As your question suggests there is little literature on the subject. Also I wrote a blog post about an algorithm I created that can sort a list of numbers using a quantum algorithm.
You can find it here.
In this article, I propose a quantum sort algorithm that allows you to sort a list of integers coded on two qubits. I have found that there is little literature on the subject. This algorithm can be used and can be used as a basis for solving sorting problems on an unsorted list of integers.
To solve this problem we will first create a quantum circuit that allows you to measure if a figure is lower than another for this the intuition is to be based on a circuit similar to a classic circuit of comparison of magnitude at 2 qubit.
The idea of my algorithm is to permute 4 digits using control qubits which will be in superimposed state.
4 Hadamard quantum gates are used to implement these permutations and take advantage of the superposition of the Qubit ‘control’ register
In other words in the circuit above we see that we start the circuit by initializing the registers $a$, $b$, $m$, $n$ with its list of unsorted numbers, here ($a = 0$, $b = 2$, $m = 3$, $n = 1$) .
The permutation circuit which is controlled by a superimposed qubit register allows the list to be permuted in quantum superposition.
Thus we obtain on the qubit q0 the quantum state $|1\rangle$ if the permuted list is sorted in a decreasing way otherwise the quantum state $|0\rangle$.
In this article you will find the implementation of the algorithm in python and in OpenQASM. If you have any further questions, please do not hesitate to contact me.