Quantum Approximate Counting with Nonadaptive Grover Iterations

• URL: http://arxiv.org/abs/2010.04370v1
• Date: Fri, 9 Oct 2020 04:48:14 GMT
• Title: Quantum Approximate Counting with Nonadaptive Grover Iterations
• Authors: Ramgopal Venkateswaran and Ryan O'Donnell
• Abstract summary: In the quantum setting, Approximate Counting can be done with $Oleft(sqrtN/epsilon, sqrtN/K/epsilonright)$ queries. We show that algorithms using only nonadaptive Grover iterations can achieve $Oleft(sqrtN/epsilonright)$ query complexity, which is tight.
• Score: 1.14219428942199
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Approximate Counting refers to the problem where we are given query access to a function $f : [N] \to \{0,1\}$, and we wish to estimate $K = #\{x : f(x) = 1\}$ to within a factor of $1+\epsilon$ (with high probability), while minimizing the number of queries. In the quantum setting, Approximate Counting can be done with $O\left(\min\left(\sqrt{N/\epsilon}, \sqrt{N/K}/\epsilon\right)\right)$ queries. It has recently been shown that this can be achieved by a simple algorithm that only uses "Grover iterations"; however the algorithm performs these iterations adaptively. Motivated by concerns of computational simplicity, we consider algorithms that use Grover iterations with limited adaptivity. We show that algorithms using only nonadaptive Grover iterations can achieve $O\left(\sqrt{N/\epsilon}\right)$ query complexity, which is tight.

Related papers

• 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)
• The Computational Complexity of Finding Stationary Points in Non-Convex Optimization [53.86485757442486]
  Finding approximate stationary points, i.e., points where the gradient is approximately zero, of non-order but smooth objective functions is a computational problem. We show that finding approximate KKT points in constrained optimization is reducible to finding approximate stationary points in unconstrained optimization but the converse is impossible.
  arXiv  Detail & Related papers  (2023-10-13T14:52:46Z)
• Robust Approximation Algorithms for Non-monotone $k$-Submodular Maximization under a Knapsack Constraint [0.0]
  Two deterministic approximation algorithms are presented for the problem of non-monotone $k$-submodular complexity under a knapsack constraint. Our algorithms provide constant approximation ratios within only $O(nk)$ query complexity for the non-monotone objective.
  arXiv  Detail & Related papers  (2023-09-21T12:42:52Z)
• Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise [50.64137465792738]
  We show that any efficient SQ algorithm for the problem requires sample complexity at least $Omega(d1/2/(maxp, epsilon)2)$. Our lower bound suggests that this quadratic dependence on $1/epsilon$ is inherent for efficient algorithms.
  arXiv  Detail & Related papers  (2023-07-13T18:59:28Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  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.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• Sketching Algorithms and Lower Bounds for Ridge Regression [65.0720777731368]
  We give a sketching-based iterative algorithm that computes $1+varepsilon$ approximate solutions for the ridge regression problem. We also show that this algorithm can be used to give faster algorithms for kernel ridge regression.
  arXiv  Detail & Related papers  (2022-04-13T22:18:47Z)
• Time and Query Optimal Quantum Algorithms Based on Decision Trees [2.492300648514128]
  We show that a quantum algorithm can be implemented in time $tilde O(sqrtGT)$. Our algorithm is based on non-binary span programs and their efficient implementation.
  arXiv  Detail & Related papers  (2021-05-18T06:51:11Z)
• Grover's search algorithm for $n$ qubits with optimal number of iterations [0.0]
  Grover's search algorithm depends on the number of iterations of the composite operation of the oracle followed by Grover's diffusion operation. General scheme for the construction of $n$-qubit Grover's search algorithm with $1 leq M leq N$ target states is presented.
  arXiv  Detail & Related papers  (2020-11-08T18:46:50Z)
• Quick Streaming Algorithms for Maximization of Monotone Submodular Functions in Linear Time [16.346069386394703]
  We consider the problem of monotone, submodular over a ground set of size $n$ subject to cardinality constraint $k$. We introduce the first deterministic algorithms with linear time complexity; these algorithms are streaming algorithms. Our single-pass algorithm obtains a constant ratio in $lceil n / c rceil + c$ oracle queries, for any $c ge 1$.
  arXiv  Detail & Related papers  (2020-09-10T16:35:54Z)
• Practical and Parallelizable Algorithms for Non-Monotone Submodular Maximization with Size Constraint [20.104148319012854]
  We present and parallelizable for a submodular function, not necessarily a monotone, with respect to a size constraint. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and nearly optimal complexity query to $0.193 - varepsilon$.
  arXiv  Detail & Related papers  (2020-09-03T22:43:55Z)
• 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)

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.