Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end

• URL: http://arxiv.org/abs/2212.01513v1
• Date: Sat, 3 Dec 2022 02:45:23 GMT
• Title: Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end
• Authors: Alexander M. Dalzell, Nicola Pancotti, Earl T. Campbell, Fernando G. S. L. Brand\~ao
• Abstract summary: We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
• Score: 114.3957763744719
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), Ising spin glasses ($p$-spin model), and $k$-local constraint satisfaction problems ($k$-CSP). We show that either (a) the algorithm finds the optimal solution in time $O^*(2^{(0.5-c)n})$ for an $n$-independent constant $c$, a $2^{cn}$ advantage over Grover's algorithm; or (b) there are sufficiently many low-cost solutions such that classical random guessing produces a $(1-\eta)$ approximation to the optimal cost value in sub-exponential time for arbitrarily small choice of $\eta$. Additionally, we show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case. The algorithm and its analysis is largely inspired by Hastings' short-path algorithm [$\textit{Quantum}$ $\textbf{2}$ (2018) 78].

Related papers

• Quantum Algorithms for Non-smooth Non-convex Optimization [30.576546266390714]
  This paper considers the problem for finding the $(,epsilon)$-Goldstein stationary point of Lipschitz continuous objective. We construct a zeroth-order quantum estimator for the surrogate oracle function.
  arXiv  Detail & Related papers  (2024-10-21T16:52:26Z)
• Near-Optimal Quantum Algorithm for Minimizing the Maximal Loss [16.91814406135565]
  We conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of $N$ convex, Lipschitz functions. We prove that quantum algorithms must take $tildeOmega(sqrtNepsilon-2/3)$ queries to a first order quantum oracle.
  arXiv  Detail & Related papers  (2024-02-20T06:23:36Z)
• A Scalable Algorithm for Individually Fair K-means Clustering [77.93955971520549]
  We present a scalable algorithm for the individually fair ($p$, $k$)-clustering problem introduced by Jung et al. and Mahabadi et al. A clustering is then called individually fair if it has centers within distance $delta(x)$ of $x$ for each $xin P$. We show empirically that not only is our algorithm much faster than prior work, but it also produces lower-cost solutions.
  arXiv  Detail & Related papers  (2024-02-09T19:01:48Z)
• An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems [76.2042837251496]
  We introduce two oblivious mirror descent algorithms based on a complementary composite setting. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. We show how to extend our framework to scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.
  arXiv  Detail & Related papers  (2023-06-30T08:34:29Z)
• Deterministic Nonsmooth Nonconvex Optimization [94.01526844386977]
  We show that randomization is necessary to obtain a dimension-free dimension-free algorithm. Our algorithm yields the first deterministic dimension-free algorithm for optimizing ReLU networks.
  arXiv  Detail & Related papers  (2023-02-16T13:57:19Z)
• Near-Optimal Lower Bounds For Convex Optimization For All Orders of Smoothness [26.71898403195793]
  We study the complexity of optimizing highly smooth convex functions. For a positive integer $p$, we want to find an $epsilon$-approximate minimum of a convex function $f$. We prove a new lower bound that matches this bound (up to log factors), and holds not only for randomized algorithms, but also for quantum algorithms.
  arXiv  Detail & Related papers  (2021-12-02T10:51:43Z)
• No quantum speedup over gradient descent for non-smooth convex optimization [22.16973542453584]
  Black-box access to a (not necessarily smooth) function $f:mathbbRn to mathbbR$ and its (sub)gradient. Our goal is to find an $epsilon$-approximate minimum of $f$ starting from a point that is distance at most $R$ from the true minimum. We show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using $O(GR/epsilon)$ quantum queries.
  arXiv  Detail & Related papers  (2020-10-05T06:32:47Z)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)
• Maximizing Determinants under Matroid Constraints [69.25768526213689]
  We study the problem of finding a basis $S$ of $M$ such that $det(sum_i in Sv_i v_i v_itop)$ is maximized. This problem appears in a diverse set of areas such as experimental design, fair allocation of goods, network design, and machine learning.
  arXiv  Detail & Related papers  (2020-04-16T19:16:38Z)

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.