Many classical randomized algorithms (e.g. approximation algorithms for #P-complete problems) utilize the following random walk algorithm for almost uniform sampling from a state space S of cardinality N : run a symmetric ergodic Markov chain P on S for long enough to obtain a random state from within total variation distance of the uniform distribution over S . The running time of this algorithm, the so-called mixing time of P , is O (δ −1 (log N +log −1 )), where δ is the spectral gap of P . We present a natural quantum version of this algorithm based on repeated measurements of the quantum walk U t = e −i Pt . We show that it samples almost uniformly from S with logarithmic dependence on −1 just as the classical walk P does; previously, no such quantum walk algorithm was known. We then outline a framework for analysing its running time and formulate two plausible conjectures which together would imply that it runs in time O (δ −1/2 log N log −1 ) when P is the standard transition matrix of a constant-degree graph. We prove each conjecture for a subclass of Cayley graphs.
Many classical randomized algorithms (e.g. approximation algorithms for #P-complete problems) utilize the following random walk algorithm for almost uniform sampling from a state space S of cardinality N : run a symmetric ergodic Markov chain P on S for long enough to obtain a random state from within total variation distance of the uniform distribution over S . The running time of this algorithm, the so-called mixing time of P , is O (δ −1 (log N +log −1 )), where δ is the spectral gap of P . We present a natural quantum version of this algorithm based on repeated measurements of the quantum walk U t = e −i Pt . We show that it samples almost uniformly from S with logarithmic dependence on −1 just as the classical walk P does; previously, no such quantum walk algorithm was known. We then outline a framework for analysing its running time and formulate two plausible conjectures which together would imply that it runs in time O (δ −1/2 log N log −1 ) when P is the standard transition matrix of a constant-degree graph. We prove each conjecture for a subclass of Cayley graphs.
Abstract Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated by walks. Quantum walks show promise for faster mixing times than classical methods but lack universal proof, especially in finite group settings. Here, we investigate the continuous-time quantum walks on Cayley graphs of the dihedral group D 2n for odd n, generated by the smallest inverse closed symmetric subset. We present a significant finding that, in contrast to the classical mixing time on these Cayley graphs, which typically takes at least order Ω ( n 2 log ( 1 / 2 ϵ ) ) , the continuous-time quantum walk mixing time on D 2n is of order O ( n ( log n ) 5 log ( 1 / ϵ ) ) , achieving a quadratic improvement over the classical case. Our paper advances the general understanding of quantum walk mixing on Cayley graphs, highlighting the improved mixing time achieved by continuous-time quantum walks on D 2n . This work has potential applications in algorithms for a class of sampling problems based on non-abelian groups.
Random walk algorithms, including sampling and approximations, have played a significant role in statistical physics and theoretical computer science. Mixing through walks is the process for a Markov chain to approximate a stationary distribution for a group. Quantum walks have shown potential advantages in mixing time over the classical case but lack general proof in the finite group case. Here, we investigate the continuous-time quantum walks on Cayley graphs of the dihedral group D2n for odd n, generated by the smallest inverse closed symmetric subset. We present a significant finding that, in contrast to the classical mixing time on these Cayley graphs, which typically takes at least order Ω(n2log(1/2ε)), the continuous-time quantum walk mixing time on D2n is of order Ω($n(log n)$5log(1/ε)), achieving a quadratic improvement over the classical case. Our paper advances the general understanding of quantum walk mixing on Cayley graphs, highlighting the improved mixing time achieved by continuous-time quantum walks on D2n. This work has potential applications in algorithms for a class of sampling problems based on non-abelian groups.
The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that, given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. There also exists a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. Recently, we provided an upper bound on the quantum mixing time for Erd\ifmmode \mbox{\H{o}}\else \H{o}\fi{}s-R\'enyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we also extend and expand upon our findings therein. Namely, we provide an intuitive understanding of the state-of-the-art random matrix theory tools used to derive our results. In particular, for our analysis we require information about macroscopic, mesoscopic, and microscopic statistics of eigenvalues of random matrices which we highlight here. Furthermore, we provide numerical simulations that corroborate our analytical findings and extend this notion of mixing from simple graphs to any ergodic, reversible, Markov chain.
Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider {\em quantum\/} walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''''. In particular, we show that after~t time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval~[−t/2,t/2]. This implies that the same walk defined on the circle mixes in {\em linear\/} time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.
Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider_quantum_ walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''. We show that after t time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval [-t/sqrt(2),t/sqrt(2)]. This implies that the same walk defined on the circle mixes in_linear_ time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.
We present a quantum algorithm that generates a uniform superposition over the spanning trees of a graph -- also known as a q-sample -- using only a sub-linear number of queries to the graph. This goes beyond previous work that focuses on generating classical samples, including a recent quantum algorithm by Apers, Gao, Ji, and Liu [ICALP 2025]. For an n-vertex, m-edge graph, our algorithm admits a pre-processing step of O~(mn+m1−δ), after which each q-sample can be generated in O~(n1+2δ) for any δ. Alternatively, k independent q-samples can be generated at a total cost of O~(kmn). We also prove a matching lower bound up to logarithmic factors, showing that our algorithm is essentially optimal. Further tradeoffs are given in the paper. In comparison, the optimal classical algorithms (Anari, Liu and Vuong [FOCS 2022]) need an O~(m) pre-processing step, after which each sample costs O~(n) operations. These results yield faster quantum walk-based algorithms for counting spanning trees or finding a marked one.
Our result is obtained via quantum walk sampling over a sequence of slowly-changing Markov chains. Each chain is an isotropized up-down walk that mixes rapidly to the spanning tree distribution of the input graph. A key ingredient is an amortized data structure supporting fast implementation of the associated quantum walk operators throughout the sequence. This data structure maintains access to a spectral sparsifier and a leverage score sampler that evolve with the underlying graph.