Reduced Quantum Genetic Algorithm:

1. Initialization:
For each individual i from 0 to m-1, initialize the pair register as follows:
    The first register (index) is set to the quantum state |ψ⟩_i^index, which is a superposition of all possible values from 0 to 2^N-1.
    The second register (fitness) is initialized to the quantum state |ψ⟩_i^fitness, which is initially set to 0.

2. Fitness Evaluation:
For each individual i from 0 to m-1, apply the unitary operation U_f_fit to the pair register.
    This operation updates the second register (fitness) to the quantum state |ψ⟩_i^fitness, which represents the fitness value of the corresponding individual.

3. Maximum Fitness Initialization:
Randomly select an integer value for max between 2^(M+1) and 2^(M+2)-1.
    This sets the initial threshold for the maximum fitness value.

4. Grover's Algorithm:
For each individual i from 0 to m-1, repeat the following steps:
    Apply the oracle O_max(f_fit(u)) to the pair register.
        This marks the basis states of |ψ⟩_i^fitness with values greater than the current maximum.
    Use Grover iterations to amplify the amplitude of the marked basis states in |ψ⟩_i^fitness.
    If a marked basis state p is found with f_fit(u) greater than the current maximum, update the maximum value to p.

5. Measurement:
After the Grover iterations, measure the index register of the individual with the highest fitness value.
    This will give you the corresponding individual (or one of the corresponding individuals if there are multiple solutions).

The measurement of the individual corresponding to the best fitness value is possible due to the fact that the correlation with the first (individual) register was not destroyed (only controlled not and Hadamard gates were used). Grover’s algorithm, interpreted as prescribed by reference, is a method of augmenting the amplitude of oracle marked basis states, and it works even if the number of such states is not known in advance.

I need you to check whether the following code is properly synchronized with the algorithm described in detail above, and ensure it should follows all the required need whichis mentioned in the above requirement.

import numpy as np
from math import pi, sqrt
from qiskit import QuantumCircuit, ClassicalRegister, transpile
from qiskit.quantum_info import Statevector
from qiskit.circuit.library import GroverOperator
from math import ceil, log2

# Parameters for population
num_params = 12  # Number of scalar variables per individual
population_size = 1000  # Number of individuals in the population

# Random population of 1000 individuals, each having 12 scalar variables
population = np.random.rand(population_size, num_params)
print(f"Population size: {len(population)}")  # Debugging print statement

# Random fitness values for testing (some valid, some invalid, denoted by NaN)
population_fitnesses = np.random.rand(population_size)
invalid_indices = np.random.choice(population_size, size=int(0.2 * population_size), replace=False)  # 20% invalid
population_fitnesses[invalid_indices] = np.nan  # Marking invalid fitnesses as NaN

# Filter valid fitness values
valid_indices = np.where(~np.isnan(population_fitnesses))[0]
valid_fitnesses = population_fitnesses[valid_indices]

# Number of qubits required (based on the valid population size)
num_qubits = ceil(log2(len(valid_fitnesses)))
print(f"Calculated number of qubits: {num_qubits}")  # Debugging print statement

# For Calculating The number of Iterations
# Calculate N
N = 2 ** num_qubits

# Calculate the number of optimal Grover iterations
optimal_iterations = int(round((pi / 4) * sqrt(N)))

print(f"Optimal number of Grover iterations: {optimal_iterations}")

# optimal num_iterations
num_iterations = optimal_iterations

# Initialize quantum population with superposition
def initialize_population(num_qubits):
    qc = QuantumCircuit(num_qubits)
    qc.h(range(num_qubits))  # Hadamard gate to create equal superposition of all states
    return qc

# Apply the fitness function to update the fitness register
def fitness_evaluation(qc, valid_fitnesses, total_qubits):
    # Consider all individuals as input and encode fitness values
    oracle_circuit = QuantumCircuit(total_qubits + 1)  # Extra qubit for validity

    for i, fitness in enumerate(valid_fitnesses):
        # Consider binary representation assuming it encodes integer scaling factor appropriately
        bin_fitness = format(int(fitness * 2**total_qubits), f'0{total_qubits}b')

        for j, bit in enumerate(bin_fitness):
            if bit == '0':
                oracle_circuit.x(j)  # Flip bits for proper two's complement

        # Mark invalid individuals differently by flipping the last bit
        oracle_circuit.x(total_qubits)
        
    return oracle_circuit

def create_two_complement_oracle(valid_fitnesses, total_qubits, current_max):
    oracle_circuit = QuantumCircuit(total_qubits + 1)  # Include a qubit for marking
    
    # Transform current_max + 1 to binary string consistent with qubit count
    threshold_bin = format(current_max + 1, f'0{total_qubits}b')
    
    # Subtract the threshold using two's complement
    for i, bit in enumerate(threshold_bin):
        if bit == '1':
            oracle_circuit.x(i)  # X-gate represents inverting the bit
    
    for i, bit in enumerate(threshold_bin):
        if bit == '0' and i < total_qubits:  # Logical comparison for qubit index
            oracle_circuit.x(i)  # X-gate phase control
     
    return oracle_circuit

# Grover's search for optimizing parameters
def run_grover_search(num_qubits, valid_fitnesses, num_iterations):
    required_qubits = ceil(log2(len(valid_fitnesses)))

    if num_qubits < required_qubits:
        raise ValueError(f"Not enough qubits to represent valid population indices. Required: {required_qubits}, but got {num_qubits}.")
    
    qc = QuantumCircuit(num_qubits + 1)  # Include fitness validity qubit

    qc.compose(initialize_population(num_qubits), inplace=True)

    max_fitness = int(max(valid_fitnesses) * (2**num_qubits - 1))  
    print(f"Initial max fitness threshold: {max_fitness}")

    for _ in range(num_iterations):
        oracle = create_two_complement_oracle(valid_fitnesses, num_qubits, max_fitness)
        qc.compose(oracle, inplace=True)

        # Proper Grover iterations go here
        
    statevector = Statevector.from_instruction(qc)
    probabilities = statevector.probabilities_dict()
    best_state = max(probabilities, key=probabilities.get)
    best_index = int(best_state, 2) % len(valid_fitnesses)

    return valid_indices[best_index], population[valid_indices[best_index]]

# Run Grover’s search to find the optimal individual
print(f"Valid population fitnesses: {valid_fitnesses}")  # Debugging print statement
best_index, optimal_solution = run_grover_search(num_qubits, valid_fitnesses, num_iterations)

# Display the optimal individual and its index
print(f"Optimal solution (best input parameters for individual {best_index}): {optimal_solution}")