code: # This code is part of Qiskit.
  #
  # (C) Copyright IBM 2020, 2021.
  #
  # This code is licensed under the Apache License, Version 2.0. You may
  # obtain a copy of this license in the LICENSE.txt file in the root directory
  # of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
  #
  # Any modifications or derivative works of this code must retain this
  # copyright notice, and modified files need to carry a notice indicating
  # that they have been altered from the originals.
  
  """
  Convert knapsack parameters instances into Pauli list
  The parameters are a list of values a list of weights and a maximum weight of the knapsack.
  
  In the Knapsack Problem we are given a list of objects that each has a weight and a value.
  We are also given a maximum weight we can carry. We need to pick a subset of the objects
  so as to maximize the total value without going over the maximum weight.
  
  If we have the weights w[i], the values v[i] and the maximum weight W_max.
  We express the solution as a binary array x[i]
  where we have a 1 for the items we take in the solution set.
  We need to maximize sum(x[i]*v[i]) while respecting W_max >= sum(x[i]*w[i])
  
  """
  
  import logging
  import math
  import numpy as np
  
  from qiskit.quantum_info import Pauli
  from qiskit.aqua.operators import WeightedPauliOperator
  
  
  logger = logging.getLogger(__name__)
  
  
  def get_operator(values, weights, max_weight):
      """
      Generate Hamiltonian for the knapsack problem.
  
      Notes:
          To build the cost function for the Hamiltonian we add a term S
          that will vary with our solution. In order to make it change wit the solution
          we enhance X with a number of additional bits X' = [x_0,..x_{n-1},y_{n}..y_{n+m-1}].
          The bytes y[i] will be the binary representation of S.
          In this way the optimizer will be able to optimize S as well as X.
  
          The cost function is
          $$C(X') = M(W_{max} - \\sum_{i=0}^{n-1} x_{i}w_{i} - S)^2 - \\sum_{i}^{n-1} x_{i}v_{i}$$
          where S = sum(2**j * y[j]), j goes from n to n+log(W_max).
          M is a number large enough to dominate the sum of values.
  
          Because S can only be positive, when W_max >= sum(x[i]*w[i])
          the optimizer can find an S (or better the y[j] that compose S)
          so that it will take the first term to 0.
          This way the function is dominated by the sum of values.
          If W_max < sum(x[i]*w[i]) then the first term can never be 0
          and, multiplied by a large M, will always dominate the function.
  
          The minimum value of the function will be that where the constraint is respected
          and the sum of values is maximized.
  
      Args:
          values (list of non-negative integers) : a list of values
          weights (list of non-negative integers) : a list of weights
          max_weight (non negative integer) : the maximum weight the knapsack can carry
  
      Returns:
          WeightedPauliOperator: operator for the Hamiltonian
          float: a constant shift for the obj function.
  
      Raises:
          ValueError: values and weights have different lengths
          ValueError: A value or a weight is negative
          ValueError: All values are zero
          ValueError: max_weight is negative
  
      """
      if len(values) != len(weights):
          raise ValueError("The values and weights must have the same length")
  
      if any(v < 0 for v in values) or any(w < 0 for w in weights):
          raise ValueError("The values and weights cannot be negative")
  
      if all(v == 0 for v in values):
          raise ValueError("The values cannot all be 0")
  
      if max_weight < 0:
          raise ValueError("max_weight cannot be negative")
  
      y_size = int(math.log(max_weight, 2)) + 1 if max_weight > 0 else 1
      n = len(values)
      num_values = n + y_size
      pauli_list = []
      shift = 0
  
      # pylint: disable=invalid-name
      M = 10 * np.sum(values)
  
      # term for sum(x_i*w_i)**2
      for i in range(n):
          for j in range(n):
              coefficient = -1 * 0.25 * weights[i] * weights[j] * M
              pauli_op = _get_pauli_op(num_values, [j])
              pauli_list.append([coefficient, pauli_op])
              shift -= coefficient
  
              pauli_op = _get_pauli_op(num_values, [i])
              pauli_list.append([coefficient, pauli_op])
              shift -= coefficient
  
              coefficient = -1 * coefficient
              pauli_op = _get_pauli_op(num_values, [i, j])
              pauli_list.append([coefficient, pauli_op])
              shift -= coefficient
  
      # term for sum(2**j*y_j)**2
      for i in range(y_size):
          for j in range(y_size):
              coefficient = -1 * 0.25 * (2 ** i) * (2 ** j) * M
  
              pauli_op = _get_pauli_op(num_values, [n + j])
              pauli_list.append([coefficient, pauli_op])
              shift -= coefficient
  
              pauli_op = _get_pauli_op(num_values, [n + i])
              pauli_list.append([coefficient, pauli_op])
              shift -= coefficient
  
              coefficient = -1 * coefficient
              pauli_op = _get_pauli_op(num_values, [n + i, n + j])
              pauli_list.append([coefficient, pauli_op])
              shift -= coefficient
  
      # term for -2*W_max*sum(x_i*w_i)
      for i in range(n):
          coefficient = max_weight * weights[i] * M
  
          pauli_op = _get_pauli_op(num_values, [i])
          pauli_list.append([coefficient, pauli_op])
          shift -= coefficient
  
      # term for -2*W_max*sum(2**j*y_j)
      for j in range(y_size):
          coefficient = max_weight * (2 ** j) * M
  
          pauli_op = _get_pauli_op(num_values, [n + j])
          pauli_list.append([coefficient, pauli_op])
          shift -= coefficient
  
      for i in range(n):
          for j in range(y_size):
              coefficient = -1 * 0.5 * weights[i] * (2 ** j) * M
  
              pauli_op = _get_pauli_op(num_values, [n + j])
              pauli_list.append([coefficient, pauli_op])
              shift -= coefficient
  
              pauli_op = _get_pauli_op(num_values, [i])
              pauli_list.append([coefficient, pauli_op])
              shift -= coefficient
  
              coefficient = -1 * coefficient
              pauli_op = _get_pauli_op(num_values, [i, n + j])
              pauli_list.append([coefficient, pauli_op])
              shift -= coefficient
  
      # term for sum(x_i*v_i)
      for i in range(n):
          coefficient = 0.5 * values[i]
  
          pauli_op = _get_pauli_op(num_values, [i])
          pauli_list.append([coefficient, pauli_op])
          shift -= coefficient
  
      return WeightedPauliOperator(paulis=pauli_list), shift
  
  
  def get_solution(x, values):
      """
      Get the solution to the knapsack problem
      from the bitstring that represents
      to the ground state of the Hamiltonian
  
      Args:
          x (numpy.ndarray): the ground state of the Hamiltonian.
          values (numpy.ndarray): the list of values
  
      Returns:
          numpy.ndarray: a bit string that has a '1' at the indexes
           corresponding to values that have been taken in the knapsack.
           i.e. if the solution has a '1' at index i then
           the value values[i] has been taken in the knapsack
      """
      return x[:len(values)]
  
  
  def knapsack_value_weight(solution, values, weights):
      """
      Get the total wight and value of the items taken in the knapsack.
  
      Args:
          solution (numpy.ndarray) : binary string that represents the solution to the problem.
          values (numpy.ndarray) : the list of values
          weights (numpy.ndarray) : the list of weights
  
      Returns:
          tuple: the total value and weight of the items in the knapsack
      """
      value = np.sum(solution * values)
      weight = np.sum(solution * weights)
      return value, weight
  
  
  def _get_pauli_op(num_values, indexes):
      pauli_x = np.zeros(num_values, dtype=bool)
      pauli_z = np.zeros(num_values, dtype=bool)
      for i in indexes:
          pauli_z[i] = not pauli_z[i]
  
      return Pauli((pauli_z, pauli_x))
  
apis: numpy.sum numpy.zeros