Abstract. 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 [Formula: see text] nodes and [Formula: see text] edges, outputs a classical description of an [Formula: see text]-spectral sparsifier in sublinear time [Formula: see text]. This contrasts with the optimal classical complexity [Formula: see text]. 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 [Formula: see text]-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.
Abstract. 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 [Formula: see text] nodes and [Formula: see text] edges, outputs a classical description of an [Formula: see text]-spectral sparsifier in sublinear time [Formula: see text]. This contrasts with the optimal classical complexity [Formula: see text]. 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 [Formula: see text]-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.
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 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 ε-spectral sparsifier in sublinear time ~O(√{mn/ε). 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 shortest paths quantum algorithm by Dürr et al. (ICALP'04) and an efficient k-wise independent hash construction by Christiani, Pagh and Thorup (STOC'15). 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.
Triangles capture higher-order structures in graphs and are fundamental to applications such as clustering and network analysis. To enable efficient use of such structures at scale, we study the problem of triangle cut sparsification, which aims to reduce the graph size while approximately preserving triangle counts across every cut. We investigate quantum algorithms for this problem, using triangle listing as our main technical ingredient. In particular, we present a quantum algorithm for triangle listing that, for a graph with n vertices, m edges, and t triangles, runs in time Tq-list=O(min(n5/4t7/12+n7/6t7/9,m+m3/4t1/2,n3/2t1/2)), improving upon the best known classical bounds over a broad range of parameters. Our algorithm is based on a heavy-light vertex partition and an extension of triangle detection via quantum walks and Grover search. Leveraging this result, we design a quantum algorithm for constructing ε-triangle cut sparsifiers of size O(n/ε2) in time O(Tq-list+mn/ε). Finally, we demonstrate applications to clustering algorithms based on triangle-related measures and prove a lower bound of Ω(n/ε2) on the size of any ε-triangle cut sparsifiers.
Graph sparsification serves as a foundation for many algorithms, such as approximation algorithms for graph cuts and Laplacian system solvers. As its natural generalization, hypergraph sparsification has recently gained increasing attention, with broad applications in graph machine learning and other areas. In this work, we propose the first quantum algorithm for hypergraph sparsification, addressing an open problem proposed by Apers and de Wolf (FOCS'20). For a weighted hypergraph with n vertices, m hyperedges, and rank r, our algorithm outputs a near-linear size ε-spectral sparsifier in time O(rmn/ε). This algorithm matches the quantum lower bound for constant r and demonstrates quantum speedup when compared with the state-of-the-art O(mr)-time classical algorithm. As applications, our algorithm implies quantum speedups for computing hypergraph cut sparsifiers, approximating hypergraph mincuts and hypergraph s-t mincuts.
In this paper, we continue a line of research initiated by Basu, Brakensiek, and Putterman [2026] studying the sparsifiability of Hamiltonians. We focus particularly on the sparsifiability of the widely-studied Quantum Cut (QC) Hamiltonians. Our main result is that in an n-qubit system, any n-qubit QC Hamiltonian can be sparsified to O(n/ε2) many terms while preserving the energy of every state up to a factor of 1±ε. Our result can be interpreted as giving an importance sampling scheme for the edges of an arbitrary graph G such that the \emph{Kikuchi} graph at level ℓ of the sampled graph is a spectral approximation to the Kikuchi graph of G. Importantly, the \emph{same} sampling scheme works simultaneously for all ℓ.
The natural approach of leverage score sampling, analyzed via matrix concentration inequalities, yields a polynomially worse bound in our setting because the underlying matrices have dimension ∼2n. Instead, our approach relies on decomposing the action of these matrices into invariant subspaces. Then, by using an operator-valued inequality of Alon and Kozma [Ann. Henri Poincaré, 2020], itself building on an \emph{octopus inequality} of Caputo, Liggett, and Richthammer [J. AMS, 2010], we extend our sparsification technique to all expander graphs. We then invoke expander decomposition to extend our sparsifier to all graphs.
Abstract. We describe a quantum algorithm based on an interior point method for solving a linear program with [Formula: see text] inequality constraints on [Formula: see text] variables. The algorithm explicitly returns a feasible solution that is [Formula: see text]-close to optimal and runs in time [Formula: see text], which is sublinear for tall linear programs (i.e., [Formula: see text]). Our algorithm speeds up the Newton step in the state-of-the-art interior point method of Lee and Sidford [ Solving Linear Programs with Sqrt(rank) Linear System Solves, 2019]. This requires us to efficiently approximate the Hessian and gradient of the barrier function, and these are our main contributions. To approximate the Hessian, we describe a quantum algorithm for the spectral approximation of [Formula: see text] for a tall matrix [Formula: see text]. The algorithm uses leverage score sampling in combination with Grover search and returns a [Formula: see text]-approximation by making [Formula: see text] row queries to [Formula: see text]. This generalizes an earlier quantum speedup for graph sparsification by Apers and de Wolf [ SIAM J. Comput., 51 (2022), pp. 1703–1742]. To approximate the gradient, we use a recent quantum algorithm for multivariate mean estimation by Cornelissen, Hamoudi, and Jerbi [ Near-optimal quantum algorithms for multivariate mean estimation, 2022]. While a naive implementation introduces a dependence on the condition number of the Hessian, we avoid this by preconditioning our random variable using our quantum algorithm for spectral approximation.
We present a quantum algorithm for sampling random spanning trees from a weighted graph in Õ(√{mn}) time, where n and m denote the number of vertices and edges, respectively. Our algorithm has sublinear runtime for dense graphs and achieves a quantum speedup over the best-known classical algorithm, which runs in Õ(m) time. The approach carefully combines, on one hand, a classical method based on "large-step" random walks for reduced mixing time and, on the other hand, quantum algorithmic techniques, including quantum graph sparsification and a sampling-without-replacement variant of Hamoudi’s multiple-state preparation. We also establish a matching lower bound, proving the optimality of our algorithm up to polylogarithmic factors. These results highlight the potential of quantum computing in accelerating fundamental graph sampling problems.