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1. Gaussian Cooling and Dikin Walks: The Interior-Point Method for Logconcave Sampling

   Kook, Yunbum; Vempala, Santosh S (July 2024, Conference on Learning Theory 2024)

   The connections between (convex) optimization and (logconcave) sampling have been considerably enriched in the past decade with many conceptual and mathematical analogies. For instance, the Langevin algorithm can be viewed as a sampling analogue of gradient descent and has condition-number-dependent guarantees on its performance. In the early 1990s, Nesterov and Nemirovski developed the Interior-Point Method (IPM) for convex optimization based on self-concordant barriers, providing efficient algorithms for structured convex optimization, often faster than the general method. This raises the following question: can we develop an analogous IPM for structured sampling problems? In 2012, Kannan and Narayanan proposed the Dikin walk for uniformly sampling polytopes, and an improved analysis was given in 2020 by Laddha-Lee-Vempala. The Dikin walk uses a local metric defined by a self-concordant barrier for linear constraints. Here we generalize this approach by developing and adapting IPM machinery together with the Dikin walk for poly-time sampling algorithms. Our IPM-based sampling framework provides an efficient warm start and goes beyond uniform distributions and linear constraints. We illustrate the approach on important special cases, in particular giving the fastest algorithms to sample uniform, exponential, or Gaussian distributions on a truncated PSD cone. The framework is general and can be applied to other sampling algorithms. 

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

   Ozgul, Guneykan; Li, Xiantao; Mahdavi, Mehrdad; Wang, Chunhao (June 2024, Proceedings of the 41st International Conference on Machine Learning)

   We present quantum algorithms for sampling from possibly non-logconcave probability distributions expressed as 𝜋(𝑥)∝exp(−𝛽𝑓(𝑥)) as well as quantum algorithms for estimating the partition function for such distributions. We also incorporate a stochastic gradient oracle that implements the quantum walk operators inexactly by only using mini-batch gradients when 𝑓 can be written as a finite sum. 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 matrix, making the quantum algorithms nontrivial to analyze. We overcame these challenges by first building a reference reversible Markov chain that converges to the target distribution, then controlling the discrepancy between our algorithm’s output and the target distribution by using the reference Markov chain as a bridge to establish the total complexity. Our quantum algorithms exhibit polynomial speedups in terms of dimension or precision dependencies when compared to best-known classical algorithms under similar assumptions. 

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3. Adaptive Quantum Simulated Annealing for Bayesian Inference and Estimating Partition Functions

   Harrow, Aram W.; Wei, AY (January 2020, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms)

   Markov chain Monte Carlo algorithms have important applications in counting problems and in machine learning problems, settings that involve estimating quantities that are difficult to compute exactly. How much can quantum computers speed up classical Markov chain algorithms? In this work we consider the problem of speeding up simulated annealing algorithms, where the stationary distributions of the Markov chains are Gibbs distributions at temperatures specified according to an annealing schedule. We construct a quantum algorithm that both adaptively constructs an annealing schedule and quantum samples at each temperature. Our adaptive annealing schedule roughly matches the length of the best classical adaptive annealing schedules and improves on nonadaptive temperature schedules by roughly a quadratic factor. Our dependence on the Markov chain gap matches other quantum algorithms and is quadratically better than what classical Markov chains achieve. Our algorithm is the first to combine both of these quadratic improvements. Like other quantum walk algorithms, it also improves on classical algorithms by producing “qsamples” instead of classical samples. This means preparing quantum states whose amplitudes are the square roots of the target probability distribution. In constructing the annealing schedule we make use of amplitude estimation, and we introduce a method for making amplitude estimation nondestructive at almost no additional cost, a result that may have independent interest. Finally we demonstrate how this quantum simulated annealing algorithm can be applied to the problems of estimating partition functions and Bayesian inference. 

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4. Strongly interacting photonic quantum walk using single atom beamsplitters

   https://doi.org/10.1103/PhysRevResearch.6.013245  

   Zheng, Xinyuan; Waks, Edo (March 2024, Physical Review Research)

   Photonics provide an efficient way to implement quantum walks, the quantum analog of classical random walks, which demonstrate rich physics with potential applications. However, most photonic quantum walks do not involve photon interactions, which limits their potential to explore strongly correlated many-body physics of light. We propose a strongly interacting discrete-time photonic quantum walk using a network of single atom beamsplitters. We calculate output statistics of the quantum walk for the case of two photons, which reveals the strongly correlated transport of photons. Particularly, the walk can exhibit either bosonlike or fermionlike statistics which is tunable by postselecting the two-photon detection time interval. Also, the walk can sort different types of two-photon bound states into distinct pairs of output ports under certain conditions. These unique phenomena show that our quantum walk is an intriguing platform to explore strongly correlated quantum many-body states of light. Finally, we propose an experimental realization based on time-multiplexed synthetic dimensions. 

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5. Convergence of Gibbs Sampling: Coordinate Hit-And-Run Mixes Fast

   https://doi.org/10.4230/LIPIcs.SoCG.2021.51  

   Laddha, Aditi; Vempala, Santosh S. (January 2021, ACM Symposium on Computational Geometry)

   Buchin, Kevin; Colin de Verdi\` (Ed.)

   The Gibbs Sampler is a general method for sampling high-dimensional distributions, dating back to 1971. In each step of the Gibbs Sampler, we pick a random coordinate and re-sample that coordinate from the distribution induced by fixing all the other coordinates. While it has become widely used over the past half-century, guarantees of efficient convergence have been elusive. We show that for a convex body K in ℝⁿ with diameter D, the mixing time of the Coordinate Hit-and-Run (CHAR) algorithm on K is polynomial in n and D. We also give a lower bound on the mixing rate of CHAR, showing that it is strictly worse than hit-and-run and the ball walk in the worst case. 

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