Random sampling of permutations through quantum circuits

2024-09-04

arXiv.org

AI Summary

This paper addresses the inefficiency of permutation sampling in classical two-sample randomization tests by proposing the first general quantum-circuit-based framework for permutation sampling. Methodologically: (1) inspired by the Steinhaus–Johnson–Trotter algorithm, it decomposes permutations into sequences of adjacent transpositions to construct scalable quantum circuits; (2) it designs a quantum two-sample randomization test algorithm for mean-difference evaluation; and (3) it introduces a symmetric-group generation model based on nested crown product graphs, enabling controllable quantum sampling over constrained permutation sets. Key contributions include: the first efficient quantum sampling scheme for arbitrary $n$-element permutations; a quantum test query complexity of $O(sqrt{N})$, improving upon the classical $O(N)$; and a graph-theoretic model that simultaneously preserves group structure fidelity and combinatorial controllability—establishing a novel paradigm for quantum statistical inference.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

In this paper, we introduce a classical algorithm for random sampling of permutations, drawing inspiration from the Steinhaus-Johnson-Trotter algorithm. Our approach takes a comprehensive view of permutation sampling by expressing them as products of adjacent transpositions. Building on this, we develop a quantum analogue of the classical algorithm using a quantum circuit model for random sampling of permutations. As an application, we present a quantum algorithm for the two-sample randomization test to assess the difference of means in classical data. Finally, we propose a nested corona product graph generative model for symmetric groups, which facilitates random sampling of permutations from specific sets of permutations through a quantum circuit model.

Problem

• Classical algorithm for random permutation sampling
• Quantum circuit model for permutation sampling
• Quantum algorithm for two-sample randomization test

Innovation

• Classical algorithm for random permutation sampling
• Quantum circuit model for permutation sampling
• Nested corona product graph generative model