Rapid Mixing of Parallel Kac's Walk: From Spheres to Stiefel Manifolds

Abstract: Kac's walk is a classical local random walk whose action on a single real unit vector in dimension $d$ mixes in total variation in $Θ(d\log d)$ sequential steps~\cite{PS17}. Lu, Qin, Song, Yao, and Zhao introduced a parallel version of Kac's walk that mixes a single quantum state in $O(\log d)$ rounds~\cite{LQSY+26}. After discretizing the randomness and replacing it by suitable pseudorandom primitives, this parallel walk gives rise to pseudorandom state scramblers, and was subsequently shown to yield pseudorandom unitaries~\cite{LQSY+25}. We study what happens when the parallel Kac's walk acts simultaneously on $k$ orthonormal quantum states. We prove that, for any $1\leq k < d$, after $O!\left((k+\log d)\log(d/\varepsilon)\right)$ steps, the joint distribution of the $k$ output states is $\varepsilon$-close, in both Wasserstein and total variation distance, to that obtained by applying a common Haar-random unitary to the same inputs. This generalizes the dispersing property of the parallel Kac's walk from a single quantum state to multiple orthonormal quantum states. Equivalently, viewing an ordered collection of $k$ orthonormal states as a point on the complex Stiefel manifold $$V_{d,k}={X\in\mathbb C^{d\times k}:X^\dagger X=I_k}$$, we show that the parallel Kac's walk mixes rapidly on $V_{d,k}$, with both Wasserstein and total variation mixing times bounded by $O!\left((k+\log d)\log(d/\varepsilon)\right)$. This extends the Wasserstein mixing result of Pillai, Smith, and Vaikuntanathan for the standard Kac's walk on real Stiefel manifolds~\cite{PSV26}.