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 \widetil O( \surd mn/\epsilon ). This contrasts with the optimal classical complexity \widetil 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.

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.

We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us to answer an open question on how the graph topology ultimately bounds their 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.

We show that the evolution of two-component particles governed by a two-dimensional spin-orbit lattice Hamiltonian can reveal transitions between topological phases. A kink in the mean width of the particle distribution signals the closing of the band gap, a prerequisite for a quantum phase transition between topological phases. Furthermore, for realistic and experimentally motivated Hamiltonians the density profile in topologically non-trivial phases displays characteristic rings in the vicinity of the origin that are absent in trivial phases. The results are expected to have immediate application to systems of ultracold atoms and photonic lattices.Topological phases have many unusual and potentially useful electronic properties, and have been proposed for fault-tolerant quantum computation and quantum memories [1][2][3][4][5]. In one dimensional systems, all topological states can be classified [6]. In higher dimensions, noninteracting systems can be classified in terms of topological invariants such as Chern numbers [7], and much work has been expended in recent years attempting to extend this classification to interacting systems [8][9][10][11]. The experimental determination of topological invariants in bulk condensed matter systems with time-reversal symmetry is not straightforward, however; topological order would generally be inferred from the existence of edge states [2,12]. In this work, the presence of non-trivial topological order is inferred from particle dynamics.The exceptional control of integrated photonic and ultracold atomic systems makes them ideal testbeds for the production and detection of topological order [13][14][15]. After the first realization of the photonic analog of the quantum Hall effect [16], topological edge modes were observed in both static and driven photonic lattices [17][18][19]. The Hofstadter Hamiltonian for neutral lattice bosons in a synthetic magnetic fields has been experimentally implemented [20,21]; with two spin components, the system is time-reversal symmetric, yielding the neutral analog of the spin-Hall effect [22]. The integer quantization of the lowest-band Chern number was determined in the time-reversal-breaking geometry using transport measurements [23]. The topological Haldane model was realized by placing ultracold fermionic atoms in a periodically modulated optical honeycomb lattice [24], and the Berry curvature was obtained using time-of-flight images of a Floquet lattice [25]. Most recently, a one-dimensional symmetry protected topological phase was realized in an ultracold atomic gas [26].Previous work has shown that particle dynamics can reveal the presence of topological order in systems that break time-reversal symmetry. Wave packets can acquire both anomalous velocities under applied forces [27] and Berry-flux phases under closed trajectories in momentum space [28]. The Berry curvature (whose integral over momentum space yields the Chern number) can be obtained directly from time-of-flight images [29,30]. Discrete-time quantum walks (i.e. dynamic...

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. The breakthrough work by Benczúr and Karger (STOC'96) and Spielman and Teng (STOC'04) showed that sparsification can be done optimally in time near-linear in the number of edges of the original graph.In this work we show that quantum algorithms allow to speed up spectral sparsification, and thereby many of the derived algorithms. Given adjacency-list access to a weighted graph with n nodes and m edges, our algorithm outputs an -spectral sparsifier in time O( √ mn/ ). We prove that this is tight up to polylog-factors. The algorithm builds on a string of existing results, most notably sparsification algorithms by Spielman and Srivastava (STOC'08) and Koutis and Xu (TOPC'16), a spanner construction by Thorup and Zwick (STOC'01), a single-source shortestpaths quantum algorithm by Dürr et al. (ICALP'04) and an efficient k-wise independent hash construction by Christiani, Pagh and Thorup (STOC'15). Combining our sparsification algorithm with existing classical algorithms yields the first quantum speedup, roughly from O(m) to O( √ mn), for approximating the max cut, min cut, min st-cut, sparsest cut and balanced separator of a graph. Combining our algorithm with a classical Laplacian solver, we demonstrate a similar speedup for Laplacian solving, for approximating effective resistances, cover times and eigenvalues of the Laplacian, and for spectral clustering.