i in self.duals:
           new_duals.append(i / other					)
        return Dual(self.real / other ,  lowerCAmelCase_					)
     raise ValueError






  def __floordiv__(    self			:		str ,  lowerCAmelCase_			:		List[Any]					)							->						Union[str, Any]:
     if not isinstance(lowerCAmelCase_ ,  lowerCAmelCase_					):
        __lowerCAmelCase							     =     []
        for i in self.duals:
           new_duals.append(i // other					)
        return Dual(self.real // other ,  lowerCAmelCase_					)
     raise ValueError






  def __pow__(    self			:		Tuple ,  lowerCAmelCase_			:		Dict					)							->						List[str]:
     if n < 0 or isinstance(lowerCAmelCase_ ,  lowerCAmelCase_					):
        raise ValueError('power must be a positive integer'					)
     if n == 0:
        return 1
     if n == 1:
        return self
     __lowerCAmelCase							     =     self
     for _ in range(n - 1					):
        x *= self
     return x





def   a_   (			lowerCAmelCase_				:      List[str],     lowerCAmelCase_				:      Optional[Any],     lowerCAmelCase_				:      Optional[Any]     ):
   if not callable(lowerCAmelCase_     ):
      raise ValueError('differentiate() requires a function as input for func'     )
   if not isinstance(lowerCAmelCase_,     (float, int)     ):
      raise ValueError('differentiate() requires a float as input for position'     )
   if not isinstance(lowerCAmelCase_,     lowerCAmelCase_     ):
      raise ValueError('differentiate() requires an int as input for order'     )
   __lowerCAmelCase							     =     Dual(lowerCAmelCase_,     1     )
   __lowerCAmelCase							     =     func(lowerCAmelCase_     )
   if order == 0:
      return result.real
   return result.duals[order - 1] * factorial(lowerCAmelCase_     )


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     def   a_   (			lowerCAmelCase_				:      int     ):
        return y**2 * y**4




     print(differentiate(f, 9, 2))

