Quantum Fisher-Yates shuffle: Unifying methods for generating uniform superpositions of permutations

Abstract: Uniform superpositions over permutations play a central role in quantum error correction, cryptography, and combinatorial optimisation. We introduce a simple yet powerful quantisation of the classical Fisher-Yates shuffle, yielding a suite of efficient quantum algorithms for preparing such superpositions on composite registers. Our method replaces classical randomness with coherent control, enabling five variants that differ in their output structure and entanglement with ancillary systems. We demonstrate that this construction achieves the best known combination of asymptotic resources among all existing approaches, requiring only $\mathcal{O}(n \log(n))$ qubits and $\mathcal{O}(n^{2} \log(n))$ gates and circuit depth. These results position the quantum Fisher-Yates shuffle as a strong candidate for optimality within this class of algorithms. Our work unifies several prior constructions under a single, transparent framework and opens up new directions for quantum state preparation using classical combinatorial insights. Our implementation in Qiskit is available as open-source code, supporting reproducibility and future exploration of quantum permutation-based algorithms.