





'''simple docstring'''




from __future__ import annotations

from math import gcd






def        _lowerCAmelCase			(  __snake_case  : int ,							__snake_case  : int = 2 ,							__snake_case  : int = 1 ,							__snake_case  : int = 3 ,							)     ->     int | None:
     # A value less than 2 can cause an infinite loop in the algorithm.
     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  : int ,							__snake_case  : int			) -> int:
          return (pow(__snake_case ,							2			) + step) % modulus

     for _ in range(__snake_case			):
          # These track the position within the cycle detection logic.
          __A						: Dict      =       seed
          __A						: int      =       seed

          while True:
               # At each iteration, the tortoise moves one step and the hare moves two.
               __A						: Tuple      =       rand_fn(__snake_case ,							__snake_case ,							__snake_case			)
               __A						: Optional[int]      =       rand_fn(__snake_case ,							__snake_case ,							__snake_case			)
               __A						: Tuple      =       rand_fn(__snake_case ,							__snake_case ,							__snake_case			)

               # 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``.
               __A						: Tuple      =       gcd(hare - tortoise ,							__snake_case			)

               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.
          __A						: Optional[Any]      =       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

  lowercase__    :      Dict						=    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''',
  )
  lowercase__    :      Optional[int]						=    parser.parse_args()

  lowercase__    :      Optional[int]						=    pollard_rho(args.num, attempts=args.attempts)
  if divisor is None:
    print(f"""{args.num} is probably prime""")
  else:
    lowercase__    :      Optional[Any]						=    args.num // divisor
    print(f"""{args.num} = {divisor} * {quotient}""")






'''simple docstring'''




import math

import qiskit






def        _lowerCAmelCase			(  __snake_case  : int = 1 ,							__snake_case  : int = 1 ,							__snake_case  : int = 1			)     ->     qiskit.result.counts.Counts:
     if (
         isinstance(__snake_case ,							__snake_case			)
         or isinstance(__snake_case ,							__snake_case			)
         or isinstance(__snake_case ,							__snake_case			)
     ):
          raise TypeError('inputs must be integers.'			)

     if (input_a < 0) or (input_a < 0) or (carry_in < 0):
          raise ValueError('inputs must be positive.'			)

     if (
         (math.floor(__snake_case			) != input_a)
         or (math.floor(__snake_case			) != input_a)
         or (math.floor(__snake_case			) != carry_in)
     ):
          raise ValueError('inputs must be exact integers.'			)

     if (input_a > 2) or (input_a > 2) or (carry_in > 2):
          raise ValueError('inputs must be less or equal to 2.'			)

     # build registers
     __A						: int      =       qiskit.QuantumRegister(4 ,							'qr'			)
     __A						: Optional[int]      =       qiskit.ClassicalRegister(2 ,							'cr'			)
     # list the entries
     __A						: Union[str, Any]      =       [input_a, input_a, carry_in]

     __A						: Dict      =       qiskit.QuantumCircuit(__snake_case ,							__snake_case			)

     for i in range(0 ,							3			):
          if entry[i] == 2:
               quantum_circuit.h(__snake_case			)  # for hadamard entries
          elif entry[i] == 1:
               quantum_circuit.x(__snake_case			)  # for 1 entries
          elif entry[i] == 0:
               quantum_circuit.i(__snake_case			)  # for 0 entries

    # build the circuit
     quantum_circuit.ccx(0 ,							1 ,							3			)  # ccx = toffoli gate
     quantum_circuit.cx(0 ,							1			)
     quantum_circuit.ccx(1 ,							2 ,							3			)
     quantum_circuit.cx(1 ,							2			)
     quantum_circuit.cx(0 ,							1			)

     quantum_circuit.measure([2, 3] ,							__snake_case			)  # measure the last two qbits

     __A						: str      =       qiskit.Aer.get_backend('aer_simulator'			)
     __A						: Any      =       qiskit.execute(__snake_case ,							__snake_case ,							shots=10_00			)

     return job.result().get_counts(__snake_case			)


if __name__ == "__main__":
  print(f"""Total sum count for state is: {quantum_full_adder(1, 1, 1)}""")