Random sampling of permutations through quantum circuits

TL;DR

The paper tackles efficient random sampling of permutations and permutation-based hypothesis testing in a quantum setting. It builds a classical SSJTT-inspired framework that expresses permutations as products of adjacent transpositions and then develops a quantum circuit model that realizes these transpositions on $n$-qubit registers using generalized Toffoli, $X$, and CNOT gates, enabling both random sampling of permutations and exact permutation implementation on a compact primary register. Key contributions include an $O(N^2)$ classical sampling method, a quantum circuit architecture with $O(N\log_2 N)$ qubits and $O(N^3\log_2 N)$ gates for general sampling, and a concrete quantum algorithm for the two-sample randomization test with potential $O(\sqrt{N})$ speedup over the classical approach. The work also introduces a nested corona product graph representation of the symmetric group to enable sampling from restricted permutation sets, broadening the applicability to structured permutation tests and post-quantum cryptographic tasks. Overall, the framework links combinatorial permutation generation with quantum circuit synthesis to enable scalable, permutation-based quantum data processing and hypothesis testing.

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.

Figures (12)

• Figure 1: Coxeter graph of $\mathcal{S}_k$.
• Figure 2: The binary tree representation $\mathfrak{P}_N$ of the process of generating all the permutations in terms of product of adjacent transpositions, for $N=2,3,4$.
• Figure 3: The quantum binary tree $\texttt{QT}_3.$ The $i$-th order nodes are labelled by $\ket{\textcolor{blue}{ q}}_i=\ket{q_{i-1}\hdots q_1q_0}_i$, where $0\leq q\leq 2^{i}-1,$$1\leq i\leq 3$ and $q=\sum_{j=0}^{i-1} q_j2^{j}, q_j\in\{0,1\}.$
• Figure 4: Generalized Toffoli gates for $3$-qubit system through standard Toffoli gate and the $X$-gate. (a) $T_{11}=(6,7)$, (b) $T_{10}=(4,5)$, (c) $T_{01}=(2,3),$ (d) $T_{00}=(0,1)$
• Figure 5: The $n$-th and $(n-1)$-th order basis states and their combinatorial connectivity through the quantum binary tree $\texttt{QT}_3.$ The labeling in blue for the $n$-th order nodes represents the coefficient for any quantum state $\ket{\psi}=\sum_{j=0}^{2^n-1} \textcolor{blue}{ a_j} \ket{j}$ with respect to the canonical ordering of the basis states in ${\mathbb C}^{2\otimes n}.$ The $n$-th qubit corresponding to odd indexed basis states of ${\mathbb C}^{2\otimes n}$ are colored red.