Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction

• URL: http://arxiv.org/abs/2504.06376v1
• Date: Tue, 08 Apr 2025 18:53:16 GMT
• Title: Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction
• Authors: Shion Fukuzawa, Michael T. Goodrich, Sandy Irani,
• Abstract summary: We introduce a quantum algorithm design paradigm called combine and conquer.<n>It is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel.
• Score: 0.984963525011891
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in $\tilde{O}(\sqrt{nh})$ time, w.h.p., where $h$ is the size of the output.

Related papers

• Quantum multi-row iteration algorithm for linear systems with non-square coefficient matrices [7.174256268278207]
  We propose a quantum algorithm inspired by the classical multi-row iteration method. Our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems.
  arXiv  Detail & Related papers  (2024-09-06T03:32:02Z)
• Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
  We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$. We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
  arXiv  Detail & Related papers  (2024-01-17T18:59:38Z)
• A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games [102.46640028830441]
  We introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $mathcalO(d/epsilon)$ to $epsilon$-Nash equilibria. This quadratic speed-up sets a new benchmark for computing $epsilon$-Nash equilibria in quantum zero-sum games.
  arXiv  Detail & Related papers  (2023-11-17T20:38:38Z)
• Making the cut: two methods for breaking down a quantum algorithm [0.0]
  It remains a major challenge to find quantum algorithms that may reach computational advantage in the present era of noisy, small-scale quantum hardware. We identify and characterize two methods of breaking down'' quantum algorithms into rounds of lower (query) depth. We show that for the first problem parallelization offers the best performance, while for the second is the better choice.
  arXiv  Detail & Related papers  (2023-05-17T18:00:06Z)
• Logarithmic-Regret Quantum Learning Algorithms for Zero-Sum Games [10.79781442303645]
  We propose the first online quantum algorithm for solving zero-sum games. Our algorithm generates classical outputs with succinct descriptions. At the heart of our algorithm is a fast quantum multi-sampling procedure for the Gibbs sampling problem.
  arXiv  Detail & Related papers  (2023-04-27T14:02:54Z)
• 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)
• Entanglement and coherence in Bernstein-Vazirani algorithm [58.720142291102135]
  Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. We analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state.
  arXiv  Detail & Related papers  (2022-05-26T20:32:36Z)
• 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)
• On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids [63.75363908696257]
  The lackadaisical quantum walk is an algorithm developed to search graph structures whose vertices have a self-loop of weight $l$. This paper addresses several issues related to applying the lackadaisical quantum walk to search for multiple solutions on grids successfully.
  arXiv  Detail & Related papers  (2021-06-11T09:43:09Z)
• Quantum Algorithm for Fidelity Estimation [8.270684567157987]
  For two unknown mixed quantum states $rho$ and $sigma$ in an $N$-dimensional space, computing their fidelity $F(rho,sigma)$ is a basic problem. We propose a quantum algorithm that solves this problem in $namepoly(log (N), r, 1/varepsilon)$ time.
  arXiv  Detail & Related papers  (2021-03-16T13:57:01Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

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.