A Tale of Two Walks: Kipnis, Marchioro and Presutti Meet Kac in a Quantum World

2026-09-29

AI Summary

This study addresses the theoretical bottleneck in efficiently approximating Haar-random rotation channels on symmetric subspaces and analyzing multi-particle mixing times. The proposed method reveals an intrinsic connection between parallel Kac walks and the KMP process, reducing the analysis of quantum rotation channels to classical Markov chain mixing problems. By leveraging a conditional product structure to establish an analogue of Aldous’ conjecture, and by integrating exact coupling with geometric analysis techniques, the work optimizes the simplex Hit-and-Run algorithm. These contributions achieve uniform convergence while reducing algorithmic complexity to near-linear scaling. Furthermore, it is established that logarithmic repetitions suffice to approximate Haar rotations with high precision, significantly lowering the computational cost of sampling in high-dimensional settings.

Abstract

We reveal an unexpected connection between the parallel Kac's walk and the Kipnis-Marchioro-Presutti (KMP) process. The twirling channel induced by the parallel Kac's walk on the symmetric subspace is exactly encoded by a classical Markov chain on partitions, which lifts to a parallel KMP process on complete graphs. This correspondence reduces the analysis of the twirling channel to the mixing of the parallel KMP process. We prove that $O(\log d+\log(1/\varepsilon))$ repetitions suffice to approximate Haar twirling on the symmetric subspace of $(\mathbb C^d)^{\otimes t}$ to error $\varepsilon$, uniformly in the number of copies $t$. For the standard KMP process on general graphs, we prove a mixing-time analogue of Aldous's conjecture: at fixed accuracy, the mixing time of the $t$-particle process is at most a constant times the single-particle mixing time multiplied by the logarithm of the number of vertices, uniformly in $t$. As an application, we improve the total variation mixing-time bound for coordinate hit-and-run on the $n$-dimensional standard simplex from $\widetilde O(n^3)$ (Kook and Vempala, 2026) to $\widetilde O(n)$, while removing the dependence on the initial distribution. Our main technical contribution is conditional product structure for both parallel and standard KMP processes. Conditioned on suitable auxiliary randomness, the labeled particles evolve independently. Combining this structure with an exact coupling yields mixing bounds uniform in the number of particles for both unlabeled KMP models. These bounds are sharp up to logarithmic factors and imply rapid convergence of the parallel Kac twirling channel on the symmetric subspace.

Problem

• Kac's walk
• KMP process
• Haar twirling
• mixing time
• Aldous's conjecture

Innovation

• Kac walk
• KMP process
• Haar twirling
• mixing time
• conditional product structure