Craig S. Hamilton, Regina Kruse, Linda Sansoni et al. · Physical Review Letters · 2017
Boson sampling has emerged as a tool to explore the advantages of quantum over classical computers as it does not require universal control over the quantum system, which favors current photonic experimental platforms. Here, we introduce Gaussian Boson sampling, a classically hard-to-solve problem that uses squeezed states as a nonclassical resource. We relate the probability to measure specific photon patterns from a general Gaussian state in the Fock basis to a matrix function called the Hafnian, which answers the last remaining question of sampling from Gaussian states. Based on this result, we design Gaussian Boson sampling, a #P hard problem, using squeezed states. This demonstrates that Boson sampling from Gaussian states is possible, with significant advantages in the photon generation probability, compared to existing protocols.
Cited by 491Open access
Claudio Conti · Quantum science and technology · 2023
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Zhong, Han-Sen, Peng, Li-Chao, Li, Yuan et al. · Figshare · 2019
Gaussian Boson sampling (GBS) provides a highly efficient approach to make use of squeezed states from parametric down-conversion to solve a classically hard-to-solve sampling problem. The GBS protocol not only significantly enhances the photon generation probability, compared to standard boson sampling with single photon Fock states, but also links to potential applications such as dense subgraph problems and molecular vibronic spectra. Here, we report the first experimental demonstration of GBS using squeezed-state sources with simultaneously high photon indistinguishability and collection efficiency. We implement and validate 3-, 4- and 5-photon GBS with high sampling rates of 832 kHz, 163 kHz and 23 kHz, respectively, which is more than 4.4, 12.0, and 29.5 times faster than the previous experiments. Further, we observe a quantum speed-up on a NP-hard optimization problem when comparing with simulated thermal sampler and uniform sampler.
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Paul Maragakis, Arjan van der Vaart, Martin Karplus · The Journal of Physical Chemistry B · 2009
We introduce the Gaussian-mixture umbrella sampling method (GAMUS) , a biased molecular dynamics technique based on adaptive umbrella sampling that efficiently escapes free energy minima in multidimensional problems. The prior simulation data are reweighted with a maximum likelihood formulation, and the new approximate probability density is fit to a Gaussian-mixture model, augmented by information about the unsampled areas. The method can be used to identify free energy minima in multidimensional reaction coordinates. To illustrate GAMUS , we apply it to the alanine dipeptide (2D reaction coordinate) and tripeptide (4D reaction coordinate).
Cited by 73Open access
Sonia López Alarcón, Federico Rueda · Research Square · 2022
Abstract Gaussian Boson Sampling is gaining relevance among the multiple technologies currently implementing quantum computing. The problem of sampling out of a boson distribution across multiple modes has a strong mathematical connection with combinatorics problems, with multiple practical applications that would classically be unfeasible to solve within reasonable time. The software stack that accompanies this GBS approach, and that takes the application from its high level description to the hardware implementation, is the main focus of this paper. The general compilation process of Gaussian Boson Samplers is described in detail. The specifics of the Strawberry Fields (Xanadu) compilation and simulation framework are discussed, along with its time profiling, and time implications on computationally significant problem sizes. Last, a compilation step to reduce the hardware description complexity is presented, demonstrating a linear reduction on the overall number of operators when the number of Gaussian operators increases.
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Les Kirkup, R. B. Frenkel · Cambridge University Press eBooks · 2006
If it is reasonable to assume that a population consists of values that have a Gaussian distribution, then what will be the distribution of a property (a ‘statistic’) of a sample drawn from this Gaussian ‘parent’? The property might be the mean, variance or standard deviation of the sample. Each of these properties has a sampling distribution, which can be described as follows.
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Han-Sen Zhong, Li-Chao Peng, Yuan Li et al. · Science Bulletin · 2019
Cited by 69Open access
Ken Stout · 1985
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2007
In this work we introduce Power Spectral Density Sampling (PSDS), a new method for building a reduced rank approximation to a stationary kernel ma-trix. We apply this method in the framework of Gaussian Processes (GP). The resulting model can be trained in an online fashion: As each training sample arrives, we can include it in the model using only O(r2) time and storage, where r is the selected number of spectral samples used in the approximation. For n training points, the overall cost is O(nr2) time and O(r2) storage. The resulting GP is properly defined and model hyperparameters can be selected by evidence maximization. Though it is especially well-suited for low dimensional problems, it can reach and even outperform other (batch) state-of-the-art sparse GP meth-ods on higher dimensional datasets if allowed to work in batch mode, learning the locations of the spectral samples. We check this possibility on some large datasets. 1 Introduction and previous work
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Lund, A. P., Laing, A., Rahimi-Keshari, S. et al. · arXiv (Cornell University) · 2013
We pose a generalized Boson Sampling problem. Strong evidence exists that such a problem becomes intractable on a classical computer as a function of the number of Bosons. We describe a quantum optical processor that can solve this problem efficiently based on Gaussian input states, a linear optical network and non-adaptive photon counting measurements. All the elements required to build such a processor currently exist. The demonstration of such a device would provide the first empirical evidence that quantum computers can indeed outperform classical computers and could lead to applications.
Cited by 2Open access
Banchi, Leonardo, Quesada, Nicolás, Arrazola, Juan Miguel · arXiv (Cornell University) · 2020
Gaussian Boson Sampling (GBS) is a near-term platform for photonic quantum computing. Applications have been developed which rely on directly programming GBS devices, but the ability to train and optimize circuits has been a key missing ingredient for developing new algorithms. In this work, we derive analytical gradient formulas for the GBS distribution, which can be used to train devices using standard methods based on gradient descent. We introduce a parametrization of the distribution that allows the gradient to be estimated by sampling from the same device that is being optimized. In the case of training using a Kullback-Leibler divergence or log-likelihood cost function, we show that gradients can be computed classically, leading to fast training. We illustrate these results with numerical experiments in stochastic optimization and unsupervised learning. As a particular example, we introduce the variational Ising solver, a hybrid algorithm for training GBS devices to sample ground states of a classical Ising model with high probability.
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Vladimir Kazakov · 2004
The case of the Sampling-Reconstruction Procedure (SRP) of stochastic process with limited number of samples is the most important variant both for theory and practice. In order to determine the optimal reconstruction function and to calculate the minimum error reconstruction function, it is necessary to apply the classical conditional mean rule. In this paper we deal with the statistical description of the SRP of multidimensional Gaussian process when the number of samples is limited. In this case we have sampling of some components of a vector Gaussian process. One can investigate the SRP of all components on the base of all sets of samples. There are two types of linear systems considerations: the first is the series system and the second is the parallel system. The output process of every filter is one component of the vector Gaussian process. The case of the series system is concerned with the linear transformations Z(t) of a given stochastic process y(t). One can see that the reconstruction algorithm based on the multidimensional presentation of the SRP provides the best results. In the same manner one can investigate the case of other linear transformations: the differentiation and the delay.
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Levon Chakhmakhchyan, Nicolas J. Cerf · Physical review. A/Physical review, A · 2017
We develop an alternative boson sampling model operating on single-photon states followed by linear interferometry and Gaussian measurements. The hardness proof for simulating such continuous-variable measurements is established in two main steps, making use of the symmetry of quantum evolution under time reversal. Namely, we first construct a twofold version of scattershot boson sampling in which, as opposed to the original proposal, both legs of a collection of two-mode squeezed vacuum states undergo parallel linear-optical transformations. This twofold scattershot model yields, as a corollary, an instance of boson sampling from Gaussian states where photon counting is hard to simulate. Then, a time-reversed setup is used to exhibit a boson sampling model in which the simulation of Gaussian measurements---namely the outcome of eight-port homodyne detection---is proven to be computationally hard. These results illustrate how the symmetry of quantum evolution under time reversal may serve as a tool for analyzing the computational complexity of novel physically motivated computational problems.
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Angshuman Karmakar, Sujoy Sinha Roy, Oscar Reparaz et al. · IEEE Transactions on Computers · 2018
Sampling from a discrete Gaussian distribution is an indispensable part of lattice-based cryptography. Several recent works have shown that the timing leakage from a non-constant-time implementation of the discrete Gaussian sampling algorithm could be exploited to recover the secret. In this paper, we propose a constant-time implementation of the Knuth-Yao random walk algorithm for performing constant-time discrete Gaussian sampling. Since the random walk is dictated by a set of input random bits, we can express the generated sample as a function of the input random bits. Hence, our constant-time implementation expresses the unique mapping of the input random-bits to the output sample-bits as a Boolean expression of the random-bits. We use bit-slicing to generate multiple samples in batches and thus increase the throughput of our constant-time sampling manifold. Our experiments on an Intel i7-Broadwell processor show that our method can be as much as 2.4 times faster than the constant-time implementation of cumulative distribution table based sampling and consumes exponentially less memory than the Knuth-Yao algorithm with shuffling for a similar level of security.
Cited by 53Open access
Paul Glasserman, et al. · 1998
This paper deals with efficient algorithms for simulating performance measures of Gaussian random vectors. Recently, we developed a simulation algorithm which consists of doing importance sampling by shifting the mean of the Gaussian random vector. Further variance reduction is obtained by stratification along a key direction. A central ingredient of this method is to compute the optimal shift of the mean for the importance sampling. The optimal shift is also a convenient, and in many cases, an effective direction for the stratification. In this paper, after giving a brief overview of the basic simulation algorithms, we focus on issues regarding the computation of the optimal change of measure. A primary application of this methodology occurs in computational finance for pricing path dependent options.
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Yongchao Huang · ArXiv.org · 2025
We present a family of \textit{Gaussian Mixture Approximation} (GMA) samplers for sampling unnormalised target densities, encompassing \textit{weights-only GMA} (W-GMA), \textit{Laplace Mixture Approximation} (LMA), \textit{expectation-maximization GMA} (EM-GMA), and further variants. GMA adopts a simple two-stage paradigm: (i) initialise a finite set of Gaussian components and draw samples from a proposal mixture; (ii) fit the mixture to the target by optimising either only the component weights or also the means and variances, via a sample-based KL divergence objective that requires only evaluations of the unnormalised density, followed by stratified resampling. The method is gradient-free, and computationally efficient: it leverages the ease of sampling from Gaussians, efficient optimisation methods (projected gradient descent, mirror descent, and EM), and the robustness of stratified resampling to produce samples faithful to the target. We show that this optimisation-resampling scheme yields consistent approximations under mild conditions, and we validate this methodology with empirical results demonstrating accuracy and speed across diverse densities.
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Alexander S. Dellios, M. D. Reid, P. D. Drummond · AAPPS bulletin · 2023
Abstract A growing cohort of experimental linear photonic networks implementing Gaussian boson sampling (GBS) have now claimed quantum advantage. However, many open questions remain on how to effectively verify these experimental results, as scalable methods are needed that fully capture the rich array of quantum correlations generated by these photonic quantum computers. In this paper, we briefly review recent theoretical methods to simulate experimental GBS networks. We focus mostly on methods that use phase-space representations of quantum mechanics, as these methods are highly scalable and can be used to validate experimental outputs and claims of quantum advantage for a variety of input states, ranging from the ideal pure squeezed vacuum state to more realistic thermalized squeezed states. A brief overview of the theory of GBS, recent experiments, and other types of methods are also presented. Although this is not an exhaustive review, we aim to provide a brief introduction to phase-space methods applied to linear photonic networks to encourage further theoretical investigations.
Cited by 12Open access
Pablo Diaz-gutierrez · 2008
Sampling of 3D meshes is at the foundation of any surface simplification technique. In this paper, we use the recent results on quantization and surface approximation theory to propose a simple, robust, linear time, output sensitive algorithm for sampling meshes with the purpose of surface approximation. Our algorithm is based on the mapping of regular sampling and triangulation of the Gaussian sphere onto a manifold surface. An interesting aspect of our algorithm is that we do not explicitly measure, minimize, or prioritize any error to simplify and do not explicitly cluster the faces to find proxies, but still achieve bounded error approximation of the shape. 1
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C. Hamilton, Igor Jex · Physical review. A/Physical review, A · 2024
Boson sampling has emerged as an important tool to demonstrate the difference between quantum and classical computers and has attracted the interest of experimentalists and theoreticians. In this work we study boson sampling from general, single-mode states using a scheme that can generate any such state by combining Gaussian states and photon number measurements. We derive a formula that can be used to calculate the output photon number probabilities of these states after they travel through a linear interferometer. This extends the Boson sampling protocol to the widest array of possible single-mode states and from this we show that the complexity scaling of all such states is similar and hence there is no complexity advantage of using complex input states over simpler ones.
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J. M. Hammersley, J. A. Nelder · Mathematical Proceedings of the Cambridge Philosophical Society · 1955
Isotropic Gaussian processes, of which we shall give a formal definition presently, arise in various practical problems. The present inquiry arose from the consideration of the variability found in the yields of plots in agricultural field experiments. Samples of such patterns of variability can be obtained from uniformity trials, whereby a piece of land is treated uniformly throughout and the crop is harvested in small units or plots. The results from such trials have been widely used to determine optimum plot sizes for future experiments with the crop concerned, but it has never been clear how valid is the generalization from a uniformity trial to future experiments on other sites. One difficulty arises from the lack of a suitable model to express the variability; another difficulty arises from the formidable analytical problems besetting any attempts to apply deductive reasoning to even simple models. We may mention two approaches that have been made to the uniformity trial problem: Fairfield Smith (3) has supplied an empirical law connecting the variance of contiguous groups of plots with the size of the group, and Quenouille (6) and Whittle (8) have considered the fitting of two-dimensional isotropic Gaussian processes to uniformity trial data. Whittle has pointed out that a satisfactory model may have to incorporate an additional random element at each point, and has outlined the difficulties of estimation which arise when such an element is introduced. The connexion between Fairfield Smith's law and the approach via two-dimensional stochastic processes, if any, seems to be quite unknown.
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