Quantum Boltzmann Machine

Amin, M. H. S.; Andriyash, Evgeny; Rolfe, Jason Tyler; Kulchytskyy, Bohdan; Melko, Roger G.

doi:10.1103/physrevx.8.021050

Cited by 440 publications

(445 citation statements)

References 36 publications

(72 reference statements)

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“…In the emerging discipline of quantum machine learning, a number of quantum algorithms for classification have been proposed 2–4 and demonstrate how to train and use models for classification on a quantum computer. It is an open question how to cast such quantum classifiers into an ensemble framework that likewise harvests the strengths of quantum computing, and this article is a first step to answering this question.…”

Section: Introductionmentioning

confidence: 99%

Quantum ensembles of quantum classifiers

Schuld

Petruccione

2018

Sci Rep

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Quantum machine learning witnesses an increasing amount of quantum algorithms for data-driven decision making, a problem with potential applications ranging from automated image recognition to medical diagnosis. Many of those algorithms are implementations of quantum classifiers, or models for the classification of data inputs with a quantum computer. Following the success of collective decision making with ensembles in classical machine learning, this paper introduces the concept of quantum ensembles of quantum classifiers. Creating the ensemble corresponds to a state preparation routine, after which the quantum classifiers are evaluated in parallel and their combined decision is accessed by a single-qubit measurement. This framework naturally allows for exponentially large ensembles in which – similar to Bayesian learning – the individual classifiers do not have to be trained. As an example, we analyse an exponentially large quantum ensemble in which each classifier is weighed according to its performance in classifying the training data, leading to new results for quantum as well as classical machine learning.

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Section: Introductionmentioning

confidence: 99%

Quantum ensembles of quantum classifiers

Schuld

Petruccione

2018

Sci Rep

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“…Through supervised trainings with a large number of data sets, neural networks 'learn' to recognize key features of a universal class. Very recently, rapid and promising development has been made from this perspective on numerical studies of condensed matter systems, including dynamical systems [2][3][4][5][6], systems undergoing phase transitions [7][8][9][10][11][12][13], as well as quantum many-body systems. Also established is the theory connection to renormalization group [14,15].…”

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confidence: 99%

Quantum Loop Topography for Machine Learning

2017

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Despite rapidly growing interest in harnessing machine learning in the study of quantum manybody systems, training neural networks to identify quantum phases is a nontrivial challenge. The key challenge is in efficiently extracting essential information from the many-body Hamiltonian or wave function and turning the information into an image that can be fed into a neural network. When targeting topological phases, this task becomes particularly challenging as topological phases are defined in terms of non-local properties. Here we introduce quantum loop topography (QLT): a procedure of constructing a multi-dimensional image from the "sample" Hamiltonian or wave function by evaluating two-point operators that form loops at independent Monte Carlo steps. The loop configuration is guided by characteristic response for defining the phase, which is Hall conductivity for the cases at hand. Feeding QLT to a fully-connected neural network with a single hidden layer, we demonstrate that the architecture can be effectively trained to distinguish Chern insulator and fractional Chern insulator from trivial insulators with high fidelity. In addition to establishing the first case of obtaining a phase diagram with topological quantum phase transition with machine learning, the perspective of bridging traditional condensed matter theory with machine learning will be broadly valuable.Introduction-Machine learning techniques have been enabling neural networks to successfully recognize and interpret big data sets of images and speeches [1]. Through supervised trainings with a large number of data sets, neural networks 'learn' to recognize key features of a universal class. Very recently, rapid and promising development has been made from this perspective on numerical studies of condensed matter systems, including dynamical systems[2-6], systems undergoing phase transitions [7][8][9][10][11][12][13], as well as quantum many-body systems. Also established is the theory connection to renormalization group [14,15]. Exciting successes in application of machine learning to symmetry broken phases [7][8][9][10] may be attributed to the locality of the defining property of the target phases: the order parameter field. The snap-shots of order parameter configuration form images that can be readily fed into neural networks that have been developed to recognize patterns in images.Unfortunately many novel states cannot be numerically detected through a local order parameter. For one, all topological phases are intrinsically defined in terms of non-local topological properties. Not only many-body localized states of growing interest [16] fit into this category, even a superconducting state fits in here since the superconducting order parameter explicitly breaks particle number conservation [17]. In order for neural networks to learn to recognize and identify such phases, we need to supply them with "images" that contain relevant non-local information. Clearly information based on single site is insufficient. One approach to detecting topologic...

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“…Examples include Refs. [48,52,[55][56][57] on (deep) Boltzmann machines. We believe that the field of quantum(-enhanced) machine learning could benefit the most from the marriage of these different ideas, and look forward to seeing more novel quantum algorithms for solving machine learning tasks.…”

Section: Discussionmentioning

confidence: 99%

Quantum algorithm for linear regression

Wang

2017

Phys. Rev. A

125

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We present a quantum algorithm for fitting a linear regression model to a given data set using the least squares approach. Different from previous algorithms which yield a quantum state encoding the optimal parameters, our algorithm outputs these numbers in the classical form. So by running it once, one completely determines the fitted model and then can use it to make predictions on new data at little cost. Moreover, our algorithm works in the standard oracle model, and can handle data sets with nonsparse design matrices. It runs in time poly(log(N ), d, κ, 1/ǫ), where N is the size of the data set, d is the number of adjustable parameters, κ is the condition number of the design matrix, and ǫ is the desired precision in the output. We also show that the polynomial dependence on d and κ is necessary. Thus, our algorithm cannot be significantly improved. Furthermore, we also give a quantum algorithm that estimates the quality of the least-squares fit (without computing its parameters explicitly). This algorithm runs faster than the one for finding this fit, and can be used to check whether the given data set qualifies for linear regression in the first place.

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Quantum Boltzmann Machine

Cited by 440 publications

References 36 publications

Quantum ensembles of quantum classifiers

Quantum ensembles of quantum classifiers

Quantum Loop Topography for Machine Learning

Quantum algorithm for linear regression

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