A quantized Fisher-Yates shuffle builds five quantum circuits for uniform permutation superpositions with O(n log n) qubits and up to O(n^2 log^2 n) gates, though the headline says O(n^2 log n).
Since the ancilla register’s state is uncomputed after each itera- tion (lines 10-12), we can reuse it’s qubits for the sub- sequent one
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Quantum Fisher-Yates shuffle: Unifying methods for generating uniform superpositions of permutations
A quantized Fisher-Yates shuffle builds five quantum circuits for uniform permutation superpositions with O(n log n) qubits and up to O(n^2 log^2 n) gates, though the headline says O(n^2 log n).