There is a new chapter describing the staggered quantum walk model. The chapter on spatial search algorithms has been rewritten to offer a more comprehensive approach and a new chapter describing the element distinctness algorithm has been added. There is a new appendix on graph theory highlighting the importance of graph theory to quantum walks.

## Quantum walks and search algorithms - CERN Document Server

As before, the reader will benefit from the pedagogical elements of the book, which include exercises and references to deepen the reader's understanding, and guidelines for the use of computer programs to simulate the evolution of quantum walks. The author proposes a series of exercises that help the reader get some working experience with the presented concepts, facilitating a better understanding. Each chapter ends with a discussion of further references, pointing the reader to major results on the topics presented in the respective chapter.

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Aucun extrait n'est disponible. Configuration requise. This book addresses an interesting area of quantum computation called quantum walks, which play an important role in building quantum algorithms, in particular search algorithms. Quantum walks are the quantum analogue of classical random walks. It is known that quantum computers have great power for searching unsorted databases. This power extends to many kinds of searches, particularly to the problem of finding a specific location in a spatial layout, which can be modeled by a graph. The goal is to find a specific node knowing that the particle uses the edges to jump from one node to the next.

This book is self-contained with main topics that include:Grover's algorithm, describing its geometrical interpretation and evolution by means of the spectral decomposition of the evolution operaterAnalytical solutions of quantum walks on important graphs like line, cycles, two-dimensional lattices, and hypercubes using Fourier transformsQuantum walks on generic graphs, describing methods to calculate the limiting distribution and mixing timeSpatial search algorithms, with emphasis on the abstract search algorithm the two-dimensional lattice is used as an example Szedgedy's quantum-walk model and a natural definition of quantum hitting time the complete graph is used as an example The reader will benefit from the pedagogical aspects of the book, learning faster and with more ease than would be possible from the primary research literature.

Exercises and references further deepen the reader's understanding, and guidelines for the use of computer programs to simulate the evolution of quantum walks are also provided.

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Trouver sur. Feoktistov Vitaliy Physical Implementation of Quantum Wa Manouchehri Kia. This problem was, to some extent, circumvented by introducing bounds on the quantum probabilities, allowing the authors to train the model efficiently by sampling.

## Quantum Walks for Computer Scientists

It is possible that a specific type of quantum Boltzmann machine has been trained in the D-Wave 2X by using a learning rule analogous to that of classical Boltzmann machines. Quantum annealing is not the only technology for sampling. In a prepare-and-measure scenario, a universal quantum computer prepares a thermal state, which is then sampled by measurements.

This can reduce the time required to train a deep restricted Boltzmann machine, and provide a richer and more comprehensive framework for deep learning than classical computing. Relying on an efficient thermal state preparation protocol starting from an arbitrary state, quantum-enhanced Markov logic networks exploit the symmetries and the locality structure of the probabilistic graphical model generated by a first-order logic template. Quantum analogues or generalizations of classical neural nets are often referred to as quantum neural networks. The term is claimed by a wide range of approaches, including the implementation and extension of neural networks using photons, layered variational circuits or quantum Ising-type models.

Quantum neural networks are often defined as an expansion on Deutsch's model of a quantum computational network. To test quantum applications in a neural network, quantum dot molecules are deposited on a substrate of GaAs or similar to record how they communicate with one another. An even distribution across the substrate in sets of two create dipoles and ultimately two spin states, up or down. Unlike the approach taken by other quantum-enhanced machine learning algorithms, HQMMs can be viewed as models inspired by quantum mechanics that can be run on classical computers as well.

Recent work has shown that these models can be successfully learned by maximizing the log-likelihood of the given data via classical optimization, and there is some empirical evidence that these models can better model sequential data compared to classical HMMs in practice, although further work is needed to determine exactly when and how these benefits are derived. In the most general case of quantum machine learning, both the learning device and the system under study, as well as their interaction, are fully quantum.

This section gives a few examples of results on this topic. One class of problem that can benefit from the fully quantum approach is that of 'learning' unknown quantum states, processes or measurements, in the sense that one can subsequently reproduce them on another quantum system.

For example, one may wish to learn a measurement that discriminates between two coherent states, given not a classical description of the states to be discriminated, but instead a set of example quantum systems prepared in these states. The naive approach would be to first extract a classical description of the states and then implement an ideal discriminating measurement based on this information. This would only require classical learning. However, one can show that a fully quantum approach is strictly superior in this case.

Going beyond the specific problem of learning states and transformations, the task of clustering also admits a fully quantum version, wherein both the oracle which returns the distance between data-points and the information processing device which runs the algorithm are quantum. The term quantum machine learning is also used for approaches that apply classical methods of machine learning to the study of quantum systems. A prime example is the use of classical learning techniques to process large amounts of experimental or calculated for example by solving Schrodinger's equation data in order to characterize an unknown quantum system for instance in the context of quantum information theory and for the development of quantum technologies or computational materials design , but there are also more exotic applications.

The ability to experimentally control and prepare increasingly complex quantum systems brings with it a growing need to turn large and noisy data sets into meaningful information. This is a problem that has already been studied extensively in the classical setting, and consequently, many existing machine learning techniques can be naturally adapted to more efficiently address experimentally relevant problems. For example, Bayesian methods and concepts of algorithmic learning can be fruitfully applied to tackle quantum state classification, [68] Hamiltonian learning, [69] and the characterization of an unknown unitary transformation.

Quantum machine learning can also be applied to dramatically accelerate the prediction of quantum properties of molecules and materials. Some examples include. Variational circuits are a family of algorithms which utilize training based on circuit parameters and an objective function. These circuits are very heavily dependent on the architecture of the proposed quantum device because parameter adjustments are adjusted based solely on the classical components within the device.

Quantum learning theory pursues a mathematical analysis of the quantum generalizations of classical learning models and of the possible speed-ups or other improvements that they may provide. The framework is very similar to that of classical computational learning theory , but the learner in this case is a quantum information processing device, while the data may be either classical or quantum.

Quantum learning theory should be contrasted with the quantum-enhanced machine learning discussed above, where the goal was to consider specific problems and to use quantum protocols to improve the time complexity of classical algorithms for these problems. Although quantum learning theory is still under development, partial results in this direction have been obtained. The starting point in learning theory is typically a concept class , a set of possible concepts.

For example, the concept class could be the set of disjunctive normal form DNF formulas on n bits or the set of Boolean circuits of some constant depth. The goal for the learner is to learn exactly or approximately an unknown target concept from this concept class.

The learner may be actively interacting with the target concept, or passively receiving samples from it. In active learning, a learner can make membership queries to the target concept c , asking for its value c x on inputs x chosen by the learner. The learner then has to reconstruct the exact target concept, with high probability. In the model of quantum exact learning , the learner can make membership queries in quantum superposition. If the complexity of the learner is measured by the number of membership queries it makes, then quantum exact learners can be polynomially more efficient than classical learners for some concept classes, but not more.

A natural model of passive learning is Valiant's probably approximately correct PAC learning. Here the learner receives random examples x,c x , where x is distributed according to some unknown distribution D. The learner has to be able to produce such an 'approximately correct' h for every D and every target concept c in its concept class. In the PAC model and the related agnostic model , this doesn't significantly reduce the number of examples needed: for every concept class, classical and quantum sample complexity are the same up to constant factors.

This passive learning type is also the most common scheme in supervised learning: a learning algorithm typically takes the training examples fixed, without the ability to query the label of unlabelled examples. Outputting a hypothesis h is a step of induction. Classically, an inductive model splits into a training and an application phase: the model parameters are estimated in the training phase, and the learned model is applied an arbitrary many times in the application phase. In the asymptotic limit of the number of applications, this splitting of phases is also present with quantum resources.

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The earliest experiments were conducted using the adiabatic D-Wave quantum computer, for instance, to detect cars in digital images using regularized boosting with a nonconvex objective function in a demonstration in Using a different annealing technology based on nuclear magnetic resonance NMR , a quantum Hopfield network was implemented in that mapped the input data and memorized data to Hamiltonians, allowing the use of adiabatic quantum computation.

The two entries of the vector are the vertical and horizontal ratio of the pixel intensity of the image. Once the vectors are defined on the feature space, the quantum support vector machine was implemented to classify the unknown input vector. Photonic implementations are attracting more attention, [] not the least because they do not require extensive cooling. Simultaneous spoken digit and speaker recognition and chaotic time-series prediction were demonstrated at data rates beyond 1 gigabyte per second in Recently, based on a neuromimetic approach, a novel ingredient has been added to the field of quantum machine learning, in the form of a so-called quantum memristor, a quantized model of the standard classical memristor.

An implementation of a quantum memristor in superconducting circuits has been proposed, [] and an experiment with quantum dots performed. In doing so, the company is encouraging software developers to pursue new algorithms through a development environment with quantum capabilities. New architectures are being explored on an experimental basis, up to 32 qbits, utilizing both trapped-ion and superconductive quantum computing methods.