Quantum neural network

Quantum neural network research is still in its infancy, and a conglomeration of proposals and ideas of varying scope and mathematical rigor have been put forward. Most of them are based on the idea of replacing classical binary or McCulloch-Pitts neurons with a qubit (which can be called a “quron”), resulting in neural units that can be in a superposition of the state ‘firing’ and ‘resting’.

A lot of proposals attempt to find a quantum equivalent for the perceptron unit from which neural nets are constructed. A problem is that nonlinear activation functions do not immediately correspond to the mathematical structure of quantum theory, since a quantum evolution is described by linear operations and leads to probabilistic observation. Ideas to imitate the perceptron activation function with a quantum mechanical formalism reach from special measurements ^[8][9] to postulating non-linear quantum operators (a mathematical framework that is disputed).^[10][11] 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.^[12]

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^[7][13] and quantum reservoir processor (quantum version of reservoir computing).^[14] 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.^[15]

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.^[16][17] The quantum network thus ‘learns’ an algorithm.

The first quantum associative memory algorithm was introduced by Dan Ventura and Tony Martinez in 1999.^[18] The authors do not attempt to translate the structure of artificial neural network models into quantum theory, but propose an algorithm for a circuit-based quantum computer that simulates associative memory. The memory states (in Hopfield neural networks saved in the weights of the neural connections) are written into a superposition, and a Grover-like quantum search algorithm retrieves the memory state closest to a given input. As such, this is not a fully content-addressable memory, since only incomplete patterns can be retrieved.

The first truly content-addressable quantum memory, which can retrieve patterns also from corrupted inputs, was proposed by Carlo A. Trugenberger.^[19][20][21] Both memories can store an exponential (in terms of n qubits) number of patterns but can be used only once due to the no-cloning theorem and their destruction upon measurement.

Trugenberger,^[20] however, has shown that his proababilistic model of quantum associative memory can be efficiently implemented and re-used multiples times for any polynomial number of stored patterns, a large advantage with respect to classical associative memories.

A substantial amount of interest has been given to a “quantum-inspired” model that uses ideas from quantum theory to implement a neural network based on fuzzy logic.^[22]

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.^[6][23] 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 (${\displaystyle U_{f}}$ ) with a dummy state qubit in a known state (Ex. ${\displaystyle |0\rangle }$  in the computational basis), also known as an Ancilla bit, the information from the qubit can be transferred to the next layer of qubits.^[7] This process adheres to the quantum operation requirement of reversibility.^[7][24]

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.^[6] In other words, no Unitary operator is acting on the entire network at any given time, meaning the number of qubits required for a given step depends on the number of inputs in a given layer. Since Quantum Computers are notorious for their ability to run multiple iterations in a short period of time, the efficiency of a quantum neural network is solely dependent on the number of qubits in any given layer, and not on the depth of the network.^[24]

To determine the effectiveness of a neural network, a cost function is used, which essentially measures the proximity of the network's output to the expected or desired output. In a Classical Neural Network, the weights (${\displaystyle w}$ ) and biases (${\displaystyle b}$ ) at each step determine the outcome of the cost function ${\displaystyle C(w,b)}$ .^[6] When training a Classical Neural network, the weights and biases are adjusted after each iteration, and given equation 1 below, where ${\displaystyle y(x)}$  is the desired output and ${\displaystyle a^{\text{out}}(x)}$  is the actual output, the cost function is optimized when ${\displaystyle C(w,b)}$ = 0. For a quantum neural network, the cost function is determined by measuring the fidelity of the outcome state (${\displaystyle \rho ^{\text{out}}}$ ) with the desired outcome state (${\displaystyle \phi ^{\text{out}}}$ ), seen in Equation 2 below. In this case, the Unitary operators are adjusted after each iteration, and the cost function is optimized when C = 1.^[6]

Equation 1 ${\displaystyle C(w,b)={1 \over N}\sum _{x}{||y(x)-a^{\text{out}}(x)|| \over 2}}$ 

Equation 2 ${\displaystyle C={1 \over N}\sum _{x}^{N}{\langle \phi ^{\text{out}}|\rho ^{\text{out}}|\phi ^{\text{out}}\rangle }}$ 

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.^[26]

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^[26] 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.

• Differentiable programming
• Optical neural network
• Holographic associative memory
• Quantum cognition
• Quantum machine learning

• Recent review of quantum neural networks by M. Schuld, I. Sinayskiy and F. Petruccione
• Review of quantum neural networks by Wei
• Article by P. Gralewicz on the plausibility of quantum computing in biological neural networks
• Training a neural net to recognize images