53 research outputs found

### Accelerating Consensus by Spectral Clustering and Polynomial Filters

It is known that polynomial filtering can accelerate the convergence towards
average consensus on an undirected network. In this paper the gain of a
second-order filtering is investigated. A set of graphs is determined for which
consensus can be attained in finite time, and a preconditioner is proposed to
adapt the undirected weights of any given graph to achieve fastest convergence
with the polynomial filter. The corresponding cost function differs from the
traditional spectral gap, as it favors grouping the eigenvalues in two
clusters. A possible loss of robustness of the polynomial filter is also
highlighted

### 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

### Expansion Testing using Quantum Fast-Forwarding and Seed Sets

Expansion testing aims to decide whether an $n$-node graph has expansion at
least $\Phi$, or is far from any such graph. We propose a quantum expansion
tester with complexity $\widetilde{O}(n^{1/3}\Phi^{-1})$. This accelerates the
$\widetilde{O}(n^{1/2}\Phi^{-2})$ classical tester by Goldreich and Ron
[Algorithmica '02], and combines the $\widetilde{O}(n^{1/3}\Phi^{-2})$ and
$\widetilde{O}(n^{1/2}\Phi^{-1})$ quantum speedups by Ambainis, Childs and Liu
[RANDOM '11] and Apers and Sarlette [QIC '19], respectively. The latter
approach builds on a quantum fast-forwarding scheme, which we improve upon by
initially growing a seed set in the graph. To grow this seed set we use a
so-called evolving set process from the graph clustering literature, which
allows to grow an appropriately local seed set.Comment: v3: final version to appear in Quantu

### 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 Ă(m1/3ÎŽâ1/3), with m the number of edges and ÎŽ 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 Ă(2n/3), surpassing the Î©(2n/2) barrier set by index erasure

### Holey graphs: very large Betti numbers are testable

We show that the graph property of having a (very) large $k$-th Betti number
$\beta_k$ for constant $k$ is testable with a constant number of queries in the
dense graph model. More specifically, we consider a clique complex defined by
an underlying graph and prove that for any $\varepsilon>0$, there exists
$\delta(\varepsilon,k)>0$ such that testing whether $\beta_k \geq (1-\delta)
d_k$ for $\delta \leq \delta(\varepsilon,k)$ reduces to tolerantly testing
$(k+2)$-clique-freeness, which is known to be testable. This complements a
result by Elek (2010) showing that Betti numbers are testable in the
bounded-degree model. Our result combines the Euler characteristic, matroid
theory and the graph removal lemma.Comment: 10 pages, 0 figure

### 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 Speedup for Graph Sparsification, Cut Approximation and Laplacian Solving

Graph sparsification underlies a large number of algorithms, ranging from
approximation algorithms for cut problems to solvers for linear systems in the
graph Laplacian. In its strongest form, "spectral sparsification" reduces the
number of edges to near-linear in the number of nodes, while approximately
preserving the cut and spectral structure of the graph. In this work we
demonstrate a polynomial quantum speedup for spectral sparsification and many
of its applications. In particular, we give a quantum algorithm that, given a
weighted graph with $n$ nodes and $m$ edges, outputs a classical description of
an $\epsilon$-spectral sparsifier in sublinear time
$\tilde{O}(\sqrt{mn}/\epsilon)$. This contrasts with the optimal classical
complexity $\tilde{O}(m)$. We also prove that our quantum algorithm is optimal
up to polylog-factors. The algorithm builds on a string of existing results on
sparsification, graph spanners, quantum algorithms for shortest paths, and
efficient constructions for $k$-wise independent random strings. Our algorithm
implies a quantum speedup for solving Laplacian systems and for approximating a
range of cut problems such as min cut and sparsest cut.Comment: v2: several small improvements to the text. An extended abstract will
appear in FOCS'20; v3: corrected a minor mistake in Appendix

### 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=PâPT its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state |vâ© and integer t returns a quantum state Ï”-close to the state Dt|vâ©/â„Dt|vâ©â„. The algorithm uses O(â„Dt|vâ©â„â1tlog(Ï”â„Dt|vâ©â„)â1â) expected quantum walk steps and O(â„Dt|vâ©â„â1) expected reflections around |vâ©. 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

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