Stochastic Quantum Sampling for Non-Logconcave Distributions and Estimating Partition Functions

• URL: http://arxiv.org/abs/2310.11445v1
• Date: Tue, 17 Oct 2023 17:55:32 GMT
• Title: Stochastic Quantum Sampling for Non-Logconcave Distributions and Estimating Partition Functions
• Authors: Guneykan Ozgul, Xiantao Li, Mehrdad Mahdavi, Chunhao Wang
• Abstract summary: We present quantum algorithms for sampling from nonlogconcave probability distributions. $f$ can be written as a finite sum $f(x):= frac1Nsum_k=1N f_k(x)$.
• Score: 13.16814860487575
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present quantum algorithms for sampling from non-logconcave probability distributions in the form of $\pi(x) \propto \exp(-\beta f(x))$. Here, $f$ can be written as a finite sum $f(x):= \frac{1}{N}\sum_{k=1}^N f_k(x)$. Our approach is based on quantum simulated annealing on slowly varying Markov chains derived from unadjusted Langevin algorithms, removing the necessity for function evaluations which can be computationally expensive for large data sets in mixture modeling and multi-stable systems. We also incorporate a stochastic gradient oracle that implements the quantum walk operators inexactly by only using mini-batch gradients. As a result, our stochastic gradient based algorithm only accesses small subsets of data points in implementing the quantum walk. One challenge of quantizing the resulting Markov chains is that they do not satisfy the detailed balance condition in general. Consequently, the mixing time of the algorithm cannot be expressed in terms of the spectral gap of the transition density, making the quantum algorithms nontrivial to analyze. To overcome these challenges, we first build a hypothetical Markov chain that is reversible, and also converges to the target distribution. Then, we quantified the distance between our algorithm's output and the target distribution by using this hypothetical chain as a bridge to establish the total complexity. Our quantum algorithms exhibit polynomial speedups in terms of both dimension and precision dependencies when compared to the best-known classical algorithms.

Related papers

• Halving the Cost of Quantum Algorithms with Randomization [0.138120109831448]
  Quantum signal processing (QSP) provides a systematic framework for implementing a transformation of a linear operator. Recent works have developed randomized compiling, a technique that promotes a unitary gate to a quantum channel. Our algorithm implements a probabilistic mixture of randomizeds, strategically chosen so that the average evolution converges to that of a target function, with an error quadratically smaller than that of an equivalent individual.
  arXiv  Detail & Related papers  (2024-09-05T17:56:51Z)
• Calculating response functions of coupled oscillators using quantum phase estimation [40.31060267062305]
  We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer. Our proposed quantum algorithm operates in the standard $s-sparse, oracle-based query access model. We show that a simple adaptation of our algorithm solves the random glued-trees problem in time.
  arXiv  Detail & Related papers  (2024-05-14T15:28:37Z)
• PAPAL: A Provable PArticle-based Primal-Dual ALgorithm for Mixed Nash Equilibrium [58.26573117273626]
  We consider the non-AL equilibrium nonconptotic objective function in two-player zero-sum continuous games. Our novel insights into the particle-based algorithms for continuous distribution strategies are presented.
  arXiv  Detail & Related papers  (2023-03-02T05:08:15Z)
• Concise and Efficient Quantum Algorithms for Distribution Closeness Testing [0.2741266294612775]
  We study the impact of quantum computation on the fundamental problem of testing the property of distributions. We propose the currently best quantum algorithms for this problem under the metrics of $l1$-distance and $l2$-distance.
  arXiv  Detail & Related papers  (2023-02-13T04:03:59Z)
• Importance sampling for stochastic quantum simulations [68.8204255655161]
  We introduce the qDrift protocol, which builds random product formulas by sampling from the Hamiltonian according to the coefficients. We show that the simulation cost can be reduced while achieving the same accuracy, by considering the individual simulation cost during the sampling stage. Results are confirmed by numerical simulations performed on a lattice nuclear effective field theory.
  arXiv  Detail & Related papers  (2022-12-12T15:06:32Z)
• 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)
• Robust Sparse Mean Estimation via Sum of Squares [42.526664955704746]
  We study the problem of high-dimensional sparse mean estimation in the presence of an $epsilon$-fraction of adversarial outliers. Our algorithms follow the Sum-of-Squares based, to algorithms approach.
  arXiv  Detail & Related papers  (2022-06-07T16:49:54Z)
• Faster quantum mixing of Markov chains in non-regular graph with fewer qubits [0.0]
  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.
  arXiv  Detail & Related papers  (2022-05-12T14:08:08Z)
• 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)
• Quantum Sampling Algorithms for Near-Term Devices [0.0]
  We introduce a family of quantum algorithms that provide unbiased samples by encoding the entire Gibbs distribution. We show that this approach leads to a speedup over a classical Markov chain algorithm. It opens the door to exploring potentially useful sampling algorithms on near-term quantum devices.
  arXiv  Detail & Related papers  (2020-05-28T14:46:20Z)
• Oracle Lower Bounds for Stochastic Gradient Sampling Algorithms [39.746670539407084]
  We consider the problem of sampling from a strongly log-concave density in $bbRd$. We prove an information theoretic lower bound on the number of gradient queries of the log density needed.
  arXiv  Detail & Related papers  (2020-02-01T23:46:35Z)

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.