'''simple docstring'''

from __future__ import annotations

from math import gcd







def      UpperCamelCase_						(					snake_case_       :       int						,		snake_case_       :       Dict = 2						,		snake_case_       :       List[str] = 1						,		snake_case_       :       str = 3						,		)					->     Any:






 '''simple docstring'''

 if num < 2:
  raise ValueError("""The input value cannot be less than 2"""      )

 # Because of the relationship between ``f(f(x))`` and ``f(x)``, this
 # algorithm struggles to find factors that are divisible by two.
 # As a workaround, we specifically check for two and even inputs.
 #   See: https://math.stackexchange.com/a/2856214/165820
 if num > 2 and num % 2 == 0:
  return 2

 # Pollard's Rho algorithm requires a function that returns pseudorandom
 # values between 0 <= X < ``num``.  It doesn't need to be random in the
 # sense that the output value is cryptographically secure or difficult
 # to calculate, it only needs to be random in the sense that all output
 # values should be equally likely to appear.
 # For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
 # However, the success of Pollard's algorithm isn't guaranteed and is
 # determined in part by the initial seed and the chosen random function.
 # To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
 # where ``C`` is a value that we can modify between each attempt.
 def rand_fn(snake_case_       :       int						,		snake_case_       :       List[str]						,		snake_case_       :       int      ) -> int:
  return (pow(_UpperCAmelCase						,		2      ) + step) % modulus

 for _ in range(_UpperCAmelCase      ):
  # These track the position within the cycle detection logic.
  __lowerCAmelCase											=						seed
  __lowerCAmelCase											=						seed

  while True:
   # At each iteration, the tortoise moves one step and the hare moves two.
   __lowerCAmelCase											=						rand_fn(_UpperCAmelCase						,		_UpperCAmelCase						,		_UpperCAmelCase      )
   __lowerCAmelCase											=						rand_fn(_UpperCAmelCase						,		_UpperCAmelCase						,		_UpperCAmelCase      )
   __lowerCAmelCase											=						rand_fn(_UpperCAmelCase						,		_UpperCAmelCase						,		_UpperCAmelCase      )

   # At some point both the tortoise and the hare will enter a cycle whose
   # length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
   # the tortoise and hare will end up on the same value modulo ``p``.
   # We can detect when this happens because the position difference between
   # the tortoise and the hare will share a common divisor with ``num``.
   __lowerCAmelCase											=						gcd(hare - tortoise						,		_UpperCAmelCase      )

   if divisor == 1:
    # No common divisor yet, just keep searching.
    continue
   else:
    # We found a common divisor!
    if divisor == num:
     # Unfortunately, the divisor is ``num`` itself and is useless.
     break
    else:
     # The divisor is a nontrivial factor of ``num``!
     return divisor

    # If we made it here, then this attempt failed.
    # We need to pick a new starting seed for the tortoise and hare
    # in addition to a new step value for the random function.
    # To keep this example implementation deterministic, the
    # new values will be generated based on currently available
    # values instead of using something like ``random.randint``.

    # We can use the hare's position as the new seed.
    # This is actually what Richard Brent's the "optimized" variant does.
  __lowerCAmelCase											=						hare

  # The new step value for the random function can just be incremented.
  # At first the results will be similar to what the old function would
  # have produced, but the value will quickly diverge after a bit.
  step += 1

 # We haven't found a divisor within the requested number of attempts.
 # We were unlucky or ``num`` itself is actually prime.
 return None


if __name__ == "__main__":
 import argparse

 _A			:      int					      =				argparse.ArgumentParser()
 parser.add_argument(
     '''num''',
     type=int,
     help='''The value to find a divisor of''',
 )
 parser.add_argument(
     '''--attempts''',
     type=int,
     default=3,
     help='''The number of attempts before giving up''',
 )
 _A			:      Optional[int]					      =				parser.parse_args()

 _A			:      List[str]					      =				pollard_rho(args.num, attempts=args.attempts)
 if divisor is None:
  print(f'{args.num} is probably prime')
 else:
  _A			:      List[str]					      =				args.num // divisor
  print(f'{args.num} = {divisor} * {quotient}')













"""simple docstring"""




import qiskit





def 						UpperCAmelCase__		(     _UpperCAmelCase ,					_UpperCAmelCase					):






  """simple docstring"""





  A_  :       Tuple										=  qiskit.Aer.get_backend('aer_simulator'					)

  A_  :       str										=  qiskit.QuantumCircuit(4 ,					2					)
  # encode inputs in qubits 0 and 1
  if bita == 1:
    qc_ha.x(0					)
  if bita == 1:
    qc_ha.x(1					)
  qc_ha.barrier()

  # use cnots to write XOR of the inputs on qubit2
  qc_ha.cx(0 ,					2					)
  qc_ha.cx(1 ,					2					)

  # use ccx / toffoli gate to write AND of the inputs on qubit3
  qc_ha.ccx(0 ,					1 ,					3					)
  qc_ha.barrier()

  # extract outputs
  qc_ha.measure(2 ,					0					)  # extract XOR value
  qc_ha.measure(3 ,					1					)  # extract AND value

  # Execute the circuit on the qasm simulator
  A_  :       Optional[Any]										=  qiskit.execute(_UpperCAmelCase ,					_UpperCAmelCase ,					shots=1000					)

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


if __name__ == "__main__":
   lowerCamelCase_     :						List[str]						=					half_adder(1, 1)
   print(F"Half Adder Output Qubit Counts: {counts}")