

import doctest

import projectq
from projectq.ops import H, Measure


def       __a       (   SCREAMING_SNAKE_CASE      )      ->     int:






    '''simple docstring'''
    __UpperCAmelCase									=			quantum_engine.allocate_qubit()
    H | qubit
    Measure | qubit
    return int(SCREAMING_SNAKE_CASE      )


if __name__ == "__main__":
       doctest.testmod()

       # initialises a new quantum backend
       A_		:			str										=   projectq.MainEngine()

       # Generate a list of 10 random numbers
       A_		:			Tuple										=   [get_random_number(quantum_engine) for _ in range(10)]

       # Flushes the quantum engine from memory
       quantum_engine.flush()

       print('Random numbers', random_numbers_list)







import numpy as np


def       __a       (   SCREAMING_SNAKE_CASE      )      ->     Tuple:






    '''simple docstring'''
    __UpperCAmelCase					,     __UpperCAmelCase									=			a.shape
    __UpperCAmelCase									=			min(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE      )
    __UpperCAmelCase									=			np.eye(SCREAMING_SNAKE_CASE      )
    __UpperCAmelCase									=			a.copy()

    for k in range(t - 1      ):
        # select a column of modified matrix A':
        __UpperCAmelCase									=			r[k:, [k]]
        # construct first basis vector
        __UpperCAmelCase									=			np.zeros_like(SCREAMING_SNAKE_CASE      )
        __UpperCAmelCase									=			1.0
        # determine scaling factor
        __UpperCAmelCase									=			np.linalg.norm(SCREAMING_SNAKE_CASE      )
        # construct vector v for Householder reflection
        __UpperCAmelCase									=			x + np.sign(x[0]      ) * alpha * ea
        v /= np.linalg.norm(SCREAMING_SNAKE_CASE      )

        # construct the Householder matrix
        __UpperCAmelCase									=			np.eye(m - k      ) - 2.0 * v @ v.T
        # pad with ones and zeros as necessary
        __UpperCAmelCase									=			np.block([[np.eye(SCREAMING_SNAKE_CASE      ), np.zeros((k, m - k)      )], [np.zeros((m - k, k)      ), q_k]]      )

        __UpperCAmelCase									=			q @ q_k.T
        __UpperCAmelCase									=			q_k @ r

    return q, r


if __name__ == "__main__":
       import doctest

       doctest.testmod()




