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Quantum Mechanics in Neural Networks

Quantum neural networks are computational neural network models which are based on the principles of quantum mechanics. The first ideas on quantum neural computation were published independently in 1995 by Subhash Kak and Ron Chrisley, engaging with the theory of quantum mind, which posits that quantum effects play a role in cognitive function. However, typical research in quantum neural networks involves combining classical artificial neural network models (which are widely used in machine learning for the important task of pattern recognition) with the advantages of quantum information in order to develop more efficient algorithms. One important motivation for these investigations is the difficulty to train classical neural networks, especially in big data applications. The hope is that features of quantum computing such as quantum parallelism or the effects of interference and entanglement can be used as resources. Since the technological implementation of a quantum computer is still in a premature stage, such quantum neural network models are mostly theoretical proposals that await their full implementation in physical experiments. Most Quantum neural networks are developed as feed forward networks. Similar to their classical counterparts, this structure intakes input from one layer of qubits, and passes that input onto another layer of qubits. This layer of qubits evaluates this information and passes on the output to the next layer. Eventually the path leads to the final layer of qubits. to postulating non linear quantum operators (a mathematical framework that is disputed). A direct implementation of the activation function using the circuit based model of quantum computation has recently been proposed by Schuld, Sinayskiy and Petruccione based on the quantum phase estimation algorithm.

At a larger scale, researchers have attempted to generalize neural networks to the quantum setting. One way of constructing a quantum neuron is to first generalise classical neurons and then generalising them further to make unitary gates. Interactions between neurons can be controlled quantumly, with unitary gates, or classically, via measurement of the network states. This high level theoretical technique can be applied broadly, by taking different types of networks and different implementations of quantum neurons, such as photonically implemented neurons and quantum reservoir processor (quantum version of reservoir computing). Most learning algorithms follow the classical model of training an artificial neural network to learn the input output function of a given training set and use classical feedback loops to update parameters of the quantum system until they converge to an optimal configuration. Learning as a parameter optimisation problem has also been approached by adiabatic models of quantum computing. Quantum neural networks can be applied to algorithmic design: given qubits with tunable mutual interactions, one can attempt to learn interactions following the classical backpropagation rule from a training set of desired input output relations, taken to be the desired output algorithm's behavior. The quantum network thus ‘learns’ an algorithm.

Quantum Neural Networks can be theoretically trained similarly to training classical/artificial neural networks. A key difference lies in communication between the layers of a neural networks. For classical neural networks, at the end of a given operation, the current perceptron copies its output to the next layer of perceptron(s) in the network. However, in a quantum neural network, where each perceptron is a qubit, this would violate the no cloning theorem. A proposed generalized solution to this is to replace the classical fan out method with an arbitrary unitary that spreads out, but does not copy, the output of one qubit to the next layer of qubits. Using this fan out Unitary with a dummy state qubit in a known state (Ex. in the computational basis), also known as an Ancilla bit, the information from the qubit can be transferred to the next layer of qubits. Using this quantum feed forward network, deep neural networks can be executed and trained efficiently. A deep neural network is essentially a network with many hidden layers, as seen in the sample model neural network above. Since the Quantum neural network being discussed uses fan out Unitary operators, and each operator only acts on its respective input, only two layers are used at any given time. Figure shows the Barren Plateau problem becomes increasingly serious as the VQA expands.]] Gradient descent is widely used and successful in classical algorithms. However, although the simplified structure is very similar to neural networks such as CNNs, QNNs perform much worse. Since the quantum space exponentially expands as the q bit grows, the observations will concentrate around the mean value at an exponential rate, where also have exponentially small gradients. This situation is known as Barren Plateaus, because most of the initial parameters are trapped on a "plateau" of almost zero gradient, which approximates random wandering rather than gradient descent. This makes the model untrainable. In fact, not only QNN, but almost all deeper VQA algorithms have this problem. In the present NISQ era, this is one of the problems that have to be solved if more applications are to be made of the various VQA algorithms, including QNN.