Quantum Algorithm for Approximating Maximum Independent Sets

• URL: http://arxiv.org/abs/2005.13089v2
• Date: Fri, 26 Feb 2021 05:52:15 GMT
• Title: Quantum Algorithm for Approximating Maximum Independent Sets
• Authors: Hongye Yu, Frank Wilczek, Biao Wu
• Abstract summary: We present a quantum algorithm for approximating maximum independent sets of a graph based on quantum non-Abelian adiabatic mixing. For sparse and dense graphs, our quantum algorithm on average can find an independent set of size very close to $alpha(G)$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a quantum algorithm for approximating maximum independent sets of a graph based on quantum non-Abelian adiabatic mixing in the sub-Hilbert space of degenerate ground states, which generates quantum annealing in a secondary Hamiltonian. For both sparse and dense graphs, our quantum algorithm on average can find an independent set of size very close to $\alpha(G)$, which is the size of the maximum independent set of a given graph $G$. Numerical results indicate that an $O(n^2)$ time complexity quantum algorithm is sufficient for finding an independent set of size $(1-\epsilon)\alpha(G)$. The best classical approximation algorithm can produce in polynomial time an independent set of size about half of $\alpha(G)$.

Related papers

• An Analysis of Quantum Annealing Algorithms for Solving the Maximum Clique Problem [49.1574468325115]
  We analyse the ability of quantum D-Wave annealers to find the maximum clique on a graph, expressed as a QUBO problem. We propose a decomposition algorithm for the complementary maximum independent set problem, and a graph generation algorithm to control the number of nodes, the number of cliques, the density, the connectivity indices and the ratio of the solution size to the number of other nodes.
  arXiv  Detail & Related papers  (2024-06-11T04:40:05Z)
• Quantum algorithms for Hopcroft's problem [45.45456673484445]
  We study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n4/3)$ time. Our results are two different quantum algorithms with time complexity $widetilde O(n5/6)$.
  arXiv  Detail & Related papers  (2024-05-02T10:29:06Z)
• Quantum Hamiltonian Algorithms for Maximum Independent Sets [6.772902928686719]
  An alternative algorithm, short named the PK algorithm, reveals that the independent sets diffuse over a media graph governed by a non-abelian gauge matrix of an emergent PXP model. Although the two are mathematically equivalent, the PK algorithm exhibits more efficient and resource-saving performance.
  arXiv  Detail & Related papers  (2023-10-23T04:00:34Z)
• Do you know what q-means? [50.045011844765185]
  Clustering is one of the most important tools for analysis of large datasets. We present an improved version of the "$q$-means" algorithm for clustering. We also present a "dequantized" algorithm for $varepsilon which runs in $Obig(frack2varepsilon2(sqrtkd + log(Nd))big.
  arXiv  Detail & Related papers  (2023-08-18T17:52:12Z)
• Private estimation algorithms for stochastic block models and mixture models [63.07482515700984]
  General tools for designing efficient private estimation algorithms. First efficient $(epsilon, delta)$-differentially private algorithm for both weak recovery and exact recovery.
  arXiv  Detail & Related papers  (2023-01-11T09:12:28Z)
• Quantum Clustering with k-Means: a Hybrid Approach [117.4705494502186]
  We design, implement, and evaluate three hybrid quantum k-Means algorithms. We exploit quantum phenomena to speed up the computation of distances. We show that our hybrid quantum k-Means algorithms can be more efficient than the classical version.
  arXiv  Detail & Related papers  (2022-12-13T16:04:16Z)
• Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants [8.453228628258778]
  We develop quantum algorithms for sampling logconcave distributions and for estimating their normalizing constants. We exploit quantum analogs of the Monte Carlo method and quantum walks. We also prove a $1/epsilon1-o(1)$ quantum lower bound for estimating normalizing constants.
  arXiv  Detail & Related papers  (2022-10-12T19:10:43Z)
• Quantum Approximate Counting for Markov Chains and Application to Collision Counting [0.0]
  We show how to generalize the quantum approximate counting technique developed by Brassard, Hoyer and Tapp [ICALP 1998] to estimating the number of marked states of a Markov chain. This makes it possible to construct quantum approximate counting algorithms from quantum search algorithms based on the powerful "quantum walk based search" framework established by Magniez, Nayak, Roland and Santha.
  arXiv  Detail & Related papers  (2022-04-06T03:04:42Z)
• Optimal Low-Degree Hardness of Maximum Independent Set [93.59919600451487]
  We study the algorithmic task of finding a large independent set in a sparse ErdHos-R'enyi random graph. We show that the class of low-degree algorithms can find independent sets of half-optimal size but no larger.
  arXiv  Detail & Related papers  (2020-10-13T17:26:09Z)
• Quasi-polynomial time algorithms for free quantum games in bounded dimension [11.56707165033]
  We give a semidefinite program of size $exp(mathcalObig(T12(log2(AT)+log(Q)log(AT))/epsilon2big)) to compute additive $epsilon$-approximations on the values of two-player free games. We make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems with linear constraints.
  arXiv  Detail & Related papers  (2020-05-18T16:55:08Z)
• The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case [6.810856082577402]
  The quantum circuit has p applications of a unitary operator that respects the locality of the graph. We focus on finding big independent sets in random graphs with dn/2 edges keeping d fixed and n large.
  arXiv  Detail & Related papers  (2020-04-20T00:48:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.