Faster quantum mixing of Markov chains in non-regular graph with fewer qubits

• URL: http://arxiv.org/abs/2205.06099v1
• Date: Thu, 12 May 2022 14:08:08 GMT
• Title: Faster quantum mixing of Markov chains in non-regular graph with fewer qubits
• Authors: Xinyin Li, Yun Shang
• Abstract summary: In quantum cases, qsampling from Markov chains can be constructed as preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution. In this paper, a new qsampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks. In non-regular graphs, the invocation of the quantum fast-forward algorithm accelerates existing state-of-the-art qsampling algorithms for both discrete-time and continuous-time cases.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Sampling from the stationary distribution is one of the fundamental tasks of Markov chain-based algorithms and has important applications in machine learning, combinatorial optimization and network science. For the quantum case, qsampling from Markov chains can be constructed as preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. In this paper, a new qsampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks and works without any limit compared with existing results. In detail, we build a qsampling algorithm that not only accelerates non-regular graphs but also keeps the speed-up of existing quantum algorithms for regular graphs. In non-regular graphs, the invocation of the quantum fast-forward algorithm accelerates existing state-of-the-art qsampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms we reduce log n, where n is the number of graph vertices. In regular graphs, our result matches other quantum algorithms, and our reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared to the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm is the first algorithm to achieve complete quadratic acceleration (without log factor) over the classical case without any limit. To enlarge success probability amplitude amplification is introduced. We construct a new reflection on stationary state with fewer ancilla qubits and think it may have independent application.

Related papers

• Comparison among Classical, Probabilistic and Quantum Algorithms for Hamiltonian Cycle problem [0.0]
  Hamiltonian cycle problem (HCP) consists of having a graph G with n nodes and m edges and finding the path that connects each node exactly once. In this paper we compare some algorithms to solve aHCP, using different models of computations and especially the probabilistic and quantum ones.
  arXiv  Detail & Related papers  (2023-11-18T02:36:10Z)
• Stochastic Quantum Sampling for Non-Logconcave Distributions and Estimating Partition Functions [13.16814860487575]
  We present quantum algorithms for sampling from nonlogconcave probability distributions. $f$ can be written as a finite sum $f(x):= frac1Nsum_k=1N f_k(x)$.
  arXiv  Detail & Related papers  (2023-10-17T17:55:32Z)
• 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 Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• 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)
• Quadratic speedup for spatial search by continuous-time quantum walk [0.0]
  Continuous-time quantum walks provide a framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. We present a new continuous-time quantum walk search algorithm that can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks.
  arXiv  Detail & Related papers  (2021-12-23T17:57:49Z)
• Correlation Clustering in Constant Many Parallel Rounds [42.10280805559555]
  Correlation clustering is a central topic in unsupervised learning, with many applications in ML and data mining. In this work we propose a massively parallel computation (MPC) algorithm for this problem that is considerably faster than prior work. Our algorithm uses machines with memory sublinear in the number of nodes in the graph and returns a constant approximation while running only for a constant number of rounds.
  arXiv  Detail & Related papers  (2021-06-15T21:45:45Z)
• Quantum algorithm for Feynman loop integrals [0.0]
  We present a novel benchmark application of a quantum algorithm to Feynman loop integrals. The two on-shell states of a Feynman propagator are identified with the two states of a qubit. A quantum algorithm is used to unfold the causal singular configurations of multiloop Feynman diagrams.
  arXiv  Detail & Related papers  (2021-05-18T17:41:56Z)
• Quantum algorithms for learning a hidden graph and beyond [0.05076419064097732]
  We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models. We additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al. for a "gapped" version of the group testing problem.
  arXiv  Detail & Related papers  (2020-11-17T13:12:43Z)
• Single-Timescale Stochastic Nonconvex-Concave Optimization for Smooth Nonlinear TD Learning [145.54544979467872]
  We propose two single-timescale single-loop algorithms that require only one data point each step. Our results are expressed in a form of simultaneous primal and dual side convergence.
  arXiv  Detail & Related papers  (2020-08-23T20:36:49Z)
• Learning to Accelerate Heuristic Searching for Large-Scale Maximum Weighted b-Matching Problems in Online Advertising [51.97494906131859]
  Bipartite b-matching is fundamental in algorithm design, and has been widely applied into economic markets, labor markets, etc. Existing exact and approximate algorithms usually fail in such settings due to either requiring intolerable running time or too much computation resource. We propose textttNeuSearcher which leverages the knowledge learned from previously instances to solve new problem instances.
  arXiv  Detail & Related papers  (2020-05-09T02:48:23Z)

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.