The archaeological evidence indicates that humans have been counting for about 50,000 years (Eves (1983)). Since 300 B.C. when early counting tools such as abacuses were used, computing machines have gone through a long path, however, the major breakthrough happened in the 1950s with developments in the semiconductor industry, which led to the invention of transistors Shockley et al. (1956). This revolutionized the computing industry and became the building block of standard computers and other digital devices and consequently human beings eventually entered the digital era. However, the transistor industry soon came to realize a fundamental question of how much the number of transistors in a dense integrated circuit (IC) can be grown, which was first addressed by Gordon Moore Moore (1964).
Kandala et al. (2017). In addition, quantum neural networks (QNNs, Farhi and Neven (2018)), and quantum convolutional neural networks (QCNN, Cong et al. (2019)) are also among the most interesting examples of such novel optimization and inference tools. While these variational algorithms on PQCs have shown great promise, the random parameters initializing the circuit become undesirable because of the exponential dimension of their Hilbert space and the resulting gradient estimation complexity. It has been shown that the gradient becomes exponentially small in the number of qubits, leading to something known as barren plateaus, failing the training, McClean et al. (2018). In fact, addressing this issue and proposing has become one of the most active fields in quantum machine learning, Cerezo et al. (2021), Marrero et al. (2021), Patti et al. (2021), and Pesah et al. (2021).
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One important category of machine learning is generative modeling that aims to learn the underlying probability distribution of a given data set and to generate new samples from it. Inspired by the probabilistic nature of quantum mechanics and novel quantum-inspired generative models, in 2018 a novel generative model known as Born machines was introduced (Cheng et al. (2018) and Han et al. (2018)) that uses quantum state representation and learns the joint probabilities over such quantum degrees of freedom according to (4.1)
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One promising class of quantum-inspired models that can be used for training the born machine are families of tensor networks, in particular, the one-dimensional factorization known as matrix product states (MPS). The complexity of the MPS arises from quantum correlations that could be generated in many-body quantum systems with a given efficient topology, thus capturing the underlying correlation of quantum data (Najafi et al. (2020)). 7
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The Development of Quantum Machine Learning Another type of quantum circuit generalization of classical machine learning models is the basis-enhanced Bayesian quantum circuit (BBQC, Gao et al. (2021)). The model is based on quantum circuits that naturally encode the Bayesian network if the output is measured in the computational basis. By generalizing the measurement basis, the resulting generative model BBQC will demonstrate quantum advantage. In particular, when being applied to sequence domain problems, it can be shown that the advantage originates from quantum contextuality, which is a generalization of the famous quantum nonlocality (the latter is confirmed by the famous Bells test, Bell (1964) and Hensen et al. (2015)).
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