

from __future__ import annotations





def      snake_case_				(__A					:						int					)      ->     list[int]:
   __lowerCAmelCase  :      Optional[Any]    =						[True] * limit
   __lowerCAmelCase  :      Tuple    =						False
   __lowerCAmelCase  :      Tuple    =						False
   __lowerCAmelCase  :      int    =						True

   for i in range(3 ,       int(limit**0.5 + 1					) ,       2					):
      __lowerCAmelCase  :      str    =						i * 2
      while index < limit:
         __lowerCAmelCase  :      str    =						False
         __lowerCAmelCase  :      Optional[int]    =						index + i

   __lowerCAmelCase  :      str    =						[2]

   for i in range(3 ,       __A ,       2					):
      if is_prime[i]:
         primes.append(__A					)

   return primes





def      snake_case_				(__A					:						int = 1_0_0_0_0_0_0					)      ->     int:
   __lowerCAmelCase  :      Optional[Any]    =						prime_sieve(__A					)
   __lowerCAmelCase  :      int    =						0
   __lowerCAmelCase  :      Optional[Any]    =						0

   for i in range(len(__A					)					):
      for j in range(i + length ,       len(__A					)					):
         __lowerCAmelCase  :      Union[str, Any]    =						sum(primes[i:j]					)
         if sol >= ceiling:
            break

         if sol in primes:
            __lowerCAmelCase  :      Dict    =						j - i
            __lowerCAmelCase  :      str    =						sol

   return largest


if __name__ == "__main__":
   print(F'{solution() = }')





import numpy as np
import qiskit





def      snake_case_				(__A					:						int = 8 ,       __A					:						int | None = None					)      ->     str:
   __lowerCAmelCase  :      List[Any]    =						np.random.default_rng(seed=__A					)

   # Roughly 25% of the qubits will contribute to the key.
   # So we take more than we need.
   __lowerCAmelCase  :      Tuple    =						6 * key_len
   # Measurement basis for Alice's qubits.
   __lowerCAmelCase  :      List[Any]    =						rng.integers(2 ,       size=__A					)
   # The set of states Alice will prepare.
   __lowerCAmelCase  :      List[str]    =						rng.integers(2 ,       size=__A					)
   # Measurement basis for Bob's qubits.
   __lowerCAmelCase  :      List[Any]    =						rng.integers(2 ,       size=__A					)

   # Quantum Circuit to simulate BB84
   __lowerCAmelCase  :      int    =						qiskit.QuantumCircuit(__A ,       name="""BB84"""					)

   # Alice prepares her qubits according to rules above.
   for index, _ in enumerate(__A					):
      if alice_state[index] == 1:
         bbaa_circ.x(__A					)
      if alice_basis[index] == 1:
         bbaa_circ.h(__A					)
   bbaa_circ.barrier()

   # Bob measures the received qubits according to rules above.
   for index, _ in enumerate(__A					):
      if bob_basis[index] == 1:
         bbaa_circ.h(__A					)

   bbaa_circ.barrier()
   bbaa_circ.measure_all()

   # Simulate the quantum circuit.
   __lowerCAmelCase  :      Optional[Any]    =						qiskit.Aer.get_backend("""aer_simulator"""					)
   # We only need to run one shot because the key is unique.
   # Multiple shots will produce the same key.
   __lowerCAmelCase  :      Optional[Any]    =						qiskit.execute(__A ,       __A ,       shots=1 ,       seed_simulator=__A					)
   # Returns the result of measurement.
   __lowerCAmelCase  :      List[Any]    =						job.result().get_counts(__A					).most_frequent()

   # Extracting the generated key from the simulation results.
   # Only keep measurement results where Alice and Bob chose the same basis.
   __lowerCAmelCase  :      Optional[int]    =						"""""".join(
       [
           result_bit
           for alice_basis_bit, bob_basis_bit, result_bit in zip(
               __A ,       __A ,       __A					)
           if alice_basis_bit == bob_basis_bit
       ]					)

   # Get final key. Pad with 0 if too short, otherwise truncate.
   __lowerCAmelCase  :      Tuple    =						gen_key[:key_len] if len(__A					) >= key_len else gen_key.ljust(__A ,       """0"""					)
   return key


if __name__ == "__main__":
   print(F'The generated key is : {bbaa(8, seed=0)}')
   from doctest import testmod

   testmod()


