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arXiv:2504.17965

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

Lennart Binkowski, Marvin Schwiering

Published Apr 24, 2025• 1 citation
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2504.17965

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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 O(nlog⁡(n))\mathcal{O}(n \log(n))O(nlog(n)) qubits and O(n2log⁡(n))\mathcal{O}(n^{2} \log(n))O(n2log(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.
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We present a quantum algorithm for random number generation that achieves a provable quadratic speedup over classical Markov chain mixing, building on the Diaconis-Shahshahani Fourier analysis of the top-to-random card shuffle. The algorithm integrates three quantum primitives into a unified mixing circuit: the Quantum Fourier Transform (QFT), which diagonalizes the Markov transition operator; controlled phase rotations, which encode the shuffle eigenvalue spectrum; and the Grover diffusion operator, which acts as a quantum analogue of the Aldous-Diaconis strong uniform stopping time by reflecting amplitudes about their mean at each iteration. For an n-qubit register, the mixing time is O(\sqrt{n \log n}) iterations. Extending to m qudits of local dimension d reduces this to O(\sqrt{\log_d N}) iterations, where N = d^m, compared to the classical O(n \log n) bound. The qudit formulation further reduces QFT circuit depth from O(\log^2 N) to O(\log_d^2 N) gates per layer by encoding the same N-state space using m = \log_d N subsystems instead of \log_2 N qubits. We validate both variants on IBM superconducting hardware.
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