Consider a quantum computing scenario where you have a set of $3$ qubits. Initially, all qubits are in the state $\text{'}0\text{'}.$ You are required to generate all possible states these qubits can take by permuting their values lexicographically.

Write a detailed algorithm that explains how to generate all permutations of the qubits' states lexicographically. Begin with the initial state of $000$ and proceed to generate all $2^3$

combinations. Use the lexicographic permutation algorithm where you identify the largest index k

such that the sequence increases right before it decreases and swap elements to continue generating the next permutation in lexicographic order.

Detail each step and provide an explanation for the reasoning behind your swaps and how you ensure that each subsequent configuration of qubit states is the next in lexicographic order.

Hints:

Start with the initial state $000.$

Identify the next state that can be achieved with the minimal change following lexicographic order.

Explain the process for determining which bits $($qubits$)$ to flip to achieve this order.

Consider the mechanism of swapping and reversing to achieve the next permutation.