Quantum machine learning

Quantum machine learning is an emerging interdisciplinary research area at the intersection of quantum physics and machine learning. The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a quantum computer, i.e. quantum-enhanced machine learning. While machine learning algorithms are used to compute immense quantities of data, quantum machine learning increases such capabilities intelligently, by creating opportunities to conduct analysis on quantum states and systems. This includes hybrid methods that involve both classical and quantum processing, where computationally difficult subroutines are outsourced to a quantum device. These routines can be more complex in nature and executed faster with the assistance of quantum devices. Furthermore, quantum algorithms can be used to analyze quantum states instead of classical data. Beyond quantum computing, the term 'quantum machine learning' is often associated with classical machine learning methods applied to data generated from quantum experiments (i.e. machine learning of quantum systems), such as learning quantum phase transitions or creating new quantum experiments. 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 classical deep learning and vice versa. Finally, researchers investigate more abstract notions of learning theory with respect to quantum information, sometimes referred to as 'quantum learning theory'. Quantum machine learning is an emerging interdisciplinary research area at the intersection of quantum physics and machine learning. The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a quantum computer, i.e. quantum-enhanced machine learning. While machine learning algorithms are used to compute immense quantities of data, quantum machine learning increases such capabilities intelligently, by creating opportunities to conduct analysis on quantum states and systems. This includes hybrid methods that involve both classical and quantum processing, where computationally difficult subroutines are outsourced to a quantum device. These routines can be more complex in nature and executed faster with the assistance of quantum devices. Furthermore, quantum algorithms can be used to analyze quantum states instead of classical data. Beyond quantum computing, the term 'quantum machine learning' is often associated with classical machine learning methods applied to data generated from quantum experiments (i.e. machine learning of quantum systems), such as learning quantum phase transitions or creating new quantum experiments. 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 classical deep learning and vice versa. Finally, researchers investigate more abstract notions of learning theory with respect to quantum information, sometimes referred to as 'quantum learning theory'. Quantum-enhanced machine learning refers to quantum algorithms 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 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 quantum computer to be tested, others have been implemented on small-scale or special purpose quantum devices. A number of quantum algorithms for machine learning are based on the idea of amplitude encoding, that is, to associate the amplitudes of a quantum state with the inputs and outputs of computations. Since a state of n {displaystyle n} qubits is described by 2 n {displaystyle 2^{n}} 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 formulate quantum algorithms whose resources grow polynomially in the number of qubits n {displaystyle n} , which amounts to a logarithmic growth in the number of amplitudes and thereby the dimension of the input. Many quantum machine learning algorithms in this category are based on variations of the quantum algorithm for linear systems of equations (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 Hamiltonian which entrywise 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 matrix inversion requires a number of operations that grows at least quadratically in the dimension of the matrix. Quantum matrix inversion can be applied to machine learning methods in which the training reduces to solving a linear system of equations, for example in least-squares linear regression, the least-squares version of support vector machines, and Gaussian processes. 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 preparation are known for specific cases, this step easily hides the complexity of the task. Another approach to improving classical machine learning with quantum information processing uses amplitude amplification methods based on Grover's search 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 k-medians and the k-nearest neighbors algorithms. Another application is a quadratic speedup in the training of perceptron. Amplitude amplification is often combined with quantum walks 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. Reinforcement learning 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 environment and occasionally receives rewards for its actions, which allows the 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 superpositions, a quantum speedup may be achieved. Implementations of these kinds of protocols in superconducting circuits and in systems of trapped ions have been proposed.