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.
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(logd+log(1/ε)) repetitions suffice to approximate Haar twirling on the symmetric subspace of (Cd)⊗t to error ε, 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 O(n3) (Kook and Vempala, 2026) to 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.
Most approximation algorithms for #P-complete problems (e.g., evaluating the partition function of a monomer-dimer or ferromagnetic Ising system) work by reduction to the problem of approximate sampling from a distribution $\ensuremath{\pi}$ over a large set S. This problem is solved using the Markov chain Monte Carlo method: a sparse, reversible Markov chain P on S with stationary distribution $\ensuremath{\pi}$ is run to near equilibrium. The running time of this random walk algorithm, the so-called mixing time of P, is $O({\ensuremath{\delta}}^{\ensuremath{-}1}\text{ }\text{log}\text{ }1∕{\ensuremath{\pi}}_{\ensuremath{\ast}})$ as shown by Aldous, where $\ensuremath{\delta}$ is the spectral gap of P and ${\ensuremath{\pi}}_{\ensuremath{\ast}}$ is the minimum value of $\ensuremath{\pi}$. A natural question is whether a speedup of this classical method to $O(\sqrt{{\ensuremath{\delta}}^{\ensuremath{-}1}}\text{ }\text{log}\text{ }1∕{\ensuremath{\pi}}_{\ensuremath{\ast}})$ is possible using quantum walks. We provide evidence for this possibility using quantum walks that decohere under repeated randomized measurements. We show that (i) decoherent quantum walks always mix, just like their classical counterparts, (ii) the mixing time is a robust quantity, essentially invariant under any smooth form of decoherence, and (iii) the mixing time of the decoherent quantum walk on a periodic lattice Znd is O(ndlogd), which is indeed $O(\sqrt{{\ensuremath{\delta}}^{\ensuremath{-}1}}\text{ }\text{log}\text{ }1∕{\ensuremath{\pi}}_{\ensuremath{\ast}})$ and is asymptotically no worse than the diameter of Znd (the obvious lower bound) up to at most a logarithmic factor.
Continuous-time quantum walks on graphs is a generalization of continuous-time Markov chains on discrete structures. Moore and Russell proved that the continuous-time quantum walk on the n-cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs Kn, the continuous-time quantum walk is neither instantaneous (except for n=2,3,4) nor average uniform mixing (except for n=2). We explore two natural {\em group-theoretic} generalizations of the n-cube as a G-circulant and as a bunkbed $G \rtimes \Int_{2}$, where G is a finite group. Analyses of these classes suggest that the n-cube might be special in having instantaneous uniform mixing and that non-uniform average mixing is pervasive, i.e., no memoryless property for the average limiting distribution; an implication of these graphs having zero spectral gap. But on the bunkbeds, we note a memoryless property with respect to the two partitions. We also analyze average mixing on complete paths, where the spectral gaps are nonzero.
Quantum mixing is a category of processes of reaching a stationary state when starting from a single initial vertex via quantum walks. In this thesis, we introduce the quantum mixing processes based on the framework of controlled quantum walks in [DH17a]. We first discuss why Szegedy's walks are not well-suited for the quantum mixing processes. We then show that the controlled quantum walk is a suitable framework for achieving a quadratic speed-up in the quantum mixing processes, because of its relationship with interpolated walks [KMOR16] and the phase estimation algorithm [CEMM98]. Accordingly, we give two definitions of quantum mixing times via controlled quantum walks: instant quantum mixing time and average quantum mixing time. We later prove the bounds of these two quantum mixing times on complete graphs. We also prove the upper bound of the average quantum mixing times on cycle graphs. To have clearer views on the bounds of instant and average quantum mixing times, we then discuss and present the simulations of quantum mixing processes, with four types of graphs (complete graphs, cycle graphs, 2-dimensional tori and dumbbell graphs) as examples. The simulations include the quantum mixing processes with the same ~θ (an important parameter in the controlled quantum walk operator) as in [DH17a], and the suitable range of ~θ that achieves the quadratic speed-up for different types of graphs. Inspired by the quantum search algorithm via controlled quantum walks in [DH17a], we adapt this algorithm to a new algorithm of quantum mixing, which generates samples from the stationary distribution of the Markov chain, with its runtime upper bounded by the quantum hitting time, which is quadratically faster than the classical hitting process.
Quantum walks have been linked to acceleration in various information processing tasks, and proposed as a possible model for quantum-enhanced behavior in biological systems.These links and acceleration claims have been made with various levels of detail.Here we consider discrete-time quantum walks, and focus on the task of mixing, i.e., distributing the state over a graph.Previous papers have observed that the so-called coined quantum walks can accelerate mixing on certain graphs with respect to the optimal classical Markov chain.We here show that the same speedup can be attained with a classical process, if a similar classical coin is added.We establish a precise correspondence between the mixing performance of quantum walks and such "lifted walks" for all (finite) graphs, and thereby improve known bounds on quantum walk mixing time.We conclude that the advantage of quantum walks with respect to classical processes is not in the mixing speed of the optimal design.However, a notable quantum advantage might reside in the fact that the mixing speed obtained with suboptimal designs, due to for instance limited graph knowledge, appears to be generically faster.
This paper gives new observations on the mixing dynamics of a continuous-time quantum walk on circulants and their bunkbeds. These bunkbeds are defined through two standard graph operators: the join G + H and the Cartesian product G \cprod H of graphs G and H. Our results include the following: (i) The quantum walk is average uniform mixing on circulants with bounded eigenvalue multiplicity; this extends a known fact about the cycles C_{n}. (ii) Explicit analysis of the probability distribution of the quantum walk on the join of circulants; this explains why complete multipartite graphs are not average uniform mixing, using the fact K_{n} = K_{1} + K_{n-1} and K_{n,\ldots,n} = \overline{K}_{n} + \ldots + \overline{K}_{n}. (iii) The quantum walk on the Cartesian product of a m-vertex path P_{m} and a circulant G, namely, P_{m} \cprod G, is average uniform mixing if G is; this highlights a difference between circulants and the hypercubes Q_{n} = P_{2} \cprod Q_{n-1}. Our proofs employ purely elementary arguments based on the spectra of the graphs.
The fundamental problem of sampling from the limiting distribution of quantum walks on networks, known as mixing, finds widespread applications in several areas of quantum information and computation. Of particular interest in most of these applications is the minimum time beyond which the instantaneous probability distribution of the quantum walk remains close to this limiting distribution, known as the "quantum mixing time". However, this quantity is only known for a handful of specific networks. In this Letter, we prove an upper bound on the quantum mixing time for almost all networks, i.e., the fraction of networks for which our bound holds, goes to one in the asymptotic limit. To this end, using several results in random matrix theory, we find the quantum mixing time of Erdös-Renyi random networks: networks of n nodes where each edge exists with probability p independently. For example, for dense random networks, where p is a constant, we show that the quantum mixing time is O(n^{3/2+o(1)}). In addition to opening avenues for the analytical study of quantum dynamics on random networks, our work could find applications beyond quantum information processing. Owing to the universality of Wigner random matrices, our results on the spectral properties of random graphs hold for general classes of random matrices that are ubiquitous in several areas of physics. In particular, our results could lead to novel insights into the equilibration times of isolated quantum systems defined by random Hamiltonians, a foundational problem in quantum statistical mechanics.