Quantum machine learning

1713:. 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. Within this model, nonlinear and irreversible gates, dissimilar to the Hamiltonian operator, are deployed to speculate the given data set. Such gates make certain phases unable to be observed and generate specific oscillations. Quantum neural networks apply the principals quantum information and quantum computation to classical neurocomputing. Current research shows that QNN can exponentially increase the amount of computing power and the degrees of freedom for a computer, which is limited for a classical computer to its size. A quantum neural network has computational capabilities to decrease the number of steps, qubits used, and computation time. The wave function to quantum mechanics is the neuron for Neural networks. 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. Each quantum dot can be referred as an island of electric activity, and when such dots are close enough (approximately 10 - 20 nm) electrons can tunnel underneath the islands. An even distribution across the substrate in sets of two create dipoles and ultimately two spin states, up or down. These states are commonly known as qubits with corresponding states of 1393:

computation to learn in a short time and also use fewer parameters than its classical counterparts. It is theoretically and numerically proven that we can approximate non-linear functions, like those used in neural networks, on quantum circuits. Due to VQCs superiority, neural network has been replaced by VQCs in Reinforcement Learning tasks and Generative Algorithms. The intrinsic nature of quantum devices towards decoherence, random gate error and measurement errors caused to have high potential to limit the training of the variation circuits. Training the VQCs on the classical devices before employing them on quantum devices helps to overcome the problem of decoherence noise that came through the number of repetitions for training.

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annealing can be used for supervised learning in classification tasks. The same device was later used to train a fully connected Boltzmann machine to generate, reconstruct, and classify down-scaled, low-resolution handwritten digits, among other synthetic datasets. In both cases, the models trained by quantum annealing had a similar or better performance in terms of quality. The ultimate question that drives this endeavour is whether there is quantum speedup in sampling applications. Experience with the use of quantum annealers for combinatorial optimization suggests the answer is not straightforward. Reverse annealing has been used as well to solve a fully connected quantum restricted Boltzmann machine.

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characterize it. The input states information are transported through the network in a feed-forward fashion, layer-to-layer transition mapping on the qubits of the two adjacent layers, as the name implies. Dissipative term also refers to the fact that the output layer is formed by the ancillary qubits while the input layers are dropped while tracing out the final layer. When performing a broad supervised learning task, DQNN are used to learn a unitary matrix connecting the input and output quantum states. The training data for this task consists of the quantum state and the corresponding classical labels.

2065:. 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. However, for learning under some fixed distribution D, quantum examples can be very helpful, for example for learning DNF under the uniform distribution. When considering time complexity, there exist concept classes that can be PAC-learned efficiently by quantum learners, even from classical examples, but not by classical learners (again, under plausible complexity-theoretic assumptions). 1882:

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. (This also relates to work on quantum pattern matching.) The problem of learning unitary transformations can be approached in a similar way.

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processor prepares quantum states and measurement is made and the optimization is done by a classical computer. VQAs are considered best for NISQ as VQAs are noise tolerant compared to other algorithms and give quantum superiority with only a few hundred qubits. Researchers have studied circuit-based algorithms to solve optimization problems and find the ground state energy of complex systems, which were difficult to solve or required a large time to perform the computation using a classical computer.

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ability when compared to the classical method of Pseudorandom Number Generators (PRNGs). However, in a more recent publication from 2021, these claims could not be reproduced for Neural Network weight initialization and no significant advantage of using QRNGs over PRNGs was found. The work also demonstrated that the generation of fair random numbers with a gate quantum computer is a non-trivial task on NISQ devices, and QRNGs are therefore typically much more difficult to use in practice than PRNGs.

1939:, 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. 1989:

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. If complexity is measured by the amount of time the learner uses, then there are concept classes that can be learned efficiently by quantum learners but not by classical learners (under plausible complexity-theoretic assumptions).

93: 2187:“Don't fall for the hype!” - Frank Zickert, who is the author of probably the most practical book related to the subject beware that ”quantum computers are far away from advancing machine learning for their representation ability”, and even speaking about evaluation and optimization for any kind of useful task quantum supremacy is not yet achieved. Furthermore, nobody among the active researchers in the field make any forecasts about when it could possibly become practical. 1050: 1996:. Here the learner receives random examples (x,c(x)), where x is distributed according to some unknown distribution D. The learner's goal is to output a hypothesis function h such that h(x)=c(x) with high probability when x is drawn according to 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. We can consider replacing the random examples by potentially more powerful quantum examples 1650:. Sampling from generic probabilistic models is hard: algorithms relying heavily on sampling are expected to remain intractable no matter how large and powerful classical computing resources become. Even though quantum annealers, like those produced by D-Wave Systems, were designed for challenging combinatorial optimization problems, it has been recently recognized as a potential candidate to speed up computations that rely on sampling by exploiting quantum effects. 9876: 8952: 9856: 36: 10188: 1857:. 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. Additionally, since classical HMMs are a particular kind of 2127:. This device can be constructed by means of a tunable resistor, weak measurements on the system, and a classical feed-forward mechanism. An implementation of a quantum memristor in superconducting circuits has been proposed, and an experiment with quantum dots performed. A quantum memristor would implement nonlinear interactions in the quantum dynamics which would aid the search for a fully functional quantum neural network. 2119:

implement an all-optical linear classifier, a perceptron model was capable of learning the classification boundary iteratively from training data through a feedback rule. A core building block in many learning algorithms is to calculate the distance between two vectors: this was first experimentally demonstrated for up to eight dimensions using entangled qubits in a photonic quantum computer in 2015.

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these filters transforms input data using a quantum circuit that can be created in an organized or randomized way. Three parts that make up the quantum convolutional filter are: the encoder, the parameterized quantum circuit (PQC), and the measurement. The quantum convolutional filter can be seen as an extension of the filter in the traditional CNN because it was designed with trainable parameters.

2134:. This platform consists of several fully operational quantum processors accessible via the IBM Web API. 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 qubits, using both trapped-ion and superconductive quantum computing methods. 1678:

of the quantum Boltzmann machine can become nontrivial. This problem was, to some extent, circumvented by introducing bounds on the quantum probabilities, allowing the authors to train the model efficiently by sampling. 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.

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distinguish hand written number ‘6’ and ‘9’ on a liquid-state quantum computer in 2015. The training data involved the pre-processing of the image which maps them to normalized 2-dimensional vectors to represent the images as the states of a qubit. 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

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a state with a corresponding element is less than the predefined one. Grover's algorithm can then find an element such that our condition is met. The minimization is initialized by some random element in our data set, and iteratively does this subroutine to find the minimum element in the data set. This minimization is notably used in quantum k-medians, and it has a speed up of at least

1413:. In quantum machine learning, classical bits are converted to qubits and they are mapped to Hilbert space; complex value data are used in a quantum binary classifier to use the advantage of Hilbert space. By exploiting the quantum mechanic properties such as superposition, entanglement, interference the quantum binary classifier produces the accurate result in short period of time. 1068:

information processing routines are applied and the result of the quantum computation is read out by measuring the quantum system. For example, the outcome of the measurement of a qubit reveals the result of a binary classification task. While many proposals of quantum machine learning algorithms are still purely theoretical and require a full-scale universal

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comprehensive framework for deep learning than classical computing. The same quantum methods also permit efficient training of full Boltzmann machines and multi-layer, fully connected models and do not have well-known classical counterparts. Relying on an efficient thermal state preparation protocol starting from an arbitrary state, quantum-enhanced

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training states from the fake states created by the generator. The relevant features of the training set are learned by the generator by alternate and adversarial training of the networks that aid in the production of sets that extend the training set. DQGAN has a fully quantum architecture and is trained in quantum data.

1903:). These efforts are often also referred to as Interpretable Machine Learning (IML, and by extension IQML). XQML/IQML can be considered as an alternative research direction instead of finding a quantum advantage. For example, XQML has been used in the context of mobile malware detection and classification. Quantum 1853:. 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. Where classical HMMs use probability vectors to represent hidden 'belief' states, HQMMs use the quantum analogue: 2069:

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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In March 2021, a team of researchers from Austria, The Netherlands, the US and Germany reported the experimental demonstration of a quantum speedup of the learning time of reinforcement learning agents interacting fully quantumly with the environment. The relevant degrees of freedom of both agent and

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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 2013. Using non-linear photonics to

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is a branch of machine learning distinct from supervised and unsupervised learning, which also admits quantum enhancements. In quantum-enhanced reinforcement learning, a quantum agent interacts with a classical or quantum environment and occasionally receives rewards for its actions, which allows the

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An example of amplitude amplification being used in a machine learning algorithm is Grover's search algorithm minimization. In which a subroutine uses Grover's search algorithm to find an element less than some previously defined element. This can be done with an oracle that determines whether or not

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and quantum operations or specialized quantum systems to improve computational speed and data storage done by algorithms in a program. This includes hybrid methods that involve both classical and quantum processing, where computationally difficult subroutines are outsourced to a quantum device. These

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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

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The quantum circuit must effectively handle spatial information in order for QCNN to function as CNN. The convolution filter is the most basic technique for making use of spatial information. One or more quantum convolutional filters make up a quantum convolutional neural network (QCNN), and each of

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Inspired by the success of Boltzmann machines based on classical Boltzmann distribution, a new machine learning approach based on quantum Boltzmann distribution of a transverse-field Ising Hamiltonian was recently proposed. Due to the non-commutative nature of quantum mechanics, the training process

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Variational Quantum Circuits also known as Parametrized Quantum Circuits (PQCs) are based on Variational Quantum Algorithms (VQAs). VQCs consist of three parts: preparation of initial states, quantum circuit, and measurement. Researchers are extensively studying VQCs, as it uses the power of quantum

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quantum computer, for instance, to detect cars in digital images using regularized boosting with a nonconvex objective function in a demonstration in 2009. Many experiments followed on the same architecture, and leading tech companies have shown interest in the potential of quantum machine learning

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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

1985:(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. 1881:

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

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Dissipative QNNs (DQNNs) are constructed from layers of qubits coupled by perceptron called building blocks, which have an arbitrary unitary design. Each node in the network layer of a DQNN is given a distinct collection of qubits, and each qubit is also given a unique quantum perceptron unitary to

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is an optimization technique used to determine the local minima and maxima of a function over a given set of candidate functions. This is a method of discretizing a function with many local minima or maxima in order to determine the observables of the function. The process can be distinguished from

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Quantum machine learning also extends to a branch of research that explores methodological and structural similarities between certain physical systems and learning systems, in particular neural networks. For example, some mathematical and numerical techniques from quantum physics are applicable to

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which explores the use of the adiabatic D-Wave quantum computer. A more recent example trained a probabilistic generative models with arbitrary pairwise connectivity, showing that their model is capable of generating handwritten digits as well as reconstructing noisy images of bars and stripes and

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have also been proposed to interpret gates within a circuit based on a game-theoretic approach. For this purpose, gates instead of features act as players in a coalitional game with a value function that depends on measurements of the quantum circuit of interest. Additionally, a quantum version of

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of the unlabeled training data . The generator and the discriminator are the two DQNNs that make up a single DQGAN. The generator's goal is to create false training states that the discriminator cannot differentiate from the genuine ones, while the discriminator's objective is to separate the real

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Quantum neural networks take advantage of the hierarchical structures, and for each subsequent layer, the number of qubits from the preceding layer is decreased by a factor of two. For n input qubits, these structure have O(log(n)) layers, allowing for shallow circuit depth. Additionally, they are

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VQAs are one of the most studied quantum algorithms as researchers expect that all the needed applications for the quantum computer will be using the VQAs and also VQAs seem to fulfill the expectation for gaining quantum supremacy. VQAs is a mixed quantum-classical approach where the quantum

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A crucial bottleneck of methods that simulate linear algebra computations with the amplitudes of quantum states is state preparation, which often requires one to initialise a quantum system in a state whose amplitudes reflect the features of the entire dataset. Although efficient methods for state

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Associative (or content-addressable memories) are able to recognize stored content on the basis of a similarity measure, rather than fixed addresses, like in random access memories. As such they must be able to retrieve both incomplete and corrupted patterns, the essential machine learning task of

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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

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that solve tasks in machine learning, thereby improving and often expediting classical machine learning techniques. Such algorithms typically require one to encode the given classical data set into a quantum computer to make it accessible for quantum information processing. Subsequently, quantum

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In October 2019, it was noted that the introduction of Quantum Random Number Generators (QRNGs) to machine learning models including Neural Networks and Convolutional Neural Networks for random initial weight distribution and Random Forests for splitting processes had a profound effect on their

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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. Finally, a general framework spanning supervised, unsupervised and reinforcement learning in the fully quantum

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is also offered to assure validity. In QCNN architecture, the pooling layer is typically placed between succeeding convolutional layers. Its function is to shrink the representation's spatial size while preserving crucial features, which allows it to reduce the number of parameters, streamline

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Many of the leading scientists that extensively publish in the field of quantum machine learning warn about the extensive hype around the topic and are very restrained if asked about its practical uses in the foreseeable future. Sophia Chen collected some of the statements made by well known

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The D-Wave 2X system hosted at NASA Ames Research Center has been recently used for the learning of a special class of restricted Boltzmann machines that can serve as a building block for deep learning architectures. Complementary work that appeared roughly simultaneously showed that quantum

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complex amplitudes, this information encoding can allow for an exponentially compact representation. Intuitively, this corresponds to associating a discrete probability distribution over binary random variables with a classical vector. The goal of algorithms based on amplitude encoding is to

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was implemented in 2009 that mapped the input data and memorized data to Hamiltonians, allowing the use of adiabatic quantum computation. NMR technology also enables universal quantum computing, and it was used for the first experimental implementation of a quantum support vector machine to

1563:. A quantum speedup of the agent's internal decision-making time has been experimentally demonstrated in trapped ions, while a quantum speedup of the learning time in a fully coherent (`quantum') interaction between agent and environment has been experimentally realized in a photonic setup. 1429:

algorithm, which has been shown to solve unstructured search problems with a quadratic speedup compared to classical algorithms. These quantum routines can be employed for learning algorithms that translate into an unstructured search task, as can be done, for instance, in the case of the

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interactions (synapses) of a real, symmetric energy matrix over a network of n artificial neurons. The encoding is such that the desired patterns are local minima of the energy functional and retrieval is done by minimizing the total energy, starting from an initial configuration.

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of the desired patterns with probability distribution peaked on the most similar pattern to an input. By its very quantum nature, the retrieval process is thus probabilistic. Because quantum associative memories are free from cross-talk, however, spurious memories are never generated.

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setting was introduced in, where it was also shown that the possibility of probing the environment in superpositions permits a quantum speedup in reinforcement learning. Such a speedup in the reinforcement-learning paradigm has been experimentally demonstrated in a photonic setup.

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neural networks, the last module is a fully connected layer with full connections to all activations in the preceding layer. Translational invariance, which requires identical blocks of parameterized quantum gates within a layer, is a distinctive feature of the QCNN architecture.

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algorithms. Another possibility is to rely on a physical process, like quantum annealing, that naturally generates samples from a Boltzmann distribution. The objective is to find the optimal control parameters that best represent the empirical distribution of a given dataset.

1128:. When too many patterns are stored, spurious memories appear which quickly proliferate, so that the energy landscape becomes disordered and no retrieval is anymore possible. The number of storable patterns is typically limited by a linear function of the number of neurons, 1053:

Four different approaches to combine the disciplines of quantum computing and machine learning. The first letter refers to whether the system under study is classical or quantum, while the second letter defines whether a classical or quantum information processing device is

1626:, the inner product is a linear process. With quantum computing, linear processes may be easily accomplished additionally, due to the simplicity of implementation, the threshold function is preferred by the majority of quantum neurons for activation functions. 6237:

Bharti, Kishor; Cervera-Lierta, Alba; Kyaw, Thi Ha; Haug, Tobias; Alperin-Lea, Sumner; Anand, Abhinav; Degroote, Matthias; Heimonen, Hermanni; Kottmann, Jakob S.; Menke, Tim; Mok, Wai-Keong; Sim, Sukin; Kwek, Leong-Chuan; Aspuru-Guzik, Alán (2022-02-15).

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template. This provides an exponential reduction in computational complexity in probabilistic inference, and, while the protocol relies on a universal quantum computer, under mild assumptions it can be embedded on contemporary quantum annealing hardware.

1306:(colloquially called HHL, after the paper's authors) which, under specific conditions, performs a matrix inversion using an amount of physical resources growing only logarithmically in the dimensions of the matrix. One of these conditions is that a 2181:"There is a lot more work that needs to be done before claiming quantum machine learning will actually work," - computer scientist Iordanis Kerenidis, the head of quantum algorithms at the Silicon Valley-based quantum computing startup QC Ware. 2162:

is a well established field of both theoretical and experimental research, quantum machine learning remains a purely theoretical field of studies. Attempts to experimentally demonstrate concepts of quantum machine learning remain insufficient.

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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

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Saggio, Valeria; Asenbeck, Beate; Hamann, Arne; Strömberg, Teodor; Schiansky, Peter; Dunjko, Vedran; Friis, Nicolai; Harris, Nicholas C.; Hochberg, Michael; Englund, Dirk; Wölk, Sabine; Briegel, Hans J.; Walther, Philip (10 March 2021).

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able to avoid "barren plateau," one of the most significant issues with PQC-based algorithms, ensuring trainability. Despite the fact that the QCNN model does not include the corresponding quantum operation, the fundamental idea of the

2184:"I have not seen a single piece of evidence that there exists a meaningful task for which it would make sense to use a quantum computer and not a classical computer," - physicist Ryan Sweke of the Free University of Berlin in Germany. 1777:

A novel design for multi-dimensional vectors that uses circuits as convolution filters is QCNN. It was inspired by the advantages of CNNs and the power of QML. It is made using a combination of a variational quantum circuit(VQC) and a

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agent to adapt its behavior—in other words, to learn what to do in order to gain more rewards. In some situations, either because of the quantum processing capability of the agent, or due to the possibility to probe the environment in

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Benedetti, Marcello; Realpe-Gómez, John; Biswas, Rupak; Perdomo-Ortiz, Alejandro (2016). "Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning".

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Sriarunothai, Theeraphot; Wölk, Sabine; Giri, Gouri Shankar; Friis, Nicolai; Dunjko, Vedran; Briegel, Hans J.; Wunderlich, Christof (2019). "Speeding-up the decision making of a learning agent using an ion trap quantum processor".

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process, by which particles tunnel through kinetic or potential barriers from a high state to a low state. Quantum annealing starts from a superposition of all possible states of a system, weighted equally. Then the time-dependent

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The need for models that can be understood by humans emerges in quantum machine learning in analogy to classical machine learning and drives the research field of explainable quantum machine learning (or XQML in analogy to

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Yu, Shang; Albarran-Arriagada, F.; Retamal, J. C.; Wang, Yi-Tao; Liu, Wei; Ke, Zhi-Jin; Meng, Yu; Li, Zhi-Peng; Tang, Jian-Shun (2018-08-28). "Reconstruction of a Photonic Qubit State with Quantum Reinforcement Learning".

221: 5264:; Jiang, Zhang; Kechezi, Kostya; Knysh, Sergey; Mandrà, Salvatore; O’Gorman, Bryan; Perdomo-Ortiz, Alejando; Pethukov, Andre; Realpe-Gómez, John; Rieffel, Eleanor; Venturelli, Davide; Vasko, Fedir; Wang, Zhihui (2016). 1877:

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.

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A paper published in December 2018 reported on an experiment using a trapped-ion system demonstrating a quantum speedup of the deliberation time of reinforcement learning agents employing internal quantum hardware.

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Clark, Lewis A.; Huang W., Wei; Barlow, Thomas H.; Beige, Almut (2015). "Hidden Quantum Markov Models and Open Quantum Systems with Instantaneous Feedback". In Sanayei, Ali; Rössler, Otto E.; Zelinka, Ivan (eds.).

1782:(DNN), fully utilizing the power of extremely parallel processing on a superposition of a quantum state with a finite number of qubits. The main strategy is to carry out an iterative optimization process in the 1538:

to achieve the same quadratic speedup. Quantum walks have been proposed to enhance Google's PageRank algorithm as well as the performance of reinforcement learning agents in the projective simulation framework.

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Korenkevych, Dmytro; Xue, Yanbo; Bian, Zhengbing; Chudak, Fabian; Macready, William G.; Rolfe, Jason; Andriyash, Evgeny (2016). "Benchmarking quantum hardware for training of fully visible Boltzmann machines".

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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

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Sampling from high-dimensional probability distributions is at the core of a wide spectrum of computational techniques with important applications across science, engineering, and society. Examples include

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guides the time evolution of the system, serving to affect the amplitude of each state as time increases. Eventually, the ground state can be reached to yield the instantaneous Hamiltonian of the system.

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Heese, Raoul; Wolter, Moritz; Mücke, Sascha; Franken, Lukas; Piatkowski, Nico (2024). "On the effects of biased quantum random numbers on the initialization of artificial neural networks".

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which entry wise corresponds to the matrix can be simulated efficiently, which is known to be possible if the matrix is sparse or low rank. For reference, any known classical algorithm for

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devices, the noise level rises, posing a significant challenge to accurately computing costs and gradients on training models. The noise tolerance will be improved by using the quantum

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Goodfellow, Ian J.; Pouget-Abadie, Jean; Mirza, Mehdi; Xu, Bing; Warde-Farley, David; Ozair, Sherjil; Courville, Aaron; Bengio, Yoshua (2014-06-10). "Generative Adversarial Networks".

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Beyond quantum computing, the term "quantum machine learning" is also associated with classical machine learning methods applied to data generated from quantum experiments (i.e.

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Benedetti, Marcello; Realpe-Gómez, John; Biswas, Rupak; Perdomo-Ortiz, Alejandro (2017-11-30). "Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models".

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Benedetti, Marcello; Realpe-Gómez, John; Biswas, Rupak; Perdomo-Ortiz, Alejandro (2017). "Quantum-assisted learning of graphical models with arbitrary pairwise connectivity".

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Bisio, Alessandro; Chiribella, Giulio; D’Ariano, Giacomo Mauro; Facchini, Stefano; Perinotti, Paolo (25 March 2010). "Optimal quantum learning of a unitary transformation".

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Cai, X.-D.; Wu, D.; Su, Z.-E.; Chen, M.-C.; Wang, X.-L.; Li, Li; Liu, N.-L.; Lu, C.-Y.; Pan, J.-W. (2015). "Entanglement-Based Machine Learning on a Quantum Computer".

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Furthermore, researchers investigate more abstract notions of learning theory with respect to quantum information, sometimes referred to as "quantum learning theory".

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Yoo, Seokwon; Bang, Jeongho; Lee, Changhyoup; Lee, Jinhyoung (2014). "A quantum speedup in machine learning: Finding a N-bit Boolean function for a classification".

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A computationally hard problem, which is key for some relevant machine learning tasks, is the estimation of averages over probabilistic models defined in terms of a

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Peddireddy, Dheeraj; Bansal, V.; Jacob, Z.; Aggarwal, V. (2022). "Tensor Ring Parametrized Variational Quantum Circuits for Large Scale Quantum Machine Learning".

2171:"I think we haven't done our homework yet. This is an extremely new scientific field," - physicist Maria Schuld of Canada-based quantum computing startup Xanadu. 1529: 1509: 1293: 1241: 7421:

Neigovzen, Rodion; Neves, Jorge L.; Sollacher, Rudolf; Glaser, Steffen J. (2009). "Quantum pattern recognition with liquid-state nuclear magnetic resonance".

1019:, i.e. quantum-enhanced machine learning. While machine learning algorithms are used to compute immense quantities of data, quantum machine learning utilizes 1786:

devices, without the negative impact of noise, which is possibly incorporated into the circuit parameter, and without the need for quantum error correction.

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Heese, Raoul; Gerlach, Thore; Mücke, Sascha; Müller, Sabine; Jakobs, Matthias; Piatkowski, Nico (22 January 2023). "Explainable Quantum Machine Learning".

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Paparo, Giuseppe Davide; Dunjko, Vedran; Makmal, Adi; Martin-Delgado, Miguel Angel; Briegel, Hans J. (2014). "Quantum Speedup for Active Learning Agents".

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Crawford, Daniel; Levit, Anna; Ghadermarzy, Navid; Oberoi, Jaspreet S.; Ronagh, Pooya (2018). "Reinforcement Learning Using Quantum Boltzmann Machines".

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Li, Ying; Holloway, Gregory W.; Benjamin, Simon C.; Briggs, G. Andrew D.; Baugh, Jonathan; Mol, Jan A. (2017). "A simple and robust quantum memristor".

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Grant, Edward; Benedetti, Marcello; Cao, Shuxiang; Hallam, Andrew; Lockhart, Joshua; Stojevic, Vid; Green, Andrew G.; Severini, Simone (December 2018).

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The starting point in learning theory is typically a concept class, a set of possible concepts. Usually a concept is a function on some domain, such as

1926:, where a quantum state is learned from measurement. Other applications include learning Hamiltonians and automatically generating quantum experiments. 8998: 1922:

The term "quantum machine learning" sometimes refers to classical machine learning performed on data from quantum systems. A basic example of this is

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Wan, Kwok-Ho; Dahlsten, Oscar; Kristjansson, Hler; Gardner, Robert; Kim, Myungshik (2017). "Quantum generalisation of feedforward neural networks".

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Huggins, William; Patel, Piyush; Whaley, K. Birgitta; Stoudenmire, E. Miles (2018-03-30). "Towards Quantum Machine Learning with Tensor Networks".

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Dunjko, Vedran; Briegel, Hans J (2018-06-19). "Machine learning & artificial intelligence in the quantum domain: a review of recent progress".

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Bang, Jeongho; Dutta, Arijit; Lee, Seung-Woo; Kim, Jaewan (2019). "Optimal usage of quantum random access memory in quantum machine learning".

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Rocutto, Lorenzo; Destri, Claudio; Prati, Enrico (2021). "Quantum Semantic Learning by Reverse Annealing of an Adiabatic Quantum Computer".

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routines can be more complex in nature and executed faster on a quantum computer. Furthermore, quantum algorithms can be used to analyze

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Berry, Dominic W.; Childs, Andrew M.; Kothari, Robin (2015). "Hamiltonian simulation with nearly optimal dependence on all parameters".

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Huembeli, Patrick; Dauphin, Alexandre; Wittek, Peter (2018). "Identifying Quantum Phase Transitions with Adversarial Neural Networks".

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Dunjko, Vedran; Friis, Nicolai; Briegel, Hans J. (2015-01-01). "Quantum-enhanced deliberation of learning agents using trapped ions".

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Biamonte, Jacob; Wittek, Peter; Nicola, Pancotti; Rebentrost, Patrick; Wiebe, Nathan; Lloyd, Seth (2017). "Quantum machine learning".

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on the state to reduce it all the way down to one qubit and then processed it in subway. The most frequently used unit type in the

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the classical technique known as LIME (Linear Interpretable Model-Agnostic Explanations) has also been proposed, known as Q-LIME.

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Lloyd, Seth; Mohseni, Masoud; Rebentrost, Patrick (2013). "Quantum algorithms for supervised and unsupervised machine learning".

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Broecker, Peter; Assaad, Fakher F.; Trebst, Simon (2017-07-03). "Quantum phase recognition via unsupervised machine learning".

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Li, Zhaokai; Liu, Xiaomei; Xu, Nanyang; Du, Jiangfeng (2015). "Experimental Realization of a Quantum Support Vector Machine".

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Cory, D. G.; Wiebe, Nathan; Ferrie, Christopher; Granade, Christopher E. (2012-07-06). "Robust Online Hamiltonian Learning".

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Nader, Bshouty H.; Jeffrey, Jackson C. (1999). "Learning DNF over the Uniform Distribution Using a Quantum Example Oracle".

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Amin, Mohammad H.; Andriyash, Evgeny; Rolfe, Jason; Kulchytskyy, Bohdan; Melko, Roger (2018). "Quantum Boltzmann machines".

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A number of quantum algorithms for machine learning are based on the idea of amplitude encoding, that is, to associate the

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Servedio, Rocco A.; Gortler, Steven J. (2004). "Equivalences and Separations Between Quantum and Classical Learnability".

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forms the basis of even the most complex brain networks. Typically, a neuron has two operations: the inner product and an

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Adachi, Steven H.; Henderson, Maxwell P. (2015). "Application of quantum annealing to training of deep neural networks".

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Schuld, Maria; Sinayskiy, Ilya; Petruccione, Francesco (2016). "Prediction by linear regression on a quantum computer".

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Rebentrost, Patrick; Mohseni, Masoud; Lloyd, Seth (2014). "Quantum Support Vector Machine for Big Data Classification".

1211:. One can thus have efficient, spurious-memory-free quantum associative memories for any polynomial number of patterns. 1179:

Correspondingly, they have a superior capacity than classical ones. The number of parameters in the unitary matrix U is

108:, as it is excessively detailed, relies heavily on primary sources, and may not provide sufficient weight to criticisms. 8489: 1015:

The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a

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Harrow, Aram W.; Hassidim, Avinatan; Lloyd, Seth (2008). "Quantum algorithm for solving linear systems of equations".

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Schuld, Maria; Killoran, Nathan (2 March 2022). "Is Quantum Advantage the Right Goal for Quantum Machine Learning?".

6216: 2111:, the quantum support vector machine was implemented to classify the unknown input vector. The readout avoids costly 1560: 234: 74: 3043:

Ghosh, Sanjib; Opala, A.; Matuszewski, M.; Paterek, T.; Liew, Timothy C. H. (2019). "Quantum reservoir processing".

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Schuld, Maria; Sinayskiy, Ilya; Petruccione, Francesco (2014-10-15). "An introduction to quantum machine learning".

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Farhi, Edward; Neven, Hartmut (2018-02-16). "Classification with Quantum Neural Networks on Near Term Processors".

1555:, a quantum speedup may be achieved. Implementations of these kinds of protocols have been proposed for systems of 323: 4056:

Zhao, Zhikuan; Fitzsimons, Jack K.; Fitzsimons, Joseph F. (2019). "Quantum assisted Gaussian process regression".

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Quantum associative memories (in their simplest realization) store patterns in a unitary matrix U acting on the

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Sasaki, Masahide; Carlini, Alberto (6 August 2002). "Quantum learning and universal quantum matching machine".

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Quantum matrix inversion can be applied to machine learning methods in which the training reduces to solving a

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Schuld, Maria; Sinayskiy, Ilya; Petruccione, Francesco (2014). "An introduction to quantum machine learning".

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Arunachalam, Srinivasan; de Wolf, Ronald (2016). "Optimal Quantum Sample Complexity of Learning Algorithms".

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Soklakov, Andrei N.; Schack, Rüdiger (2006). "Efficient state preparation for a register of quantum bits".

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Since 2016, IBM has launched an online cloud-based platform for quantum software developers, called the

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Chen, Samuel Yen-Chi; Yang, Chao-Han Huck; Qi, Jun; Chen, Pin-Yu; Ma, Xiaoli; Goan, Hsi-Sheng (2020).

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Griol-Barres, Israel; Milla, Sergio; Cebrián, Antonio; Mansoori, Yashar; Millet, José (January 2021).

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Dunjko, Vedran; Taylor, Jacob M.; Briegel, Hans J. (2016-09-20). "Quantum-Enhanced Machine Learning".

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itself is now not only a research field but an economically significant and fast growing industry and

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Aïmeur, Esma; Brassard, Gilles; Gambs, Sébastien (1 January 2007). "Quantum clustering algorithms".

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Beer, Kerstin; Müller, Gabriel (2021-12-11). "Dissipative quantum generative adversarial networks".

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Dong, Daoyi; Chen, Chunlin; Li, Hanxiong; Tarn, Tzyh-Jong (2008). "Quantum Reinforcement Learning".

2819: 2684:(2014). "Quantum Algorithms for Nearest-Neighbor Methods for Supervised and Unsupervised Learning". 2619: 1861:, an exciting aspect of HQMMs is that the techniques used show how we can perform quantum-analogous 10142: 10137: 10041: 10026: 10021: 10016: 9786: 9043: 8687: 8636: 8599: 8390: 8323: 8258: 8243: 8213: 8136: 7590:"Parallel photonic information processing at gigabyte per second data rates using transient states" 7341:"NIPS 2009 Demonstration: Binary Classification using Hardware Implementation of Quantum Annealing" 7256: 7198: 6889: 4025: 1666: 1272: 6370:"Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus" 3993:

Lloyd, Seth; Mohseni, Masoud; Rebentrost, Patrick (2014). "Quantum principal component analysis".

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Another approach to improving classical machine learning with quantum information processing uses

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Some research groups have recently explored the use of quantum annealing hardware for training

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Arunachalam, Srinivasan; de Wolf, Ronald (2017-01-24). "A Survey of Quantum Learning Theory".

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to be tested, others have been implemented on small-scale or special purpose quantum devices.

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Mercaldo, F.; Ciaramella, G.; Iadarola, G.; Storto, M.; Martinelli, F.; Santone, A.o (2022).

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Giovannetti, Vittorio; Lloyd, Seth; MacCone, Lorenzo (2008). "Quantum Random Access Memory".

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preparation are known for specific cases, this step easily hides the complexity of the task.

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Knott, Paul (2016-03-22). "A search algorithm for quantum state engineering and metrology".

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environment were realized on a compact and fully tunable integrated nanophotonic processor.

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Monràs, Alex; Sentís, Gael; Wittek, Peter (2017). "Inductive supervised quantum learning".

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Interdisciplinary research area at the intersection of quantum physics and machine learning

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Tezak, Nikolas; Mabuchi, Hideo (2015). "A coherent perceptron for all-optical learning".

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Many quantum machine learning algorithms in this category are based on variations of the

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Arunachalam, Srinivasan; de Wolf, Ronald (2017). "A Survey of Quantum Learning Theory".

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Henderson, Maxwell; Shakya, Samriddhi; Pradhan, Shashindra; Cook, Tristan (2020-02-27).

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Wiebe, Nathan; Braun, Daniel; Lloyd, Seth (2012). "Quantum Algorithm for Data Fitting".

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Unfortunately, classical associative memories are severely limited by the phenomenon of

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network computing, and manage over-fitting. Such process can be accomplished applying

1709:

Quantum analogues or generalizations of classical neural nets are often referred to as

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Going beyond the specific problem of learning states and transformations, the task of

10101: 9989: 9959: 9954: 9867: 9855: 9659: 9311: 9182: 9175: 8917: 8562: 8469: 8426: 8357: 8273: 8253: 8208: 8168: 8146: 7969: 7932: 7875: 7808: 7733: 7627: 7521: 7509: 7460: 7314: 7010: 6902: 6865: 6812: 6689: 6675: 6588: 6543: 6531: 6476: 6464: 6419: 6407: 6297: 6285: 6212: 6169: 6104: 6096: 6061: 6049: 5976: 5919: 5898: 5886: 5837: 5790: 5750: 5738: 5686: 5569: 5516: 5504: 5380: 5247: 5235: 5217: 5146: 5089: 5026: 5014: 4932: 4817: 4698: 4658: 4646: 4617:"Variational Quantum Classifier for Binary Classification: Real vs Synthetic Dataset" 4601: 4589: 4552: 4540: 4487: 4475: 4418: 4252: 4197: 3973: 3924: 3875: 3819: 3807: 3743: 3671: 3611: 3572: 3471: 3351: 3343: 3219: 3167: 3155: 3082: 2893: 2844: 2832: 2657: 2636: 2564: 2517: 2456: 2448: 2324: 2212: 2202: 2159: 1691: 1654: 1577: 1572: 1405:

is one of the tools or algorithms to find patterns. Binary classification is used in

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by reading out the final state in terms of direction (up/down) of the NMR signal.

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Pruning convolution neural network (SqueezeNet) for efficient hardware deployment

5786: 5591: 5261: 5042:"Basic protocols in quantum reinforcement learning with superconducting circuits" 3557: 2828: 2628: 2174:“When mixing machine learning with ‘quantum,’ you catalyse a hype-condensate.” - 1866: 1845:(HMMs), which are typically used to model sequential data in various fields like 1841:

Hidden quantum Markov models (HQMMs) are a quantum-enhanced version of classical

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of a quantum state with the inputs and outputs of computations. Since a state of

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It may require cleanup to comply with Knowledge's content policies, particularly

7588:

Brunner, Daniel; Soriano, Miguel C.; Mirasso, Claudio R.; Fischer, Ingo (2013).

7002: 6434: 6076: 6003: 5856: 5291: 4641: 4616: 4471: 2975: 10096: 10086: 9699: 9664: 9654: 9479: 9237: 9063: 8819: 8796: 8763: 8567: 8444: 8010: 7964: 7947: 7924: 7452: 6857: 6804: 6460: 6369: 6312: 6092: 6037: 5938: 5882: 5372: 5213: 5142: 5075: 4437: 4297: 4140: 4087: 3867: 3457: 3400: 3329: 3151: 2175: 2078: 1854: 1828:, dissipative quantum generative adversarial network (DQGAN) is introduced for 1662: 1367:, for example in least-squares linear regression, the least-squares version of 1075: 894: 854: 834: 804: 784: 734: 700: 550: 540: 333: 8091: 7566: 7265: 7207: 6592: 6526: 6491: 6403: 6346: 6131: 5972: 5923: 5561: 5455: 4928: 4873: 4693: 4676: 4568:"Binary classification of single qubits using quantum machine learning method" 3074: 2889: 2560: 2384: 10208: 10147: 10046: 9644: 9624: 9541: 9220: 8641: 8459: 8385: 7973: 6535: 6468: 6411: 6289: 6173: 6165: 6100: 6053: 5980: 5890: 5841: 5742: 5508: 5221: 5179: 5150: 5018: 4702: 4650: 4593: 4544: 4479: 4422: 3347: 3159: 2897: 2836: 2452: 2222: 2108: 1904: 1865:, which should allow for the general construction of the quantum versions of 1795: 1636: 1416: 1167: 1025: 954: 949: 879: 849: 819: 690: 636: 363: 338: 216:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle } 7759:

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Pattern reorganization is one of the important tasks of machine learning,

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and the quantum algorithm on the currently accessible quantum hardware.

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Trainability of Dissipative Perceptron-Based Quantum Neural Networks

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for future technological implementations. In 2013, Google Research,

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in the number of amplitudes and thereby the dimension of the input.

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is max pooling, although there are other types as well. Similar to

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Future Directions for Intelligent Systems and Information Sciences

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2996: 2944:; Wiebe, Nathan (2020). "Circuit-centric quantum classifiers". 1615: 1084:

Typical classical associative memories store p patterns in the

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Paparo, Giuseppe Davide; Martin-Delgado, Miguel Angel (2012).

4438:"Variational Quantum Circuits for Deep Reinforcement Learning" 3598:. Lecture Notes in Computer Science. Vol. 4013. pp. 2481: 2353:

Trugenberger, Carlo A. (2002). "Quantum Pattern Recognition".

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Bird, Jordan J.; Ekárt, Anikó; Faria, Diego R. (2019-10-28).

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Sentís, Gael; Guţă, Mădălin; Adesso, Gerardo (9 July 2015).

6236: 3042: 2077:

The earliest experiments were conducted using the adiabatic

1622:. As opposed to the activation function, which is typically 1076:

Quantum associative memories and quantum pattern recognition

9546: 8922: 8395: 8328: 6492:"Structure optimization for parameterized quantum circuits" 5937:

Cong, Iris; Choi, Soonwon; Lukin, Mikhail D. (2019-08-26).

4320:"Quantum Computing: The Next Big Thing For Finance By 2024" 2083: 7958:(12). Springer Science and Business Media LLC: 9243–9256. 7179: 7177: 7175: 6708:

Srinivasan, Siddarth; Gordon, Geoff; Boots, Byron (2018).

6001: 5854: 4503:"The theory of the quantum kernel-based binary classifier" 3835: 3695: 3693: 2786: 2586: 1542: 1417:

Quantum machine learning algorithms based on Grover search

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ISCS 2014: Interdisciplinary Symposium on Complex Systems

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exploit the symmetries and the locality structure of the

1039:

of a quantum system or creating new quantum experiments.

8079:"Can quantum machine learning move beyond its own hype?" 8050:"Can quantum machine learning move beyond its own hype?" 7892: 6640: 5396: 4055: 2939: 1378: 7986: 7172: 6932: 6310: 5260: 4161: 3690: 3530: 1893: 7104: 6433:

Hur, Tak; Kim, Leeseok; Park, Daniel K. (2022-02-10).

3992: 3699: 3631: 3629: 3627: 2712: 2058:{\displaystyle \sum _{x}{\sqrt {D(x)}}|x,c(x)\rangle } 1387: 1316:

more than quadratically in the dimension of the matrix

44:

A major contributor to this article appears to have a

6707: 6074: 5639:"Quantum Enhanced Inference in Markov Logic Networks" 3119: 2002: 1981:. For example, the concept class could be the set of 1948: 1815: 1772: 1747: 1719: 1517: 1497: 1456: 1324: 1281: 1249: 1229: 1185: 1134: 1090: 154: 7056: 6075:

Hochreiter, Sepp; Schmidhuber, Jürgen (1997-11-01).

4837: 4835: 4833: 4831: 4767: 4500: 3831: 3829: 3767: 3765: 3095: 1614:

A regular connection of similar components known as

1353:{\displaystyle O{\mathord {\left(n^{2.373}\right)}}} 8366: 6878: 4674: 3888: 3635: 3624: 3589: 2675: 2673: 2241:Ventura, Dan (2000). "Quantum Associative Memory". 2178:

a contributor to the theory of quantum computation.

1491:compared to classical versions of k-medians, where 1360:), but they are not restricted to sparse matrices. 7828:"Quantum Memristors with Superconducting Circuits" 7278: 7220: 7157: 7083:On the Interpretability of Quantum Neural Networks 5476: 4978: 3949: 3509: 2411:"Phase Transitions in Quantum Pattern Recognition" 2072: 2057: 1973: 1761: 1733: 1523: 1503: 1483: 1352: 1287: 1262: 1235: 1203: 1155: 1112: 215: 7392:"NASA Quantum Artificial Intelligence Laboratory" 5419: 5417: 5415: 5392: 5390: 5335: 5333: 5172: 5170: 5168: 4828: 3826: 3762: 3533:"A new Quantum approach to binary classification" 3421: 2542: 2208:Quantum algorithm for linear systems of equations 1992:A natural model of passive learning is Valiant's 1872: 1304:quantum algorithm for linear systems of equations 1215:Linear algebra simulation with quantum amplitudes 10206: 6729: 6203:, Cambridge University Press, pp. 509–518, 5611: 4715: 2679: 2670: 2098:Using a different annealing technology based on 1603:As the depth of the quantum circuit advances on 1598: 8121: 7183: 5315: 4265: 4223:Aaronson, Scott (2015). "Read the fine print". 3771: 3583: 3179: 3177: 1836: 1824:Inspired by the extremely successful classical 1534:Amplitude amplification is often combined with 7945: 6970: 6132:"Quantum Computing in the NISQ era and beyond" 5412: 5387: 5330: 5165: 4894: 4566:Yi, Teng; Wang, Jie; Xu, Fufang (2021-08-01). 4435: 4108: 2543:Schuld, Maria; Petruccione, Francesco (2018). 1912:Classical learning applied to quantum problems 1629: 9916: 8992: 8107: 6772: 6240:"Noisy intermediate-scale quantum algorithms" 5936: 5636: 5311: 5309: 4709: 3297: 2733: 1994:probably approximately correct (PAC) learning 982: 9006: 7644: 7241: 6605:: CS1 maint: multiple names: authors list ( 5766: 4677:"Quantum speed-up for unsupervised learning" 3238: 3174: 2408: 2352: 2288: 2052: 1962: 1949: 1756: 1728: 1396: 1063:Quantum-enhanced machine learning refers to 210: 184: 7697: 7081:Pira, Lirandë; Ferrie, Chris (2024-04-18), 6928: 6926: 4494: 1314:requires a number of operations that grows 10187: 9923: 9909: 8999: 8985: 8114: 8100: 7528: 7080: 6559: 6432: 5637:Wittek, Peter; Gogolin, Christian (2017). 5306: 4340:"Classical Computing vs Quantum Computing" 4049: 2918: 2545:Supervised Learning with Quantum Computers 1929: 1698: 1275:grow polynomially in the number of qubits 1170:of n qubits. Retrieval is realized by the 989: 975: 9930: 8000: 7963: 7906: 7869: 7843: 7802: 7776: 7711: 7658: 7621: 7548: 7487: 7473: 7434: 7292: 7255: 7226: 7197: 7163: 7118: 7090: 7066: 7041: 6984: 6938: 6888: 6839: 6786: 6757: 6747: 6661: 6651: 6625: 6565: 6525: 6507: 6450: 6385: 6368:Zhao, Chen; Gao, Xiao-Shan (2021-06-04). 6328: 6255: 6147: 6019: 5954: 5872: 5776: 5724: 5680: 5654: 5621: 5543: 5490: 5437: 5403: 5354: 5321: 5281: 5229: 5195: 5124: 5083: 5057: 4992: 4963: 4918: 4908: 4855: 4811: 4785: 4752: 4731: 4692: 4640: 4583: 4559: 4518: 4453: 4412: 4402: 4369: 4279: 4175: 4122: 4069: 4024: 4006: 3959: 3902: 3849: 3785: 3713: 3649: 3566: 3556: 3515: 3494: 3465: 3439: 3390: 3337: 3311: 3258: 3197: 3133: 3101: 3056: 3011: 2957: 2924: 2871: 2818: 2800: 2747: 2718: 2697: 2618: 2600: 2495: 2426: 2366: 2302: 2254: 75:Learn how and when to remove this message 6923: 6129: 5710: 4565: 4222: 1048: 1043:classical deep learning and vice versa. 8632:Continuous-variable quantum information 6710:"Learning Hidden Quantum Markov Models" 6367: 6197:"Experimental quantum error correction" 5939:"Quantum convolutional neural networks" 5713:Journal of Computer and System Sciences 2240: 2088:Universities Space Research Association 1543:Quantum-enhanced reinforcement learning 1059:Machine learning with quantum computers 14: 10207: 5815: 5813: 5762: 5760: 5706: 5704: 5702: 5700: 5039: 3115: 3113: 2649: 9904: 8980: 8095: 6732:"Quantum learning of coherent states" 6703: 6701: 6699: 6555: 6553: 6194: 4670: 4668: 4572:Journal of Physics: Conference Series 3592:"Machine Learning in a Quantum World" 3244: 3183: 2912: 2686:Quantum Information & Computation 1379:Variational quantum algorithms (VQAs) 9837:Generative adversarial network (GAN) 3488: 2582: 2580: 2404: 2402: 2348: 2346: 2284: 2282: 1894:Explainable quantum machine learning 1566: 86: 29: 5930: 5911: 5810: 5757: 5697: 4746: 3596:Advances in Artificial Intelligence 3110: 2780: 2092:Quantum Artificial Intelligence Lab 1826:Generative adversarial network(GAN) 1388:Variational quantum circuits (VQCs) 1271:formulate quantum algorithms whose 1033:machine learning of quantum systems 24: 6953: 6696: 6580: 6550: 6313:"Hierarchical quantum classifiers" 4665: 4378: 4357: 3291: 2547:. Quantum Science and Technology. 2409:Trugenberger, C. A. (2002-12-19). 1816:Dissipative Quantum Neural Network 1773:Quantum Convolution Neural Network 521:Sum-over-histories (path integral) 207: 181: 137:Part of a series of articles about 25: 10241: 7398:. 31 January 2017. Archived from 5616:(2014). "Quantum deep learning". 2577: 2399: 2343: 2279: 1511:is the number of data points and 1295:, which amounts to a logarithmic 158: 10186: 9875: 9874: 9854: 8961: 8960: 8951: 8950: 8071: 8042: 8017: 7980: 7939: 7886: 7819: 7752: 7691: 7638: 7581: 7467: 7414: 7384: 7358: 7333: 7272: 7235: 7214: 7151: 7098: 1484:{\displaystyle O({\sqrt {n/k}})} 91: 55:. Please discuss further on the 34: 7677:10.1140/epjqt/s40507-015-0023-3 7074: 7050: 7017: 6964: 6947: 6872: 6819: 6766: 6759:10.1140/epjqt/s40507-015-0030-4 6723: 6634: 6613: 6574: 6483: 6426: 6361: 6304: 6230: 6188: 6123: 6068: 5995: 5905: 5848: 5630: 5612:Wiebe, Nathan; Kapoor, Ashish; 5605: 5576: 5523: 5470: 5254: 5033: 4972: 4951: 4888: 4761: 4740: 4716:Wiebe, Nathan; Kapoor, Ashish; 4585:10.1088/1742-6596/2006/1/012020 4429: 4332: 4312: 4259: 4216: 4155: 4102: 3986: 3943: 3882: 3524: 3503: 3482: 3415: 3370: 3089: 3036: 2990: 2940:Schuld, Maria; Bocharov, Alex; 2933: 2851: 2727: 2706: 2680:Wiebe, Nathan; Kapoor, Ashish; 2073:Implementations and experiments 9787:Recurrent neural network (RNN) 9777:Differentiable neural computer 7730:10.1103/PhysRevLett.114.110504 7506:10.1103/PhysRevLett.114.140504 7366:"Google Quantum A.I. Lab Team" 7346:. Static.googleusercontent.com 7311:10.1103/PhysRevLett.118.190503 7137:10.1088/1367-2630/14/10/103013 6957:Interpretable Machine Learning 5113:Quantum Science and Technology 4537:10.1016/j.physleta.2020.126422 4194:10.1103/PhysRevLett.100.160501 3921:10.1103/PhysRevLett.103.150502 3804:10.1103/PhysRevLett.109.050505 3732:10.1103/PhysRevLett.113.130503 3668:10.1103/PhysRevLett.117.130501 3379:Quantum Science and Technology 3300:Reports on Progress in Physics 3216:10.1103/PhysRevLett.116.090405 2766:10.1088/1367-2630/16/10/103014 2643: 2536: 2475: 2355:Quantum Information Processing 2234: 2049: 2043: 2030: 2024: 2018: 1873:Fully quantum machine learning 1867:probabilistic graphical models 1749: 1721: 1478: 1460: 1436:k-nearest neighbors algorithms 1198: 1189: 1174:of a fixed initial state to a 1150: 1144: 1107: 1094: 671:Relativistic quantum mechanics 203: 196: 177: 13: 1: 9832:Variational autoencoder (VAE) 9792:Long short-term memory (LSTM) 9059:Computational learning theory 8627:Adiabatic quantum computation 6130:Preskill, John (2018-08-06). 5834:10.1016/S0020-0255(00)00056-6 5479:Advanced Quantum Technologies 5011:10.1088/1367-2630/17/2/023006 4770:"Google in a Quantum Network" 3277:10.1088/1367-2630/18/7/073033 3000:Advanced Quantum Technologies 2445:10.1103/physrevlett.89.277903 2321:10.1103/PhysRevLett.87.067901 2265:10.1016/S0020-0255(99)00101-2 2228: 2149: 1937:computational learning theory 1688:probabilistic graphical model 1599:NISQ Circuit as Quantum Model 711:Quantum statistical mechanics 10225:Theoretical computer science 9812:Convolutional neural network 8678:Topological quantum computer 8056:. 2020-05-04. Archived from 6672:10.1007/978-3-319-10759-2_16 6439:Quantum Machine Intelligence 6274:10.1103/revmodphys.94.015004 6209:10.1017/cbo9781139034807.023 5861:Quantum Machine Intelligence 5787:10.1007/978-3-7908-1856-7_11 5590:. 2014-01-13. Archived from 3558:10.1371/journal.pone.0216224 2829:10.1080/00107514.2014.964942 2629:10.1080/00107514.2014.964942 1837:Hidden quantum Markov models 7: 10220:Quantum information science 9807:Multilayer perceptron (MLP) 8956:Quantum information science 8123:Quantum information science 7003:10.1103/PRXQuantum.3.030101 5292:10.1016/j.parco.2016.11.002 4642:10.1109/ACCESS.2021.3139323 4472:10.1109/ACCESS.2020.3010470 2976:10.1103/PhysRevA.101.032308 2191: 1974:{\displaystyle \{0,1\}^{n}} 1918:Machine learning in physics 1851:natural language processing 1630:Quantum sampling techniques 1531:is the number of clusters. 1028:instead of classical data. 681:Quantum information science 104:to comply with Knowledge's 10: 10246: 10170:Thermoacoustic heat engine 9883:Artificial neural networks 9797:Gated recurrent unit (GRU) 9023:Differentiable programming 8351:quantum gate teleportation 8011:10.1007/s10994-023-06490-y 7965:10.1007/s00500-019-04450-0 7925:10.1103/PhysRevB.96.075446 7453:10.1103/PhysRevA.79.042321 6858:10.1103/PhysRevA.81.032324 6805:10.1103/PhysRevA.66.022303 6461:10.1007/s42484-021-00061-x 6195:Bacon, Dave (2013-09-12), 6093:10.1162/neco.1997.9.8.1735 6038:10.1103/physreva.98.032309 6004:"Quantum circuit learning" 5883:10.1007/s42484-020-00012-y 5373:10.1103/PhysRevA.94.022308 5214:10.1038/s41586-021-03242-7 5076:10.1038/s41598-017-01711-6 4298:10.1103/PhysRevA.99.012326 4141:10.1103/PhysRevA.73.012307 4088:10.1103/PhysRevA.99.052331 3868:10.1103/PhysRevA.94.022342 3458:10.1038/s41467-018-07520-3 3152:10.1103/PhysRevB.97.134109 2198:Differentiable programming 2100:nuclear magnetic resonance 1915: 1762:{\displaystyle |1\rangle } 1734:{\displaystyle |0\rangle } 1702: 1570: 1371:, and Gaussian processes. 1365:linear system of equations 1156:{\displaystyle p\leq O(n)} 10182: 10155:Immersive virtual reality 10115: 9945: 9938: 9850: 9764: 9708: 9637: 9570: 9442: 9342: 9335: 9289: 9253: 9216:Artificial neural network 9196: 9072: 9039:Automatic differentiation 9012: 8946: 8889: 8852: 8818: 8795: 8762: 8753: 8686: 8615: 8553: 8513: 8480:Quantum Fourier transform 8425: 8376:Post-quantum cryptography 8319:Entanglement distillation 8292: 8201: 8129: 7567:10.1038/s41534-017-0032-4 7266:10.1137/S0097539795293123 7244:SIAM Journal on Computing 7208:10.1137/S0097539704412910 7186:SIAM Journal on Computing 6527:10.22331/q-2021-01-28-391 6404:10.22331/q-2021-06-04-466 6347:10.1038/s41534-018-0116-9 6244:Reviews of Modern Physics 5973:10.1038/s41567-019-0648-8 5562:10.1103/PhysRevX.8.021050 5456:10.1103/PhysRevX.7.041052 4929:10.1109/TSMCB.2008.925743 4874:10.1103/PhysRevX.4.031002 4723:Quantum Perceptron Models 4694:10.1007/s10994-012-5316-5 3075:10.1038/s41534-019-0149-8 2890:10.1103/PhysRevX.7.041052 2561:10.1007/978-3-319-96424-9 2167:scientists in the field: 1809:conventional feed-forward 1641:probabilistic programming 1397:Quantum binary classifier 10138:Digital scent technology 9044:Neuromorphic engineering 9007:Differentiable computing 8966:Quantum mechanics topics 8661:Quantum machine learning 8637:One-way quantum computer 8490:Quantum phase estimation 8391:Quantum key distribution 8324:Monogamy of entanglement 6201:Quantum Error Correction 6166:10.22331/q-2018-08-06-79 6077:"Long Short-Term Memory" 5143:10.1088/2058-9565/aaef5e 3401:10.1088/2058-9565/aaea94 3330:10.1088/1361-6633/aab406 1924:quantum state tomography 1667:Markov chain Monte Carlo 1561:superconducting circuits 1113:{\displaystyle O(n^{2})} 1035:), such as learning the 1002:Quantum machine learning 716:Quantum machine learning 469:Wheeler's delayed-choice 117:may contain suggestions. 102:may need to be rewritten 18:Quantum Machine Learning 9817:Residual neural network 9233:Artificial Intelligence 8573:Randomized benchmarking 8435:Amplitude amplification 7700:Physical Review Letters 7537:npj Quantum Information 7476:Physical Review Letters 7281:Physical Review Letters 6899:10.1145/1273496.1273497 6317:npj Quantum Information 5135:2019QS&T....4a5014S 4164:Physical Review Letters 3891:Physical Review Letters 3774:Physical Review Letters 3702:Physical Review Letters 3638:Physical Review Letters 3186:Physical Review Letters 3045:npj Quantum Information 2415:Physical Review Letters 2385:10.1023/A:1024022632303 2291:Physical Review Letters 1983:disjunctive normal form 1930:Quantum learning theory 1711:quantum neural networks 1699:Quantum neural networks 1442:and the computation of 1423:amplitude amplification 1369:support vector machines 1243:qubits is described by 426:Leggett–Garg inequality 10160:Magnetic refrigeration 8673:Quantum Turing machine 8666:quantum neural network 8413:Quantum secret sharing 7647:EPJ Quantum Technology 7107:New Journal of Physics 6736:EPJ Quantum Technology 5735:10.1006/jcss.2001.1769 5501:10.1002/qute.202000133 5040:Lamata, Lucas (2017). 4981:New Journal of Physics 3247:New Journal of Physics 3022:10.1002/qute.201800074 2736:New Journal of Physics 2650:Wittek, Peter (2014). 2218:Quantum neural network 2059: 1975: 1763: 1735: 1705:Quantum neural network 1648:Boltzmann distribution 1548:Reinforcement learning 1525: 1505: 1485: 1354: 1289: 1264: 1237: 1205: 1157: 1114: 1055: 1004:is the integration of 217: 10133:Cloak of invisibility 9932:Emerging technologies 9772:Neural Turing machine 9360:Human image synthesis 8745:Entanglement-assisted 8706:quantum convolutional 8381:Quantum coin flipping 8346:Quantum teleportation 8307:entanglement-assisted 8137:DiVincenzo's criteria 7594:Nature Communications 3428:Nature Communications 2060: 1976: 1916:Further information: 1830:unsupervised learning 1764: 1736: 1684:Markov logic networks 1526: 1506: 1486: 1411:unsupervised learning 1403:binary classification 1355: 1290: 1265: 1263:{\displaystyle 2^{n}} 1238: 1206: 1204:{\displaystyle O(pn)} 1176:quantum superposition 1158: 1115: 1081:pattern recognition. 1052: 411:Elitzur–Vaidman 401:Davisson–Germer 218: 53:neutral point of view 9863:Computer programming 9842:Graph neural network 9417:Text-to-video models 9395:Text-to-image models 9243:Large language model 9228:Scientific computing 9034:Statistical manifold 9029:Information geometry 8556:processor benchmarks 8485:Quantum optimization 8368:Quantum cryptography 8179:physical vs. logical 7761:"Quantum memristors" 7043:10.3390/app122312025 5822:Information Sciences 3970:10.1109/FOCS.2015.54 2789:Contemporary Physics 2589:Contemporary Physics 2243:Information Sciences 2095:handwritten digits. 2000: 1946: 1843:Hidden Markov Models 1745: 1717: 1665:techniques, such as 1659:deep neural networks 1592:Schrödinger equation 1515: 1495: 1454: 1322: 1279: 1247: 1227: 1183: 1132: 1088: 676:Quantum field theory 588:Consistent histories 225:Schrödinger equation 152: 10230:Quantum programming 10165:Phased-array optics 10123:Acoustic levitation 9209:In-context learning 9049:Pattern recognition 8269:Quantum speed limit 8164:Quantum programming 8159:Quantum information 7917:2017PhRvB..96g5446L 7854:2017NatSR...742044S 7787:2016NatSR...629507P 7722:2015PhRvL.114k0504C 7669:2015arXiv150101608T 7606:2013NatCo...4.1364B 7559:2017npjQI...3...36W 7498:2015PhRvL.114n0504L 7445:2009PhRvA..79d2321N 7303:2017PhRvL.118s0503M 7129:2012NJPh...14j3013G 6995:2022PRXQ....3c0101S 6954:Molnar, Christoph. 6850:2010PhRvA..81c2324B 6797:2002PhRvA..66b2303S 6518:2021Quant...5..391O 6396:2021Quant...5..466Z 6339:2018npjQI...4...65G 6266:2022RvMP...94a5004B 6158:2018Quant...2...79P 6030:2018PhRvA..98c2309M 5965:2019NatPh..15.1273C 5665:2017NatSR...745672W 5554:2018PhRvX...8b1050A 5448:2017PhRvX...7d1052B 5365:2016PhRvA..94b2308B 5206:2021Natur.591..229S 5068:2017NatSR...7.1609L 5003:2015NJPh...17b3006D 4866:2014PhRvX...4c1002P 4796:2012NatSR...2E.444P 4633:2022IEEEA..10.3705M 4529:2020PhLA..38426422P 4464:2020IEEEA...8n1007C 4404:10.3390/app11146427 4290:2019PhRvA..99a2326B 4237:2015NatPh..11..291A 4186:2008PhRvL.100p0501G 4133:2006PhRvA..73a2307S 4080:2019PhRvA..99e2331Z 4017:2014NatPh..10..631L 3913:2009PhRvL.103o0502H 3860:2016PhRvA..94b2342S 3796:2012PhRvL.109e0505W 3724:2014PhRvL.113m0503R 3660:2016PhRvL.117m0501D 3608:10.1007/11766247_37 3549:2019PLoSO..1416224S 3450:2018NatCo...9.5322C 3322:2018RPPh...81g4001D 3269:2016NJPh...18g3033K 3208:2016PhRvL.116i0405K 3144:2018PhRvB..97m4109H 3067:2019npjQI...5...35G 2968:2020PhRvA.101c2308S 2882:2017PhRvX...7d1052B 2811:2015ConPh..56..172S 2758:2014NJPh...16j3014Y 2611:2015ConPh..56..172S 2553:2018slqc.book.....S 2514:10.1038/nature23474 2506:2017Natur.549..195B 2437:2002PhRvL..89A7903T 2377:2002QuIP....1..471T 2313:2001PhRvL..87f7901T 1780:deep neural network 1769:in Dirac notation. 1620:activation function 1583:Simulated annealing 1407:supervised learning 464:Stern–Gerlach 261:Classical mechanics 9802:Echo state network 9690:Jürgen Schmidhuber 9385:Facial recognition 9380:Speech recognition 9290:Software libraries 8918:Forest/Rigetti QCS 8654:quantum logic gate 8440:Bernstein–Vazirani 8427:Quantum algorithms 8302:Classical capacity 8186:Quantum processors 8169:Quantum simulation 8083:quantamagazine.org 7832:Scientific Reports 7765:Scientific Reports 7614:10.1038/ncomms2368 7402:on 1 February 2017 6081:Neural Computation 5912:Gaikwad, Akash S. 5643:Scientific Reports 5270:Parallel Computing 5046:Scientific Reports 4774:Scientific Reports 2656:. Academic Press. 2113:quantum tomography 2055: 2012: 1971: 1863:Bayesian inference 1759: 1731: 1655:Boltzmann machines 1521: 1501: 1481: 1350: 1285: 1260: 1233: 1201: 1153: 1110: 1065:quantum algorithms 1056: 1006:quantum algorithms 652:Von Neumann–Wigner 632:Objective-collapse 431:Mach–Zehnder 421:Leggett inequality 416:Franck–Hertz 266:Old quantum theory 213: 10202: 10201: 10178: 10177: 9985:complexity theory 9970:cellular automata 9898: 9897: 9660:Stephen Grossberg 9633: 9632: 8974: 8973: 8885: 8884: 8782:Linear optical QC 8563:Quantum supremacy 8517:complexity theory 8470:Quantum annealing 8421: 8420: 8358:Superdense coding 8147:Quantum computing 7895:Physical Review B 7862:10.1038/srep42044 7795:10.1038/srep29507 7423:Physical Review A 7372:. 31 January 2017 6908:978-1-59593-793-3 6828:Physical Review A 6775:Physical Review A 6681:978-3-319-10759-2 6008:Physical Review A 5949:(12): 1273–1278. 5796:978-3-7908-2470-4 5673:10.1038/srep45672 5538:(21050): 021050. 5426:Physical Review X 5343:Physical Review A 5190:(7849): 229–233. 4844:Physical Review X 4804:10.1038/srep00444 4507:Physics Letters A 4448:: 141007–141024. 4268:Physical Review A 4245:10.1038/nphys3272 4111:Physical Review A 4058:Physical Review A 4035:10.1038/nphys3029 3979:978-1-4673-8191-8 3838:Physical Review A 3617:978-3-540-34628-9 3122:Physical Review B 2946:Physical Review A 2860:Physical Review X 2663:978-0-12-800953-6 2570:978-3-319-96423-2 2490:(7671): 195–202. 2213:Quantum annealing 2203:Quantum computing 2160:quantum computing 2102:(NMR), a quantum 2027: 2003: 1692:first-order logic 1587:Quantum tunneling 1578:Quantum annealing 1573:Quantum annealing 1567:Quantum annealing 1524:{\displaystyle k} 1504:{\displaystyle n} 1476: 1425:methods based on 1288:{\displaystyle n} 1236:{\displaystyle n} 1172:unitary evolution 1037:phase transitions 999: 998: 706:Scattering theory 686:Quantum computing 459:Schrödinger's cat 391:Bell's inequality 199: 174: 143:Quantum mechanics 132: 131: 106:quality standards 85: 84: 77: 48:with its subject. 16:(Redirected from 10237: 10215:Machine learning 10190: 10189: 10067:machine learning 10042:key distribution 10027:image processing 10017:error correction 9943: 9942: 9925: 9918: 9911: 9902: 9901: 9888:Machine learning 9878: 9877: 9858: 9613:Action selection 9603:Self-driving car 9410:Stable Diffusion 9375:Speech synthesis 9340: 9339: 9204:Machine learning 9080:Gradient descent 9001: 8994: 8987: 8978: 8977: 8964: 8963: 8954: 8953: 8760: 8759: 8690:error correction 8619:computing models 8585:Relaxation times 8475:Quantum counting 8364: 8363: 8312:quantum capacity 8259:No-teleportation 8244:No-communication 8116: 8109: 8102: 8093: 8092: 8087: 8086: 8075: 8069: 8068: 8066: 8065: 8046: 8040: 8039: 8037: 8035: 8021: 8015: 8014: 8004: 7995:(3): 1189–1217. 7989:Machine Learning 7984: 7978: 7977: 7967: 7943: 7937: 7936: 7910: 7890: 7884: 7883: 7873: 7847: 7838:(42044): 42044. 7823: 7817: 7816: 7806: 7780: 7756: 7750: 7749: 7715: 7695: 7689: 7688: 7662: 7642: 7636: 7635: 7625: 7585: 7579: 7578: 7552: 7532: 7526: 7525: 7491: 7471: 7465: 7464: 7438: 7418: 7412: 7411: 7409: 7407: 7388: 7382: 7381: 7379: 7377: 7362: 7356: 7355: 7353: 7351: 7345: 7337: 7331: 7330: 7296: 7276: 7270: 7269: 7259: 7250:(3): 1136–1153. 7239: 7233: 7232: 7230: 7218: 7212: 7211: 7201: 7192:(5): 1067–1092. 7181: 7170: 7169: 7167: 7155: 7149: 7148: 7122: 7102: 7096: 7095: 7094: 7078: 7072: 7071: 7070: 7054: 7048: 7047: 7045: 7030:Applied Sciences 7021: 7015: 7014: 6988: 6968: 6962: 6961: 6951: 6945: 6944: 6942: 6930: 6921: 6920: 6892: 6883:. pp. 1–8. 6876: 6870: 6869: 6843: 6823: 6817: 6816: 6790: 6788:quant-ph/0202173 6770: 6764: 6763: 6761: 6751: 6727: 6721: 6720: 6714: 6705: 6694: 6693: 6665: 6655: 6638: 6632: 6631: 6629: 6617: 6611: 6610: 6604: 6596: 6578: 6572: 6571: 6569: 6557: 6548: 6547: 6529: 6511: 6487: 6481: 6480: 6454: 6430: 6424: 6423: 6389: 6365: 6359: 6358: 6332: 6308: 6302: 6301: 6259: 6234: 6228: 6227: 6226: 6225: 6192: 6186: 6185: 6151: 6127: 6121: 6120: 6087:(8): 1735–1780. 6072: 6066: 6065: 6023: 5999: 5993: 5992: 5958: 5934: 5928: 5927: 5909: 5903: 5902: 5876: 5852: 5846: 5845: 5828:(3–4): 257–269. 5817: 5808: 5807: 5780: 5764: 5755: 5754: 5728: 5726:quant-ph/0201144 5708: 5695: 5694: 5684: 5658: 5649:(45672): 45672. 5634: 5628: 5627: 5625: 5614:Svore, Krysta M. 5609: 5603: 5602: 5600: 5599: 5580: 5574: 5573: 5547: 5527: 5521: 5520: 5494: 5474: 5468: 5467: 5441: 5421: 5410: 5409: 5407: 5394: 5385: 5384: 5358: 5337: 5328: 5327: 5325: 5313: 5304: 5303: 5285: 5258: 5252: 5251: 5233: 5199: 5174: 5163: 5162: 5128: 5107: 5098: 5097: 5087: 5061: 5037: 5031: 5030: 4996: 4976: 4970: 4969: 4967: 4955: 4949: 4948: 4922: 4912: 4903:(5): 1207–1220. 4892: 4886: 4885: 4859: 4839: 4826: 4825: 4815: 4789: 4765: 4759: 4758: 4756: 4744: 4738: 4737: 4735: 4718:Svore, Krysta M. 4713: 4707: 4706: 4696: 4681:Machine Learning 4672: 4663: 4662: 4644: 4612: 4606: 4605: 4587: 4563: 4557: 4556: 4522: 4498: 4492: 4491: 4457: 4433: 4427: 4426: 4416: 4406: 4391:Applied Sciences 4382: 4376: 4375: 4373: 4361: 4355: 4354: 4352: 4351: 4336: 4330: 4329: 4327: 4326: 4316: 4310: 4309: 4283: 4263: 4257: 4256: 4220: 4214: 4213: 4179: 4159: 4153: 4152: 4126: 4124:quant-ph/0408045 4106: 4100: 4099: 4073: 4053: 4047: 4046: 4028: 4010: 3990: 3984: 3983: 3963: 3947: 3941: 3940: 3906: 3886: 3880: 3879: 3853: 3833: 3824: 3823: 3789: 3769: 3760: 3759: 3717: 3697: 3688: 3687: 3653: 3633: 3622: 3621: 3587: 3581: 3580: 3570: 3560: 3528: 3522: 3521: 3519: 3507: 3501: 3500: 3498: 3486: 3480: 3479: 3469: 3443: 3419: 3413: 3412: 3394: 3374: 3368: 3367: 3341: 3315: 3295: 3289: 3288: 3262: 3242: 3236: 3235: 3201: 3181: 3172: 3171: 3137: 3117: 3108: 3107: 3105: 3093: 3087: 3086: 3060: 3040: 3034: 3033: 3015: 3006:(7–8): 1800074. 2994: 2988: 2987: 2961: 2937: 2931: 2930: 2928: 2916: 2910: 2909: 2875: 2855: 2849: 2848: 2822: 2804: 2784: 2778: 2777: 2751: 2731: 2725: 2724: 2722: 2710: 2704: 2703: 2701: 2692:(3): 0318–0358. 2677: 2668: 2667: 2647: 2641: 2640: 2622: 2604: 2584: 2575: 2574: 2540: 2534: 2533: 2499: 2479: 2473: 2472: 2430: 2428:quant-ph/0204115 2406: 2397: 2396: 2370: 2368:quant-ph/0210176 2350: 2341: 2340: 2306: 2304:quant-ph/0012100 2286: 2277: 2276: 2258: 2256:quant-ph/9807053 2249:(1–4): 273–296. 2238: 2156:machine learning 2132:IBM Q Experience 2104:Hopfield network 2064: 2062: 2061: 2056: 2033: 2028: 2014: 2011: 1980: 1978: 1977: 1972: 1970: 1969: 1855:density matrices 1768: 1766: 1765: 1760: 1752: 1740: 1738: 1737: 1732: 1724: 1530: 1528: 1527: 1522: 1510: 1508: 1507: 1502: 1490: 1488: 1487: 1482: 1477: 1472: 1464: 1359: 1357: 1356: 1351: 1349: 1348: 1347: 1343: 1342: 1312:matrix inversion 1294: 1292: 1291: 1286: 1269: 1267: 1266: 1261: 1259: 1258: 1242: 1240: 1239: 1234: 1210: 1208: 1207: 1202: 1162: 1160: 1159: 1154: 1119: 1117: 1116: 1111: 1106: 1105: 1070:quantum computer 1017:quantum computer 1010:machine learning 991: 984: 977: 618:Superdeterminism 271:Bra–ket notation 222: 220: 219: 214: 206: 201: 200: 192: 180: 175: 173: 162: 134: 133: 127: 124: 118: 95: 87: 80: 73: 69: 66: 60: 46:close connection 38: 37: 30: 21: 10245: 10244: 10240: 10239: 10238: 10236: 10235: 10234: 10205: 10204: 10203: 10198: 10174: 10111: 10022:finite automata 9934: 9929: 9899: 9894: 9846: 9760: 9726:Google DeepMind 9704: 9670:Geoffrey Hinton 9629: 9566: 9492:Project Debater 9438: 9336:Implementations 9331: 9285: 9249: 9192: 9134:Backpropagation 9068: 9054:Tensor calculus 9008: 9005: 8975: 8970: 8942: 8892: 8881: 8854:Superconducting 8848: 8814: 8805:Neutral atom QC 8797:Ultracold atoms 8791: 8756:implementations 8755: 8749: 8689: 8682: 8649:Quantum circuit 8617: 8611: 8605: 8595: 8555: 8549: 8516: 8509: 8465:Hidden subgroup 8417: 8406:other protocols 8362: 8339:quantum network 8334:Quantum channel 8294: 8288: 8234:No-broadcasting 8224:Gottesman–Knill 8197: 8125: 8120: 8090: 8077: 8076: 8072: 8063: 8061: 8048: 8047: 8043: 8033: 8031: 8023: 8022: 8018: 7985: 7981: 7944: 7940: 7891: 7887: 7824: 7820: 7771:(2016): 29507. 7757: 7753: 7696: 7692: 7643: 7639: 7586: 7582: 7533: 7529: 7472: 7468: 7419: 7415: 7405: 7403: 7390: 7389: 7385: 7375: 7373: 7364: 7363: 7359: 7349: 7347: 7343: 7339: 7338: 7334: 7277: 7273: 7240: 7236: 7219: 7215: 7182: 7173: 7156: 7152: 7103: 7099: 7079: 7075: 7055: 7051: 7022: 7018: 6969: 6965: 6952: 6948: 6931: 6924: 6909: 6877: 6873: 6824: 6820: 6771: 6767: 6728: 6724: 6712: 6706: 6697: 6682: 6663:10.1.1.749.3332 6639: 6635: 6618: 6614: 6598: 6597: 6579: 6575: 6558: 6551: 6488: 6484: 6431: 6427: 6366: 6362: 6309: 6305: 6235: 6231: 6223: 6221: 6219: 6193: 6189: 6128: 6124: 6073: 6069: 6000: 5996: 5935: 5931: 5910: 5906: 5853: 5849: 5818: 5811: 5797: 5778:10.1.1.683.5972 5765: 5758: 5709: 5698: 5635: 5631: 5610: 5606: 5597: 5595: 5582: 5581: 5577: 5528: 5524: 5475: 5471: 5422: 5413: 5395: 5388: 5338: 5331: 5314: 5307: 5259: 5255: 5175: 5166: 5108: 5101: 5038: 5034: 4977: 4973: 4956: 4952: 4920:10.1.1.243.5369 4893: 4889: 4840: 4829: 4766: 4762: 4745: 4741: 4714: 4710: 4673: 4666: 4613: 4609: 4564: 4560: 4499: 4495: 4434: 4430: 4383: 4379: 4362: 4358: 4349: 4347: 4338: 4337: 4333: 4324: 4322: 4318: 4317: 4313: 4264: 4260: 4221: 4217: 4160: 4156: 4107: 4103: 4054: 4050: 3991: 3987: 3980: 3948: 3944: 3887: 3883: 3834: 3827: 3770: 3763: 3698: 3691: 3634: 3625: 3618: 3588: 3584: 3543:(5): e0216224. 3529: 3525: 3508: 3504: 3487: 3483: 3420: 3416: 3375: 3371: 3296: 3292: 3243: 3239: 3182: 3175: 3118: 3111: 3094: 3090: 3041: 3037: 2995: 2991: 2938: 2934: 2917: 2913: 2856: 2852: 2820:10.1.1.740.5622 2785: 2781: 2732: 2728: 2711: 2707: 2678: 2671: 2664: 2648: 2644: 2620:10.1.1.740.5622 2585: 2578: 2571: 2541: 2537: 2480: 2476: 2407: 2400: 2351: 2344: 2287: 2280: 2239: 2235: 2231: 2194: 2152: 2075: 2029: 2013: 2007: 2001: 1998: 1997: 1965: 1961: 1947: 1944: 1943: 1932: 1920: 1914: 1896: 1875: 1839: 1818: 1801:full Tomography 1775: 1748: 1746: 1743: 1742: 1720: 1718: 1715: 1714: 1707: 1701: 1690:generated by a 1632: 1601: 1575: 1569: 1545: 1516: 1513: 1512: 1496: 1493: 1492: 1468: 1463: 1455: 1452: 1451: 1427:Grover's search 1419: 1399: 1390: 1381: 1338: 1334: 1330: 1329: 1328: 1323: 1320: 1319: 1297:time complexity 1280: 1277: 1276: 1254: 1250: 1248: 1245: 1244: 1228: 1225: 1224: 1217: 1184: 1181: 1180: 1133: 1130: 1129: 1101: 1097: 1089: 1086: 1085: 1078: 1061: 995: 966: 965: 964: 729: 721: 720: 666: 665:Advanced topics 658: 657: 656: 608:Hidden-variable 598:de Broglie–Bohm 577: 575:Interpretations 567: 566: 565: 535: 527: 526: 525: 483: 475: 474: 473: 440: 396:CHSH inequality 385: 377: 376: 375: 304:Complementarity 298: 290: 289: 288: 256: 227: 202: 191: 190: 176: 166: 161: 153: 150: 149: 128: 122: 119: 109: 96: 81: 70: 64: 61: 50: 39: 35: 28: 23: 22: 15: 12: 11: 5: 10243: 10233: 10232: 10227: 10222: 10217: 10200: 10199: 10197: 10196: 10183: 10180: 10179: 10176: 10175: 10173: 10172: 10167: 10162: 10157: 10152: 10151: 10150: 10140: 10135: 10130: 10125: 10119: 10117: 10113: 10112: 10110: 10109: 10104: 10099: 10094: 10089: 10084: 10082:neural network 10079: 10074: 10069: 10064: 10059: 10054: 10049: 10044: 10039: 10034: 10029: 10024: 10019: 10014: 10009: 10004: 10003: 10002: 9992: 9987: 9982: 9977: 9972: 9967: 9962: 9957: 9951: 9949: 9940: 9936: 9935: 9928: 9927: 9920: 9913: 9905: 9896: 9895: 9893: 9892: 9891: 9890: 9885: 9872: 9871: 9870: 9865: 9851: 9848: 9847: 9845: 9844: 9839: 9834: 9829: 9824: 9819: 9814: 9809: 9804: 9799: 9794: 9789: 9784: 9779: 9774: 9768: 9766: 9762: 9761: 9759: 9758: 9753: 9748: 9743: 9738: 9733: 9728: 9723: 9718: 9712: 9710: 9706: 9705: 9703: 9702: 9700:Ilya Sutskever 9697: 9692: 9687: 9682: 9677: 9672: 9667: 9665:Demis Hassabis 9662: 9657: 9655:Ian Goodfellow 9652: 9647: 9641: 9639: 9635: 9634: 9631: 9630: 9628: 9627: 9622: 9621: 9620: 9610: 9605: 9600: 9595: 9590: 9585: 9580: 9574: 9572: 9568: 9567: 9565: 9564: 9559: 9554: 9549: 9544: 9539: 9534: 9529: 9524: 9519: 9514: 9509: 9504: 9499: 9494: 9489: 9484: 9483: 9482: 9472: 9467: 9462: 9457: 9452: 9446: 9444: 9440: 9439: 9437: 9436: 9431: 9430: 9429: 9424: 9414: 9413: 9412: 9407: 9402: 9392: 9387: 9382: 9377: 9372: 9367: 9362: 9357: 9352: 9346: 9344: 9337: 9333: 9332: 9330: 9329: 9324: 9319: 9314: 9309: 9304: 9299: 9293: 9291: 9287: 9286: 9284: 9283: 9278: 9273: 9268: 9263: 9257: 9255: 9251: 9250: 9248: 9247: 9246: 9245: 9238:Language model 9235: 9230: 9225: 9224: 9223: 9213: 9212: 9211: 9200: 9198: 9194: 9193: 9191: 9190: 9188:Autoregression 9185: 9180: 9179: 9178: 9168: 9166:Regularization 9163: 9162: 9161: 9156: 9151: 9141: 9136: 9131: 9129:Loss functions 9126: 9121: 9116: 9111: 9106: 9105: 9104: 9094: 9089: 9088: 9087: 9076: 9074: 9070: 9069: 9067: 9066: 9064:Inductive bias 9061: 9056: 9051: 9046: 9041: 9036: 9031: 9026: 9018: 9016: 9010: 9009: 9004: 9003: 8996: 8989: 8981: 8972: 8971: 8969: 8968: 8958: 8947: 8944: 8943: 8941: 8940: 8938:many others... 8935: 8930: 8925: 8920: 8911: 8897: 8895: 8887: 8886: 8883: 8882: 8880: 8879: 8874: 8869: 8864: 8858: 8856: 8850: 8849: 8847: 8846: 8841: 8836: 8831: 8825: 8823: 8816: 8815: 8813: 8812: 8810:Trapped-ion QC 8807: 8801: 8799: 8793: 8792: 8790: 8789: 8784: 8779: 8774: 8768: 8766: 8764:Quantum optics 8757: 8751: 8750: 8748: 8747: 8742: 8741: 8740: 8733: 8728: 8723: 8718: 8713: 8708: 8703: 8694: 8692: 8684: 8683: 8681: 8680: 8675: 8670: 8669: 8668: 8658: 8657: 8656: 8646: 8645: 8644: 8634: 8629: 8623: 8621: 8613: 8612: 8610: 8609: 8608: 8607: 8603: 8597: 8593: 8582: 8581: 8580: 8570: 8568:Quantum volume 8565: 8559: 8557: 8551: 8550: 8548: 8547: 8542: 8537: 8532: 8527: 8521: 8519: 8511: 8510: 8508: 8507: 8502: 8497: 8492: 8487: 8482: 8477: 8472: 8467: 8462: 8457: 8452: 8447: 8445:Boson sampling 8442: 8437: 8431: 8429: 8423: 8422: 8419: 8418: 8416: 8415: 8410: 8409: 8408: 8403: 8398: 8388: 8383: 8378: 8372: 8370: 8361: 8360: 8355: 8354: 8353: 8343: 8342: 8341: 8331: 8326: 8321: 8316: 8315: 8314: 8309: 8298: 8296: 8290: 8289: 8287: 8286: 8281: 8279:Solovay–Kitaev 8276: 8271: 8266: 8261: 8256: 8251: 8246: 8241: 8236: 8231: 8226: 8221: 8216: 8211: 8205: 8203: 8199: 8198: 8196: 8195: 8194: 8193: 8183: 8182: 8181: 8171: 8166: 8161: 8156: 8155: 8154: 8144: 8139: 8133: 8131: 8127: 8126: 8119: 8118: 8111: 8104: 8096: 8089: 8088: 8070: 8041: 8016: 7979: 7952:Soft Computing 7938: 7885: 7818: 7751: 7706:(11): 110504. 7690: 7637: 7580: 7527: 7482:(14): 140504. 7466: 7413: 7383: 7357: 7332: 7287:(19): 190503. 7271: 7257:10.1.1.23.5709 7234: 7213: 7199:10.1.1.69.6555 7171: 7150: 7113:(10): 103013. 7097: 7073: 7049: 7016: 6963: 6946: 6922: 6907: 6890:10.1.1.80.9513 6871: 6818: 6765: 6722: 6695: 6680: 6633: 6612: 6573: 6549: 6482: 6425: 6360: 6303: 6229: 6217: 6187: 6122: 6067: 5994: 5943:Nature Physics 5929: 5904: 5847: 5809: 5795: 5756: 5719:(3): 355–383. 5696: 5629: 5604: 5575: 5522: 5485:(2): 2000133. 5469: 5411: 5386: 5329: 5305: 5253: 5164: 5099: 5032: 4971: 4950: 4887: 4827: 4760: 4739: 4708: 4687:(2): 261–287. 4664: 4607: 4558: 4513:(21): 126422. 4493: 4428: 4377: 4356: 4331: 4311: 4258: 4231:(4): 291–293. 4225:Nature Physics 4215: 4170:(16): 160501. 4154: 4101: 4048: 4026:10.1.1.746.480 3995:Nature Physics 3985: 3978: 3942: 3897:(15): 150502. 3881: 3825: 3761: 3708:(13): 130503. 3689: 3644:(13): 130501. 3623: 3616: 3582: 3523: 3502: 3481: 3414: 3369: 3290: 3237: 3173: 3128:(13): 134109. 3109: 3088: 3035: 2989: 2932: 2911: 2850: 2795:(2): 172–185. 2779: 2742:(10): 103014. 2726: 2705: 2669: 2662: 2642: 2595:(2): 172–185. 2576: 2569: 2535: 2474: 2421:(27): 277903. 2398: 2361:(6): 471–493. 2342: 2278: 2232: 2230: 2227: 2226: 2225: 2220: 2215: 2210: 2205: 2200: 2193: 2190: 2189: 2188: 2185: 2182: 2179: 2176:Jacob Biamonte 2172: 2151: 2148: 2074: 2071: 2054: 2051: 2048: 2045: 2042: 2039: 2036: 2032: 2026: 2023: 2020: 2017: 2010: 2006: 1968: 1964: 1960: 1957: 1954: 1951: 1931: 1928: 1913: 1910: 1905:Shapley values 1895: 1892: 1874: 1871: 1838: 1835: 1817: 1814: 1774: 1771: 1758: 1755: 1751: 1730: 1727: 1723: 1703:Main article: 1700: 1697: 1631: 1628: 1600: 1597: 1571:Main article: 1568: 1565: 1553:superpositions 1544: 1541: 1520: 1500: 1480: 1475: 1471: 1467: 1462: 1459: 1418: 1415: 1398: 1395: 1389: 1386: 1380: 1377: 1346: 1341: 1337: 1333: 1327: 1284: 1257: 1253: 1232: 1216: 1213: 1200: 1197: 1194: 1191: 1188: 1152: 1149: 1146: 1143: 1140: 1137: 1109: 1104: 1100: 1096: 1093: 1077: 1074: 1060: 1057: 1026:quantum states 997: 996: 994: 993: 986: 979: 971: 968: 967: 963: 962: 957: 952: 947: 942: 937: 932: 927: 922: 917: 912: 907: 902: 897: 892: 887: 882: 877: 872: 867: 862: 857: 852: 847: 842: 837: 832: 827: 822: 817: 812: 807: 802: 797: 792: 787: 782: 777: 772: 767: 762: 757: 752: 747: 742: 737: 731: 730: 727: 726: 723: 722: 719: 718: 713: 708: 703: 701:Density matrix 698: 693: 688: 683: 678: 673: 667: 664: 663: 660: 659: 655: 654: 649: 644: 639: 634: 629: 624: 623: 622: 621: 620: 605: 600: 595: 590: 585: 579: 578: 573: 572: 569: 568: 564: 563: 558: 553: 548: 543: 537: 536: 533: 532: 529: 528: 524: 523: 518: 513: 508: 503: 498: 492: 491: 490: 484: 481: 480: 477: 476: 472: 471: 466: 461: 455: 454: 453: 452: 451: 449:Delayed-choice 444:Quantum eraser 439: 438: 433: 428: 423: 418: 413: 408: 403: 398: 393: 387: 386: 383: 382: 379: 378: 374: 373: 372: 371: 361: 356: 351: 346: 341: 336: 334:Quantum number 331: 326: 321: 316: 311: 306: 300: 299: 296: 295: 292: 291: 287: 286: 281: 275: 274: 273: 268: 263: 257: 254: 253: 250: 249: 248: 247: 242: 237: 229: 228: 223: 212: 209: 205: 198: 195: 189: 186: 183: 179: 172: 169: 165: 160: 157: 146: 145: 139: 138: 130: 129: 99: 97: 90: 83: 82: 65:September 2018 42: 40: 33: 26: 9: 6: 4: 3: 2: 10242: 10231: 10228: 10226: 10223: 10221: 10218: 10216: 10213: 10212: 10210: 10195: 10194: 10185: 10184: 10181: 10171: 10168: 10166: 10163: 10161: 10158: 10156: 10153: 10149: 10148:Plasma window 10146: 10145: 10144: 10141: 10139: 10136: 10134: 10131: 10129: 10126: 10124: 10121: 10120: 10118: 10114: 10108: 10107:teleportation 10105: 10103: 10100: 10098: 10095: 10093: 10090: 10088: 10085: 10083: 10080: 10078: 10075: 10073: 10070: 10068: 10065: 10063: 10060: 10058: 10055: 10053: 10050: 10048: 10045: 10043: 10040: 10038: 10035: 10033: 10030: 10028: 10025: 10023: 10020: 10018: 10015: 10013: 10010: 10008: 10005: 10001: 9998: 9997: 9996: 9993: 9991: 9988: 9986: 9983: 9981: 9978: 9976: 9973: 9971: 9968: 9966: 9963: 9961: 9958: 9956: 9953: 9952: 9950: 9948: 9944: 9941: 9937: 9933: 9926: 9921: 9919: 9914: 9912: 9907: 9906: 9903: 9889: 9886: 9884: 9881: 9880: 9873: 9869: 9866: 9864: 9861: 9860: 9857: 9853: 9852: 9849: 9843: 9840: 9838: 9835: 9833: 9830: 9828: 9825: 9823: 9820: 9818: 9815: 9813: 9810: 9808: 9805: 9803: 9800: 9798: 9795: 9793: 9790: 9788: 9785: 9783: 9780: 9778: 9775: 9773: 9770: 9769: 9767: 9765:Architectures 9763: 9757: 9754: 9752: 9749: 9747: 9744: 9742: 9739: 9737: 9734: 9732: 9729: 9727: 9724: 9722: 9719: 9717: 9714: 9713: 9711: 9709:Organizations 9707: 9701: 9698: 9696: 9693: 9691: 9688: 9686: 9683: 9681: 9678: 9676: 9673: 9671: 9668: 9666: 9663: 9661: 9658: 9656: 9653: 9651: 9648: 9646: 9645:Yoshua Bengio 9643: 9642: 9640: 9636: 9626: 9625:Robot control 9623: 9619: 9616: 9615: 9614: 9611: 9609: 9606: 9604: 9601: 9599: 9596: 9594: 9591: 9589: 9586: 9584: 9581: 9579: 9576: 9575: 9573: 9569: 9563: 9560: 9558: 9555: 9553: 9550: 9548: 9545: 9543: 9542:Chinchilla AI 9540: 9538: 9535: 9533: 9530: 9528: 9525: 9523: 9520: 9518: 9515: 9513: 9510: 9508: 9505: 9503: 9500: 9498: 9495: 9493: 9490: 9488: 9485: 9481: 9478: 9477: 9476: 9473: 9471: 9468: 9466: 9463: 9461: 9458: 9456: 9453: 9451: 9448: 9447: 9445: 9441: 9435: 9432: 9428: 9425: 9423: 9420: 9419: 9418: 9415: 9411: 9408: 9406: 9403: 9401: 9398: 9397: 9396: 9393: 9391: 9388: 9386: 9383: 9381: 9378: 9376: 9373: 9371: 9368: 9366: 9363: 9361: 9358: 9356: 9353: 9351: 9348: 9347: 9345: 9341: 9338: 9334: 9328: 9325: 9323: 9320: 9318: 9315: 9313: 9310: 9308: 9305: 9303: 9300: 9298: 9295: 9294: 9292: 9288: 9282: 9279: 9277: 9274: 9272: 9269: 9267: 9264: 9262: 9259: 9258: 9256: 9252: 9244: 9241: 9240: 9239: 9236: 9234: 9231: 9229: 9226: 9222: 9221:Deep learning 9219: 9218: 9217: 9214: 9210: 9207: 9206: 9205: 9202: 9201: 9199: 9195: 9189: 9186: 9184: 9181: 9177: 9174: 9173: 9172: 9169: 9167: 9164: 9160: 9157: 9155: 9152: 9150: 9147: 9146: 9145: 9142: 9140: 9137: 9135: 9132: 9130: 9127: 9125: 9122: 9120: 9117: 9115: 9112: 9110: 9109:Hallucination 9107: 9103: 9100: 9099: 9098: 9095: 9093: 9090: 9086: 9083: 9082: 9081: 9078: 9077: 9075: 9071: 9065: 9062: 9060: 9057: 9055: 9052: 9050: 9047: 9045: 9042: 9040: 9037: 9035: 9032: 9030: 9027: 9025: 9024: 9020: 9019: 9017: 9015: 9011: 9002: 8997: 8995: 8990: 8988: 8983: 8982: 8979: 8967: 8959: 8957: 8949: 8948: 8945: 8939: 8936: 8934: 8931: 8929: 8926: 8924: 8921: 8919: 8915: 8912: 8910: 8906: 8902: 8899: 8898: 8896: 8894: 8888: 8878: 8875: 8873: 8870: 8868: 8865: 8863: 8860: 8859: 8857: 8855: 8851: 8845: 8842: 8840: 8837: 8835: 8834:Spin qubit QC 8832: 8830: 8827: 8826: 8824: 8821: 8817: 8811: 8808: 8806: 8803: 8802: 8800: 8798: 8794: 8788: 8785: 8783: 8780: 8778: 8775: 8773: 8770: 8769: 8767: 8765: 8761: 8758: 8752: 8746: 8743: 8739: 8738: 8734: 8732: 8729: 8727: 8724: 8722: 8719: 8717: 8714: 8712: 8709: 8707: 8704: 8702: 8699: 8698: 8696: 8695: 8693: 8691: 8685: 8679: 8676: 8674: 8671: 8667: 8664: 8663: 8662: 8659: 8655: 8652: 8651: 8650: 8647: 8643: 8642:cluster state 8640: 8639: 8638: 8635: 8633: 8630: 8628: 8625: 8624: 8622: 8620: 8614: 8606: 8602: 8598: 8596: 8592: 8588: 8587: 8586: 8583: 8579: 8576: 8575: 8574: 8571: 8569: 8566: 8564: 8561: 8560: 8558: 8552: 8546: 8543: 8541: 8538: 8536: 8533: 8531: 8528: 8526: 8523: 8522: 8520: 8518: 8512: 8506: 8503: 8501: 8498: 8496: 8493: 8491: 8488: 8486: 8483: 8481: 8478: 8476: 8473: 8471: 8468: 8466: 8463: 8461: 8458: 8456: 8453: 8451: 8450:Deutsch–Jozsa 8448: 8446: 8443: 8441: 8438: 8436: 8433: 8432: 8430: 8428: 8424: 8414: 8411: 8407: 8404: 8402: 8399: 8397: 8394: 8393: 8392: 8389: 8387: 8386:Quantum money 8384: 8382: 8379: 8377: 8374: 8373: 8371: 8369: 8365: 8359: 8356: 8352: 8349: 8348: 8347: 8344: 8340: 8337: 8336: 8335: 8332: 8330: 8327: 8325: 8322: 8320: 8317: 8313: 8310: 8308: 8305: 8304: 8303: 8300: 8299: 8297: 8295:communication 8291: 8285: 8282: 8280: 8277: 8275: 8272: 8270: 8267: 8265: 8262: 8260: 8257: 8255: 8252: 8250: 8247: 8245: 8242: 8240: 8237: 8235: 8232: 8230: 8227: 8225: 8222: 8220: 8217: 8215: 8212: 8210: 8207: 8206: 8204: 8200: 8192: 8189: 8188: 8187: 8184: 8180: 8177: 8176: 8175: 8172: 8170: 8167: 8165: 8162: 8160: 8157: 8153: 8150: 8149: 8148: 8145: 8143: 8140: 8138: 8135: 8134: 8132: 8128: 8124: 8117: 8112: 8110: 8105: 8103: 8098: 8097: 8094: 8085:. 2018-01-22. 8084: 8080: 8074: 8060:on 2020-10-27 8059: 8055: 8051: 8045: 8030: 8029:New Scientist 8026: 8020: 8012: 8008: 8003: 7998: 7994: 7990: 7983: 7975: 7971: 7966: 7961: 7957: 7953: 7949: 7942: 7934: 7930: 7926: 7922: 7918: 7914: 7909: 7904: 7901:(7): 075446. 7900: 7896: 7889: 7881: 7877: 7872: 7867: 7863: 7859: 7855: 7851: 7846: 7841: 7837: 7833: 7829: 7822: 7814: 7810: 7805: 7800: 7796: 7792: 7788: 7784: 7779: 7774: 7770: 7766: 7762: 7755: 7747: 7743: 7739: 7735: 7731: 7727: 7723: 7719: 7714: 7709: 7705: 7701: 7694: 7686: 7682: 7678: 7674: 7670: 7666: 7661: 7656: 7652: 7648: 7641: 7633: 7629: 7624: 7619: 7615: 7611: 7607: 7603: 7599: 7595: 7591: 7584: 7576: 7572: 7568: 7564: 7560: 7556: 7551: 7546: 7542: 7538: 7531: 7523: 7519: 7515: 7511: 7507: 7503: 7499: 7495: 7490: 7485: 7481: 7477: 7470: 7462: 7458: 7454: 7450: 7446: 7442: 7437: 7432: 7429:(4): 042321. 7428: 7424: 7417: 7401: 7397: 7393: 7387: 7371: 7367: 7361: 7342: 7336: 7328: 7324: 7320: 7316: 7312: 7308: 7304: 7300: 7295: 7290: 7286: 7282: 7275: 7267: 7263: 7258: 7253: 7249: 7245: 7238: 7229: 7224: 7217: 7209: 7205: 7200: 7195: 7191: 7187: 7180: 7178: 7176: 7166: 7161: 7154: 7146: 7142: 7138: 7134: 7130: 7126: 7121: 7116: 7112: 7108: 7101: 7093: 7088: 7084: 7077: 7069: 7064: 7060: 7053: 7044: 7039: 7036:(23): 12025. 7035: 7031: 7027: 7020: 7012: 7008: 7004: 7000: 6996: 6992: 6987: 6982: 6979:(3): 030101. 6978: 6974: 6967: 6959: 6958: 6950: 6941: 6936: 6929: 6927: 6918: 6914: 6910: 6904: 6900: 6896: 6891: 6886: 6882: 6875: 6867: 6863: 6859: 6855: 6851: 6847: 6842: 6837: 6834:(3): 032324. 6833: 6829: 6822: 6814: 6810: 6806: 6802: 6798: 6794: 6789: 6784: 6781:(2): 022303. 6780: 6776: 6769: 6760: 6755: 6750: 6745: 6741: 6737: 6733: 6726: 6718: 6711: 6704: 6702: 6700: 6691: 6687: 6683: 6677: 6673: 6669: 6664: 6659: 6654: 6649: 6645: 6637: 6628: 6623: 6616: 6608: 6602: 6594: 6590: 6586: 6585: 6577: 6568: 6563: 6556: 6554: 6545: 6541: 6537: 6533: 6528: 6523: 6519: 6515: 6510: 6505: 6501: 6497: 6493: 6486: 6478: 6474: 6470: 6466: 6462: 6458: 6453: 6448: 6444: 6440: 6436: 6429: 6421: 6417: 6413: 6409: 6405: 6401: 6397: 6393: 6388: 6383: 6379: 6375: 6371: 6364: 6356: 6352: 6348: 6344: 6340: 6336: 6331: 6326: 6322: 6318: 6314: 6307: 6299: 6295: 6291: 6287: 6283: 6279: 6275: 6271: 6267: 6263: 6258: 6253: 6250:(1): 015004. 6249: 6245: 6241: 6233: 6220: 6218:9780521897877 6214: 6210: 6206: 6202: 6198: 6191: 6183: 6179: 6175: 6171: 6167: 6163: 6159: 6155: 6150: 6145: 6141: 6137: 6133: 6126: 6118: 6114: 6110: 6106: 6102: 6098: 6094: 6090: 6086: 6082: 6078: 6071: 6063: 6059: 6055: 6051: 6047: 6043: 6039: 6035: 6031: 6027: 6022: 6017: 6014:(3): 032309. 6013: 6009: 6005: 5998: 5990: 5986: 5982: 5978: 5974: 5970: 5966: 5962: 5957: 5952: 5948: 5944: 5940: 5933: 5925: 5921: 5917: 5916: 5908: 5900: 5896: 5892: 5888: 5884: 5880: 5875: 5870: 5866: 5862: 5858: 5851: 5843: 5839: 5835: 5831: 5827: 5823: 5816: 5814: 5806: 5802: 5798: 5792: 5788: 5784: 5779: 5774: 5770: 5763: 5761: 5752: 5748: 5744: 5740: 5736: 5732: 5727: 5722: 5718: 5714: 5707: 5705: 5703: 5701: 5692: 5688: 5683: 5678: 5674: 5670: 5666: 5662: 5657: 5652: 5648: 5644: 5640: 5633: 5624: 5619: 5615: 5608: 5594:on 2014-01-13 5593: 5589: 5585: 5579: 5571: 5567: 5563: 5559: 5555: 5551: 5546: 5541: 5537: 5533: 5526: 5518: 5514: 5510: 5506: 5502: 5498: 5493: 5488: 5484: 5480: 5473: 5465: 5461: 5457: 5453: 5449: 5445: 5440: 5435: 5432:(4): 041052. 5431: 5427: 5420: 5418: 5416: 5406: 5401: 5393: 5391: 5382: 5378: 5374: 5370: 5366: 5362: 5357: 5352: 5349:(2): 022308. 5348: 5344: 5336: 5334: 5324: 5319: 5312: 5310: 5301: 5297: 5293: 5289: 5284: 5279: 5275: 5271: 5267: 5263: 5262:Biswas, Rupak 5257: 5249: 5245: 5241: 5237: 5232: 5227: 5223: 5219: 5215: 5211: 5207: 5203: 5198: 5193: 5189: 5185: 5181: 5173: 5171: 5169: 5160: 5156: 5152: 5148: 5144: 5140: 5136: 5132: 5127: 5122: 5119:(1): 015014. 5118: 5114: 5106: 5104: 5095: 5091: 5086: 5081: 5077: 5073: 5069: 5065: 5060: 5055: 5051: 5047: 5043: 5036: 5028: 5024: 5020: 5016: 5012: 5008: 5004: 5000: 4995: 4990: 4987:(2): 023006. 4986: 4982: 4975: 4966: 4961: 4954: 4946: 4942: 4938: 4934: 4930: 4926: 4921: 4916: 4911: 4906: 4902: 4898: 4891: 4883: 4879: 4875: 4871: 4867: 4863: 4858: 4853: 4850:(3): 031002. 4849: 4845: 4838: 4836: 4834: 4832: 4823: 4819: 4814: 4809: 4805: 4801: 4797: 4793: 4788: 4783: 4779: 4775: 4771: 4764: 4755: 4750: 4743: 4734: 4729: 4725: 4724: 4719: 4712: 4704: 4700: 4695: 4690: 4686: 4682: 4678: 4671: 4669: 4660: 4656: 4652: 4648: 4643: 4638: 4634: 4630: 4627:: 3705–3715. 4626: 4622: 4618: 4611: 4603: 4599: 4595: 4591: 4586: 4581: 4578:(1): 012020. 4577: 4573: 4569: 4562: 4554: 4550: 4546: 4542: 4538: 4534: 4530: 4526: 4521: 4516: 4512: 4508: 4504: 4497: 4489: 4485: 4481: 4477: 4473: 4469: 4465: 4461: 4456: 4451: 4447: 4443: 4439: 4432: 4424: 4420: 4415: 4410: 4405: 4400: 4396: 4392: 4388: 4381: 4372: 4367: 4360: 4345: 4344:GeeksforGeeks 4341: 4335: 4321: 4315: 4307: 4303: 4299: 4295: 4291: 4287: 4282: 4277: 4274:(1): 012326. 4273: 4269: 4262: 4254: 4250: 4246: 4242: 4238: 4234: 4230: 4226: 4219: 4211: 4207: 4203: 4199: 4195: 4191: 4187: 4183: 4178: 4173: 4169: 4165: 4158: 4150: 4146: 4142: 4138: 4134: 4130: 4125: 4120: 4117:(1): 012307. 4116: 4112: 4105: 4097: 4093: 4089: 4085: 4081: 4077: 4072: 4067: 4064:(5): 052331. 4063: 4059: 4052: 4044: 4040: 4036: 4032: 4027: 4022: 4018: 4014: 4009: 4004: 4000: 3996: 3989: 3981: 3975: 3971: 3967: 3962: 3957: 3953: 3946: 3938: 3934: 3930: 3926: 3922: 3918: 3914: 3910: 3905: 3900: 3896: 3892: 3885: 3877: 3873: 3869: 3865: 3861: 3857: 3852: 3847: 3844:(2): 022342. 3843: 3839: 3832: 3830: 3821: 3817: 3813: 3809: 3805: 3801: 3797: 3793: 3788: 3783: 3780:(5): 050505. 3779: 3775: 3768: 3766: 3757: 3753: 3749: 3745: 3741: 3737: 3733: 3729: 3725: 3721: 3716: 3711: 3707: 3703: 3696: 3694: 3685: 3681: 3677: 3673: 3669: 3665: 3661: 3657: 3652: 3647: 3643: 3639: 3632: 3630: 3628: 3619: 3613: 3609: 3605: 3601: 3597: 3593: 3586: 3578: 3574: 3569: 3564: 3559: 3554: 3550: 3546: 3542: 3538: 3534: 3527: 3518: 3513: 3506: 3497: 3492: 3485: 3477: 3473: 3468: 3463: 3459: 3455: 3451: 3447: 3442: 3437: 3433: 3429: 3425: 3418: 3410: 3406: 3402: 3398: 3393: 3388: 3385:(2): 024001. 3384: 3380: 3373: 3365: 3361: 3357: 3353: 3349: 3345: 3340: 3335: 3331: 3327: 3323: 3319: 3314: 3309: 3306:(7): 074001. 3305: 3301: 3294: 3286: 3282: 3278: 3274: 3270: 3266: 3261: 3256: 3253:(7): 073033. 3252: 3248: 3241: 3233: 3229: 3225: 3221: 3217: 3213: 3209: 3205: 3200: 3195: 3192:(9): 090405. 3191: 3187: 3180: 3178: 3169: 3165: 3161: 3157: 3153: 3149: 3145: 3141: 3136: 3131: 3127: 3123: 3116: 3114: 3104: 3099: 3092: 3084: 3080: 3076: 3072: 3068: 3064: 3059: 3054: 3050: 3046: 3039: 3031: 3027: 3023: 3019: 3014: 3009: 3005: 3001: 2993: 2985: 2981: 2977: 2973: 2969: 2965: 2960: 2955: 2952:(3): 032308. 2951: 2947: 2943: 2942:Svore, Krysta 2936: 2927: 2922: 2915: 2907: 2903: 2899: 2895: 2891: 2887: 2883: 2879: 2874: 2869: 2866:(4): 041052. 2865: 2861: 2854: 2846: 2842: 2838: 2834: 2830: 2826: 2821: 2816: 2812: 2808: 2803: 2798: 2794: 2790: 2783: 2775: 2771: 2767: 2763: 2759: 2755: 2750: 2745: 2741: 2737: 2730: 2721: 2716: 2709: 2700: 2695: 2691: 2687: 2683: 2682:Svore, Krysta 2676: 2674: 2665: 2659: 2655: 2654: 2646: 2638: 2634: 2630: 2626: 2621: 2616: 2612: 2608: 2603: 2598: 2594: 2590: 2583: 2581: 2572: 2566: 2562: 2558: 2554: 2550: 2546: 2539: 2531: 2527: 2523: 2519: 2515: 2511: 2507: 2503: 2498: 2493: 2489: 2485: 2478: 2470: 2466: 2462: 2458: 2454: 2450: 2446: 2442: 2438: 2434: 2429: 2424: 2420: 2416: 2412: 2405: 2403: 2394: 2390: 2386: 2382: 2378: 2374: 2369: 2364: 2360: 2356: 2349: 2347: 2338: 2334: 2330: 2326: 2322: 2318: 2314: 2310: 2305: 2300: 2297:(6): 067901. 2296: 2292: 2285: 2283: 2274: 2270: 2266: 2262: 2257: 2252: 2248: 2244: 2237: 2233: 2224: 2223:Quantum image 2221: 2219: 2216: 2214: 2211: 2209: 2206: 2204: 2201: 2199: 2196: 2195: 2186: 2183: 2180: 2177: 2173: 2170: 2169: 2168: 2164: 2161: 2157: 2147: 2143: 2139: 2135: 2133: 2128: 2126: 2120: 2116: 2114: 2110: 2109:feature space 2105: 2101: 2096: 2093: 2090:launched the 2089: 2085: 2080: 2070: 2066: 2046: 2040: 2037: 2034: 2021: 2015: 2008: 2004: 1995: 1990: 1986: 1984: 1966: 1958: 1955: 1952: 1940: 1938: 1927: 1925: 1919: 1909: 1906: 1902: 1891: 1888: 1883: 1879: 1870: 1868: 1864: 1860: 1856: 1852: 1848: 1844: 1834: 1831: 1827: 1822: 1813: 1810: 1806: 1805:pooling layer 1802: 1797: 1796:pooling layer 1791: 1787: 1785: 1781: 1770: 1753: 1725: 1712: 1706: 1696: 1693: 1689: 1685: 1679: 1675: 1671: 1668: 1664: 1660: 1656: 1651: 1649: 1644: 1642: 1638: 1637:deep learning 1627: 1625: 1621: 1617: 1612: 1610: 1606: 1596: 1593: 1588: 1584: 1579: 1574: 1564: 1562: 1558: 1554: 1549: 1540: 1537: 1536:quantum walks 1532: 1518: 1498: 1473: 1469: 1465: 1457: 1447: 1445: 1441: 1437: 1433: 1428: 1424: 1414: 1412: 1408: 1404: 1394: 1385: 1376: 1372: 1370: 1366: 1361: 1344: 1339: 1335: 1331: 1325: 1317: 1313: 1309: 1305: 1300: 1298: 1282: 1274: 1255: 1251: 1230: 1222: 1212: 1195: 1192: 1186: 1177: 1173: 1169: 1168:Hilbert space 1164: 1147: 1141: 1138: 1135: 1127: 1122: 1102: 1098: 1091: 1082: 1073: 1071: 1066: 1051: 1047: 1044: 1040: 1038: 1034: 1029: 1027: 1022: 1018: 1013: 1011: 1007: 1003: 992: 987: 985: 980: 978: 973: 972: 970: 969: 961: 958: 956: 953: 951: 948: 946: 943: 941: 938: 936: 933: 931: 928: 926: 923: 921: 918: 916: 913: 911: 908: 906: 903: 901: 898: 896: 893: 891: 888: 886: 883: 881: 878: 876: 873: 871: 868: 866: 863: 861: 858: 856: 853: 851: 848: 846: 843: 841: 838: 836: 833: 831: 828: 826: 823: 821: 818: 816: 813: 811: 808: 806: 803: 801: 798: 796: 793: 791: 788: 786: 783: 781: 778: 776: 773: 771: 768: 766: 763: 761: 758: 756: 753: 751: 748: 746: 743: 741: 738: 736: 733: 732: 725: 724: 717: 714: 712: 709: 707: 704: 702: 699: 697: 694: 692: 691:Quantum chaos 689: 687: 684: 682: 679: 677: 674: 672: 669: 668: 662: 661: 653: 650: 648: 647:Transactional 645: 643: 640: 638: 637:Quantum logic 635: 633: 630: 628: 625: 619: 616: 615: 614: 611: 610: 609: 606: 604: 601: 599: 596: 594: 591: 589: 586: 584: 581: 580: 576: 571: 570: 562: 559: 557: 554: 552: 549: 547: 544: 542: 539: 538: 531: 530: 522: 519: 517: 514: 512: 509: 507: 504: 502: 499: 497: 494: 493: 489: 486: 485: 479: 478: 470: 467: 465: 462: 460: 457: 456: 450: 447: 446: 445: 442: 441: 437: 434: 432: 429: 427: 424: 422: 419: 417: 414: 412: 409: 407: 404: 402: 399: 397: 394: 392: 389: 388: 381: 380: 370: 367: 366: 365: 364:Wave function 362: 360: 357: 355: 352: 350: 347: 345: 344:Superposition 342: 340: 337: 335: 332: 330: 327: 325: 322: 320: 317: 315: 312: 310: 307: 305: 302: 301: 294: 293: 285: 282: 280: 277: 276: 272: 269: 267: 264: 262: 259: 258: 252: 251: 246: 243: 241: 238: 236: 233: 232: 231: 230: 226: 193: 187: 170: 167: 163: 155: 148: 147: 144: 141: 140: 136: 135: 126: 116: 112: 107: 103: 100:This article 98: 94: 89: 88: 79: 76: 68: 58: 54: 49: 47: 41: 32: 31: 19: 10191: 10128:Anti-gravity 10072:metamaterial 10066: 10000:post-quantum 9995:cryptography 9731:Hugging Face 9695:David Silver 9343:Audio–visual 9197:Applications 9176:Augmentation 9021: 8862:Charge qubit 8787:KLM protocol 8736: 8660: 8600: 8590: 8284:Purification 8214:Eastin–Knill 8082: 8073: 8062:. Retrieved 8058:the original 8053: 8044: 8032:. Retrieved 8028: 8019: 7992: 7988: 7982: 7955: 7951: 7941: 7898: 7894: 7888: 7835: 7831: 7821: 7768: 7764: 7754: 7703: 7699: 7693: 7650: 7646: 7640: 7597: 7593: 7583: 7540: 7536: 7530: 7479: 7475: 7469: 7426: 7422: 7416: 7404:. Retrieved 7400:the original 7395: 7386: 7374:. Retrieved 7369: 7360: 7348:. Retrieved 7335: 7284: 7280: 7274: 7247: 7243: 7237: 7216: 7189: 7185: 7153: 7110: 7106: 7100: 7082: 7076: 7058: 7052: 7033: 7029: 7019: 6976: 6972: 6966: 6956: 6949: 6880: 6874: 6831: 6827: 6821: 6778: 6774: 6768: 6739: 6735: 6725: 6716: 6643: 6636: 6615: 6583: 6576: 6499: 6495: 6485: 6442: 6438: 6428: 6377: 6373: 6363: 6320: 6316: 6306: 6282:10356/161272 6247: 6243: 6232: 6222:, retrieved 6200: 6190: 6139: 6135: 6125: 6084: 6080: 6070: 6011: 6007: 5997: 5946: 5942: 5932: 5914: 5907: 5864: 5860: 5850: 5825: 5821: 5768: 5716: 5712: 5646: 5642: 5632: 5607: 5596:. Retrieved 5592:the original 5587: 5578: 5535: 5532:Phys. Rev. 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482:Formulations 319:Energy level 314:Entanglement 297:Fundamentals 284:Interference 235:Introduction 120: 111:You can help 101: 71: 62: 43: 10143:Force field 10092:programming 10052:logic clock 10037:information 10012:electronics 9879:Categories 9827:Autoencoder 9782:Transformer 9650:Alex Graves 9598:OpenAI Five 9502:IBM Watsonx 9124:Convolution 9102:Overfitting 8893:programming 8872:Phase qubit 8777:Circuit QED 8249:No-deleting 8191:cloud-based 7370:Google Plus 7350:26 November 6973:PRX Quantum 6046:11094/77645 5052:(1): 1609. 4621:IEEE Access 4442:IEEE Access 3434:(1): 5322. 1308:Hamiltonian 935:von Neumann 920:Schrödinger 696:EPR paradox 627:Many-worlds 561:Schrödinger 516:Schrödinger 511:Phase-space 501:Interaction 406:Double-slit 384:Experiments 359:Uncertainty 329:Nonlocality 324:Measurement 309:Decoherence 279:Hamiltonian 10209:Categories 10057:logic gate 9955:algorithms 9868:Technology 9721:EleutherAI 9680:Fei-Fei Li 9675:Yann LeCun 9588:Q-learning 9571:Decisional 9497:IBM Watson 9405:Midjourney 9297:TensorFlow 9144:Activation 9097:Regression 9092:Clustering 8933:libquantum 8867:Flux qubit 8772:Cavity QED 8721:Bacon–Shor 8711:stabilizer 8239:No-cloning 8064:2020-10-27 8002:2108.13329 7908:1612.08409 7845:1603.04487 7778:1511.02192 7660:1501.01608 7550:1612.01045 7543:(36): 36. 7406:31 January 7376:31 January 7294:1605.07541 7228:1607.00932 7165:1701.06806 7092:2308.11098 7068:1602.04938 6986:2203.01340 6940:2301.09138 6593:1228410830 6567:2112.06088 6509:1905.09692 6452:2108.00661 6387:2102.01828 6330:1804.03680 6257:2101.08448 6224:2022-11-23 6149:1801.00862 6021:1803.00745 5956:1810.03787 5924:1197735354 5874:1904.04767 5656:1611.08104 5598:2018-12-07 5588:archive.is 5545:1601.02036 5492:2003.11945 5439:1609.02542 5405:1611.04528 5356:1510.07611 5323:1510.06356 5283:1704.04836 5197:2103.06294 5126:1709.01366 5059:1701.05131 4965:1612.05695 4754:2307.08045 4733:1602.04799 4520:2004.03489 4455:1907.00397 4371:2201.08878 4350:2023-06-17 4325:2023-06-17 4281:1809.04814 4071:1512.03929 4001:(9): 631. 3961:1501.01715 3851:1601.07823 3651:1610.08251 3517:1701.06806 3441:1802.09558 3392:1803.11537 3339:1887/71084 3313:1709.02779 3260:1511.05327 3199:1509.02749 3135:1710.08382 3103:1707.00663 3058:1811.10335 3051:(35): 35. 3013:1808.09241 2959:1804.00633 2926:1802.06002 2873:1609.02542 2497:1611.09347 2229:References 2150:Skepticism 2086:, and the 1887:clustering 1609:perceptron 1440:perceptron 1221:amplitudes 1126:cross-talk 1012:programs. 930:Sommerfeld 845:Heisenberg 840:Gutzwiller 780:de Broglie 728:Scientists 642:Relational 593:Copenhagen 496:Heisenberg 354:Tunnelling 255:Background 10102:simulator 9990:computing 9960:amplifier 9751:MIT CSAIL 9716:Anthropic 9685:Andrew Ng 9583:AlphaZero 9427:VideoPoet 9390:AlphaFold 9327:MindSpore 9281:SpiNNaker 9276:Memristor 9183:Diffusion 9159:Rectifier 9139:Batchnorm 9119:Attention 9114:Adversary 8839:NV center 8274:Threshold 8254:No-hiding 8219:Gleason's 8034:31 August 7974:1432-7643 7933:119454549 7713:1409.7770 7522:119182770 7489:1410.1054 7461:119115625 7436:0802.1592 7252:CiteSeerX 7194:CiteSeerX 7120:1207.1655 7011:247222732 6885:CiteSeerX 6866:119289138 6841:0903.0543 6813:119383508 6749:1410.8700 6690:119226526 6658:CiteSeerX 6653:1406.5847 6627:1406.2661 6601:cite book 6544:231719244 6536:2521-327X 6477:236772493 6469:2524-4906 6420:231786346 6412:2521-327X 6323:(1): 65. 6298:231662441 6290:0034-6861 6174:2521-327X 6101:0899-7667 6062:117542570 6054:2469-9926 5981:1745-2473 5899:104291950 5891:2524-4906 5842:0020-0255 5773:CiteSeerX 5751:206569020 5743:0022-0000 5623:1412.3489 5570:119198869 5517:214667224 5509:2511-9044 5381:118602077 5276:: 81–98. 5248:232185235 5222:1476-4687 5151:2058-9565 5027:119292539 5019:1367-2630 4994:1407.2830 4915:CiteSeerX 4910:0810.3828 4857:1401.4997 4787:1112.2079 4703:0885-6125 4659:245614428 4651:2169-3536 4602:237286847 4594:1742-6588 4553:215238793 4545:0375-9601 4488:195767325 4480:2169-3536 4423:2076-3417 4253:122167250 4177:0708.1879 4021:CiteSeerX 4008:1307.0401 3904:0811.3171 3876:118459345 3820:118439810 3787:1204.5242 3715:1307.0471 3496:1301.3124 3348:0034-4885 3168:125593239 3160:2469-9950 3083:119197635 2898:2160-3308 2845:119263556 2837:0010-7514 2815:CiteSeerX 2802:1409.3097 2749:1303.6055 2720:1307.0411 2699:1401.2142 2637:119263556 2615:CiteSeerX 2602:1409.3097 2453:0031-9007 2125:memristor 2053:⟩ 2005:∑ 1859:Bayes net 1757:⟩ 1729:⟩ 1624:nonlinear 1444:attention 1432:k-medians 1273:resources 1139:≤ 960:Zeilinger 805:Ehrenfest 534:Equations 211:⟩ 208:Ψ 197:^ 185:⟩ 182:Ψ 159:ℏ 123:July 2023 115:talk page 57:talk page 10007:dynamics 9859:Portals 9618:Auto-GPT 9450:Word2vec 9254:Hardware 9171:Datasets 9073:Concepts 8901:OpenQASM 8877:Transmon 8754:Physical 8554:Quantum 8455:Grover's 8229:Holevo's 8202:Theorems 8152:timeline 8142:NISQ era 8054:Protocol 7880:28195193 7813:27381511 7746:44769024 7738:25839250 7685:28568346 7632:23322052 7600:: 1364. 7575:51685660 7514:25910101 7319:28548536 6445:(1): 3. 6355:55479810 6182:44098998 5989:53642483 5691:28422093 5464:55331519 5300:27547901 5240:33692560 5094:28487535 4945:17768796 4937:18784007 4882:54652978 4822:22685626 4720:(2016). 4306:62841090 4202:18518173 4149:17318769 4096:18303333 4043:11553314 3929:19905613 3812:23006156 3748:25302877 3684:12698722 3676:27715099 3577:31071129 3537:PLOS ONE 3476:30552316 3356:29504942 3232:20182586 3224:26991161 3030:85529734 2984:49577148 2906:55331519 2530:64536201 2522:28905917 2469:33065081 2461:12513243 2337:23325931 2329:11497863 2192:See also 1847:robotics 1663:sampling 1434:and the 885:Millikan 810:Einstein 795:Davisson 750:Blackett 735:Aharonov 603:Ensemble 583:Bayesian 488:Overview 369:Collapse 349:Symmetry 240:Glossary 10097:sensing 10077:network 10062:machine 10032:imaging 9980:circuit 9975:channel 9947:Quantum 9741:Meta AI 9578:AlphaGo 9562:PanGu-Σ 9532:ChatGPT 9507:Granite 9455:Seq2seq 9434:Whisper 9355:WaveNet 9350:AlexNet 9322:Flux.jl 9302:PyTorch 9154:Sigmoid 9149:Softmax 9014:General 8891:Quantum 8829:Kane QC 8688:Quantum 8616:Quantum 8545:PostBQP 8515:Quantum 8500:Simon's 8293:Quantum 8130:General 7913:Bibcode 7871:5307327 7850:Bibcode 7804:4933948 7783:Bibcode 7718:Bibcode 7665:Bibcode 7623:3562454 7602:Bibcode 7555:Bibcode 7494:Bibcode 7441:Bibcode 7327:6521971 7299:Bibcode 7145:9928389 7125:Bibcode 6991:Bibcode 6917:4357684 6846:Bibcode 6793:Bibcode 6717:Aistats 6514:Bibcode 6502:: 391. 6496:Quantum 6392:Bibcode 6380:: 466. 6374:Quantum 6335:Bibcode 6262:Bibcode 6154:Bibcode 6136:Quantum 6117:1915014 6109:9377276 6026:Bibcode 5961:Bibcode 5805:9099722 5682:5395824 5661:Bibcode 5550:Bibcode 5444:Bibcode 5361:Bibcode 5231:7612051 5202:Bibcode 5159:2429346 5131:Bibcode 5085:5431677 5064:Bibcode 4999:Bibcode 4862:Bibcode 4813:3370332 4792:Bibcode 4629:Bibcode 4525:Bibcode 4460:Bibcode 4286:Bibcode 4233:Bibcode 4182:Bibcode 4129:Bibcode 4076:Bibcode 4013:Bibcode 3937:5187993 3909:Bibcode 3856:Bibcode 3792:Bibcode 3756:5503025 3720:Bibcode 3656:Bibcode 3600:431–442 3568:6508868 3545:Bibcode 3467:6294148 3446:Bibcode 3409:4531946 3364:3681629 3318:Bibcode 3285:2721958 3265:Bibcode 3204:Bibcode 3140:Bibcode 3063:Bibcode 2964:Bibcode 2878:Bibcode 2807:Bibcode 2774:4956424 2754:Bibcode 2607:Bibcode 2549:Bibcode 2502:Bibcode 2433:Bibcode 2393:1928001 2373:Bibcode 2309:Bibcode 2273:7232952 1901:XAI/XML 1616:neurons 1585:by the 1409:and in 1008:within 925:Simmons 915:Rydberg 880:Moseley 860:Kramers 850:Hilbert 835:Glauber 830:Feynman 815:Everett 785:Compton 556:Rydberg 245:History 10087:optics 9939:Fields 9756:Huawei 9736:OpenAI 9638:People 9608:MuZero 9470:Gemini 9465:Claude 9400:DALL-E 9312:Theano 8909:IBM QX 8905:Qiskit 8844:NMR QC 8822:-based 8726:Steane 8697:Codes 8495:Shor's 8401:SARG04 8209:Bell's 7972: 7931: 7878: 7868: 7811: 7801: 7744: 7736: 7683: 7630: 7620: 7573: 7520: 7512: 7459: 7325: 7317: 7254: 7196: 7143: 7009: 6915: 6905: 6887: 6864: 6811: 6688: 6678: 6660: 6591: 6542: 6534: 6475: 6467: 6418: 6410: 6353: 6296: 6288: 6215: 6180: 6172: 6142:: 79. 6115: 6107: 6099: 6060: 6052: 5987: 5979: 5922: 5897: 5889: 5840: 5803: 5793: 5775: 5749: 5741: 5689: 5679: 5568: 5515: 5507: 5462: 5379: 5298: 5246: 5238: 5228: 5220: 5184:Nature 5157: 5149: 5092: 5082: 5025: 5017: 4943: 4935: 4917: 4880: 4820: 4810: 4701: 4657: 4649: 4600: 4592: 4551: 4543: 4486: 4478: 4421: 4304: 4251: 4210:570390 4208: 4200: 4147: 4094: 4041: 4023: 3976: 3935: 3927: 3874: 3818: 3810: 3754: 3746: 3682: 3674: 3614: 3575: 3565: 3474: 3464: 3407: 3362: 3354: 3346: 3283: 3230: 3222: 3166: 3158: 3081: 3028: 2982: 2904: 2896: 2843: 2835: 2817: 2772: 2660: 2635: 2617: 2567: 2528: 2520: 2484:Nature 2467: 2459: 2451: 2391: 2335: 2327: 2271: 2154:While 2079:D-Wave 1318:(e.g. 1021:qubits 955:Zeeman 950:Wigner 900:Planck 870:Landau 855:Jordan 506:Matrix 436:Popper 113:. The 10116:Other 10047:logic 9822:Mamba 9593:SARSA 9557:LLaMA 9552:BLOOM 9537:GPT-J 9527:GPT-4 9522:GPT-3 9517:GPT-2 9512:GPT-1 9475:LaMDA 9307:Keras 8731:Toric 8174:Qubit 7997:arXiv 7929:S2CID 7903:arXiv 7840:arXiv 7773:arXiv 7742:S2CID 7708:arXiv 7681:S2CID 7655:arXiv 7571:S2CID 7545:arXiv 7518:S2CID 7484:arXiv 7457:S2CID 7431:arXiv 7344:(PDF) 7323:S2CID 7289:arXiv 7223:arXiv 7160:arXiv 7141:S2CID 7115:arXiv 7087:arXiv 7063:arXiv 7007:S2CID 6981:arXiv 6935:arXiv 6913:S2CID 6862:S2CID 6836:arXiv 6809:S2CID 6783:arXiv 6744:arXiv 6742:(1). 6713:(PDF) 6686:S2CID 6648:arXiv 6622:arXiv 6562:arXiv 6540:S2CID 6504:arXiv 6473:S2CID 6447:arXiv 6416:S2CID 6382:arXiv 6351:S2CID 6325:arXiv 6294:S2CID 6252:arXiv 6178:S2CID 6144:arXiv 6113:S2CID 6058:S2CID 6016:arXiv 5985:S2CID 5951:arXiv 5895:S2CID 5869:arXiv 5867:(1). 5801:S2CID 5747:S2CID 5721:arXiv 5651:arXiv 5618:arXiv 5566:S2CID 5540:arXiv 5513:S2CID 5487:arXiv 5460:S2CID 5434:arXiv 5400:arXiv 5377:S2CID 5351:arXiv 5318:arXiv 5296:S2CID 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