Simulation of quantum walks and fast mixing with classical processes

We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains (LMCs), that is, classical Markov chains with added memory. We show that LMCs can simulate the mixing behavior of any quantum walk, under a commonly satisfied invariance condition. This allows us to answer an open question on how the graph topology ultimately bounds a quantum walk's mixing performance, and that of any stochastic local evolution. The results highlight that speedups in mixing and transport phenomena are not necessarily diagnostic of quantum effects, although superdiffusive spreading is more prominent with quantum walks. The general simulating LMC construction may lead to large memory, yet we show that for the main graphs under study (i.e., lattices) this memory can be brought down to the same size employed in the quantum walks proposed in the literature

Quantum walk sampling by growing seed sets

This work describes a new algorithm for creating a superposition over the edge set of a graph, encoding a quantum sample of the random walk stationary distribution. The algorithm requires a number of quantum walk steps scaling as Õ(m1/3δ−1/3), with m the number of edges and δ the random walk spectral gap. This improves on existing strategies by initially growing a classical seed set in the graph, from which a quantum walk is then run. The algorithm leads to a number of improvements: (i) it provides a new bound on the setup cost of quantum walk search algorithms, (ii) it yields a new algorithm for st-connectivity, and (iii) it allows to create a superposition over the isomorphisms of an n-node graph in time Õ(2n/3), surpassing the Ω(2n/2) barrier set by index erasure

A Unified Framework of Quantum Walk Search

Many quantum algorithms critically rely on quantum walk search, or the use of quantum walks to speed up search problems on graphs. However, the main results on quantum walk search are scattered over different, incomparable frameworks, such as the hitting time framework, the MNRS framework, and the electric network framework. As a consequence, a number of pieces are currently missing. For example, recent work by Ambainis et al. (STOC\u2720) shows how quantum walks starting from the stationary distribution can always find elements quadratically faster. In contrast, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements. We present a new quantum walk search framework that unifies and strengthens these frameworks, leading to a number of new results. For example, the new framework effectively finds marked elements in the electric network setting. The new framework also allows to interpolate between the hitting time framework, minimizing the number of walk steps, and the MNRS framework, minimizing the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. In addition to quantum walks and phase estimation, our new algorithm makes use of quantum fast-forwarding, similar to the recent results by Ambainis et al. This perspective also enables us to derive more general complexity bounds on the quantum walk algorithms, e.g., based on Monte Carlo type bounds of the corresponding classical walk. As a final result, we show how in certain cases we can avoid the use of phase estimation and quantum fast-forwarding, answering an open question of Ambainis et al

Quantum fast-forwarding: Markov chains and graph property testing

We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D=P∘PT its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state |v⟩ and integer t returns a quantum state ϵ-close to the state Dt|v⟩/∥Dt|v⟩∥. The algorithm uses O(∥Dt|v⟩∥−1tlog(ϵ∥Dt|v⟩∥)−1√) expected quantum walk steps and O(∥Dt|v⟩∥−1) expected reflections around |v⟩. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks

A (simple) classical algorithm for estimating Betti numbers

We describe a simple algorithm for estimating the $k$-th normalized Betti number of a simplicial complex over $n$ elements using the path integral Monte Carlo method. For a general simplicial complex, the running time of our algorithm is $n^{O\left(\frac{1}{\sqrt{\gamma}}\log\frac{1}{\varepsilon}\right)}$ with $\gamma$ measuring the spectral gap of the combinatorial Laplacian and $\varepsilon \in (0,1)$ the additive precision. In the case of a clique complex, the running time of our algorithm improves to $\left(n/\lambda_{\max}\right)^{O\left(\frac{1}{\sqrt{\gamma}}\log\frac{1}{\varepsilon}\right)}$ with $\lambda_{\max} \geq k$, where $\lambda_{\max}$ is the maximum eigenvalue of the combinatorial Laplacian. Our algorithm provides a classical benchmark for a line of quantum algorithms for estimating Betti numbers. On clique complexes it matches their running time when, for example, $\gamma \in \Omega(1)$ and $k \in \Omega(n)$.Comment: v2: improved gap dependency by using Chebyshev polynomial

(Quantum) complexity of testing signed graph clusterability

This study examines clusterability testing for a signed graph in the bounded-degree model. Our contributions are two-fold. First, we provide a quantum algorithm with query complexity $\tilde{O}(N^{1/3})$ for testing clusterability, which yields a polynomial speedup over the best classical clusterability tester known [arXiv:2102.07587]. Second, we prove an $\tilde{\Omega}(\sqrt{N})$ classical query lower bound for testing clusterability, which nearly matches the upper bound from [arXiv:2102.07587]. This settles the classical query complexity of clusterability testing, and it shows that our quantum algorithm has an advantage over any classical algorithm