A Derivative-free Method for Quantum Perceptron Training in Multi-layered Neural Networks

Quantum computing algorithms often exhibit significant increases in efficiency, in some cases exponentially, compared to their classical counterparts. This is particularly relevant to machine learning, which has had a growing importance in recent years. This is since machine learning methods tend to be computationally intensive. Thus, recently, there have been numerous research studies aiming to investigate the promise of quantum computers for machine learning [Lloyd2014, Jeswal2019, Biamonte2017, Chen2017].

Moreover, it has been suggested that quantum computers may be an ideal platform for the implementation of artificial neural networks [Biamonte2017]. In recent years ongoing attempts have been made to implement artificial neural networks (ANN) in quantum computers. Grover et al. [Grover96] attempted to emulate quantum computation on classical computers to perform search quadratically faster than its classic equivalent in an unordered dataset. The computational power of quantum computing in terms of efficiency and effectiveness for ANNs as compared to that of classical computers has also been explored in [Jeswal2019]. Nonetheless, the incorporation of quantum computation into ANNs is still an open and challenging research direction [Chen2017].

This is further compelled by the fact that deep learning is an algorithmic class within the wider category of machine learning algorithms with their own practical and architectural properties. Deep nets are used primarily to classify patterns on a specific data set and/or to produce new data that imitates these patterns. At heart, there are three main components in neural network algorithms. Firstly, the model, comprised by a parametric functional hypothesis class, typically set up in a network of layered composition of simpler parametric functions. Secondly, a cost function, which determines how well the prediction based upon the input data fits a specific hypothesis. Thirdly, the optimizer. This is an algorithmic technique used to minimise the loss function based upon the parameters in the network. This is often done by backward error propagation, also known as the backpropagation algorithm. This is at the core of ANN training.

Moreover, the cost (error) function of ANNs is often purely a function of the network’s output. The backpropagation algorithm is the most common method in both, quantum and classical computing to train ANNs[Wang2012]. This is used to train the network using the cost function gradient (in relation to the network parameters), beginning with the output layer going layer-by-layer towards the input one. In 1986, the backpropagation algorithm was proposed by Rumelhart and Mcllelland to solve the non-linear continuous function weight adjustment problems in the area of the neural multi-layer feedforward network as a back error method [Rumelhart1986].

Since the development of the back-propagation method for neural networks, a lot of research has been carried out on the choice of activation function, design of structure parameters and characterise the loss function. A lot of research has also been carried out to improve the efficiency of the back-propagation methods in neural networks. Sun et al. [Sun2014] have developed an improved prediction model of back-propagation neural network and quantitatively researched related parameters. Xiao et al. [Xiao2009] present a short-term load forecast method for Neural Network Prediction.

Here we note that, although the back-propagation algorithm is the most widely used one in artificial neural networks in both classical as well as quantum domains, it does exhibit the following drawbacks:

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  It can fall into local extremum points if there are multiple points on the loss space with zero gradient [Zhang2009].

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  Its speed of convergence is dependent on two aspects. Firstly, the learning rate. Secondly, the magnitude of the gradient associated to the excitation function [Xu2015]. This can result in slow convergence rates. This is further complicated by the fact that the magnitude of the learning rate is not a straightforward parameter to set. If the learning rate is too large, the back propagation solver will often suffer from “over shooting”. If the learning rate is too small, the network may fail to converge at all.

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  Its computational complexity is dependent on the network structure, increasing with the number of hidden layers and the number of neurons per layer.

In this paper, we turn out attention to the evaluation of the quantum perceptron as a means to tackle the drawbacks above in the training of neural networks. To do this, we depart from the concept of a single quantum bit, i.e. a qubit, and examine the strategies for evaluating the equivalent in quantum computing of the perceptron in machine learning. We then focus on measurable operators to model the evolution of an artificial neural network as the action of a unitary transformation on the network’s present firing states. This has the advantage that, for training an artificial neural network, the forward pass is given by a measurement of the transformed state whereas the training can be effected using a gradient-free strategy whose complexity is devoid of the number of layers in the network. This, together with the natural efficiency of quantum neural nets, is a promising trait that can greatly speed up both training and testing of neural networks in quantum computing. This unitary transformation would, of course, have to include information on the probabilities of transition between each basic state.

In this section, we briefly introduce the concepts of quantum computing that are necessary for the remainder of the paper. For a detailed introduction on quantum computing, we would like to remit the interested reader to [Biamonte2017].

As mentioned earlier, the qubit is often represented using the Bra-ket notation. In this manner, following the notion that $\mathbf{x}_{i}$ can be represented using the notation $\left|{x_{1}}\rangle$, consider the training set written in the form $\left\{{\left({\left|{{x_{1}}}\right\rangle,\left|{{y_{1}}}\right\rangle}\right),\ldots,\left({\left|{{x_{N}}}\right\rangle,\left|{{y_{N}}}\right\rangle}\right)}\right\}$ where ${\left|{{x_{j}}}\right\rangle}$ is an input input and ${\left|{{y_{j}}}\right\rangle}$ is the corresponding label.

Recall that, in order to obtain the weight vector ${\left|{{{w}_{j}}}\right\rangle}$, as suggested in [8631025], a tensor product can be used, which yields $\mathrm{W}_{i}=\left|y_{i}\right\rangle\otimes\left\langle x_{i}\right|$. After calculating the vector of weights using the expression above, these can be added to get the final weight vector given by $\mathrm{\hat{W}}=\sum_{i=1}^{N}\mathrm{W}_{i}$. Note that the matrix $\hat{W}$ is, in general, not unitary. Thus, to preserve the quantum properties requiring a unitary matrix, $\hat{w}$ is decomposed into three unitary matrices using the singular value decomposition (SVD) given by $\mathbf{\hat{W}}=\mathbf{U}\matbhf{\Sigma}\mathbf{V}^{*}$.

As noted by Liu et al. [8631025], the diagonal matrix $\matbhf{\Sigma}$ can be substituted, without any loss of generality, with a unitary matrix with ones in diagonal and zeros elsewhere, i.e. and identity matrix. This yields, using the notation commonly employed in quantum machine learning texts where the unitary matrix $\mathbf{U}$ is denoted by $\mathrm{\hat{F}}$, the quantum perceptron output given by $\mathrm{\hat{Y}}=\mathrm{\hat{F}}\lvert x_{j}\rangle$. Here, we follow Zak and Wiliams [Zak1998], who viewed an $n$-neuron network as a dynamic system that obeys the differential equation given by

where $\tau_{i}$ is a positive time constant, $\mathrm{Z}_{i}$ is the activation of the $i^{th}$ neuron and $\mathrm{W}_{j}$ are synaptic weights analogue to those elaborated upon previously that feed the activation function of the neuron indexed $j$ to the activation function $f(\cdot)$.

Equation 2 can be used to formulate the update of the quantum perceptron weights as follows

where $\mathrm{W}^{old}$ is the current, i.e. old, weights, $\mathrm{U}$ is a unitary matrix that acts on the state vector $\mathrm{\hat{Y}}$, $M$ is a measurable operator that project states of $\mathrm{U}\mathrm{\hat{Y}}$ into some eigenstate of $M$. The use of measurable operators naturally provides a link to statistics and quantum measurements. The equation 3 shows the update of the perceptron weights by using the state vector projected upon U (the orthonormal basis spanned by the SVD of the sum of outer products in $\mathrm{\hat{W}}=\sum_{i=1}^{N}\mathrm{W}_{i}$).

In this section, we examine closer the role of the the measurable operator $M$. This is due to the fact that it can open-up several opportunities as related to the design of neural networks, specially as related to approaches elsewhere in the literature that employ backpropagation methods for training. Note that $\mathrm{U}\mathrm{\hat{Y}}$ can be viewed as a representation of a sequence of measurable states that define a Markov process with a transition probability matrix. Moreover, Quantum probability is a non-commutative extension of classical probability which represents random variables as self-adjoint operators that act on a complex Hilbert space whereby the underlying probability is measured by a unitary vector.

Furthermore, note that $M$ is, by definition, a self-adjoint operator. In quantum mechanics, self-adjoint operators form a Dirac–Von Neumann formulation of quantum mechanics. Self-adjoint operators represent the physical observables such as spin, momentum, angular momentum and position on a Hilbert space. Recall that, by definition, probability distributions are non-negative and normalised to unity. Since self-adjoint operators are are unitarily equivalent to real-valued multiplication operators, they can be easily generalised to theoretically unbounded operators on infinite-dimensional spaces. Here, we will also require $M$ to be hermitian. This is important since then, by definition, has an orthonormal set of eignevectors $\lvert\xi\rangle$ with real eigenvales ${\lambda_{i}}$.

Note that, if the elements of $\mathrm{U}$ are appropriately chosen, then any desired Markov chain can be simulated without a random number generator. This is possible due to the inherent randomness of quantum measurement processes. The modulus square of the probability amplitude represents the probability of a network in the $j^{th}$ transitions to the $i^{th}$ state. This can be expressed using the following expression

where $u_{ij}$ is the entry indexed $i,j$ of $\mathrm{U}$ and we have used the notation $\mathrm{Y}^{new}$ and $\mathrm{Y}^{old}$ to denote the current and previous states of the network. It is worth noting in passing that, since $\mathrm{U}$ is a unitary matrix, there will be appropriate constraints on the possible values of $\lvert\lvert u_{ij}\lvert\lvert^{2}$ so as to avoid divergent behavior. We remit the interested reader to [Zak1998] for further reading on this.

Note that it is not unusual that millions of iterations of back-propagation may be required to train an ANN with stochastic gradient descent. For every iteration, the forward pass calculates the output of the network and, in the backward pass, calculates the gradient with respect to the weights of the network. The network weights are then updated by an amount proportional to the gradient. This computational demands makes training ANNs one of the key drivers of increasing demand for high performance computing. This makes particularly compelling the investigation of methods for forward and backward steps that can be speeded up by quantum computations. Moreover, derivative-free strategies in quantum computing can remove some of these steps altogether. One of the advantages of using strategies like the one presented here is that they require a forward pass but does not involve a back-propagation step. This makes these methods 2 to 3 times faster than those based upon back-propagation in a traditional computer. Moreover, an additional advantage is that, by using these evaluation strategies, a large number of non-differentiable excitation functions can be explored.

This also applies to the objective functions of quantum networks, which can potentially be discontinuous or even non-compact in neural networks trained using back-propagation methods. The strategies such as that presented here can also be very effective when solutions are known to be within an elliptical domain, i.e. around a fixed point. Moreover, gradient descent, despite effective, can be inefficient, particularly if the choice of starting point is poor, whereby the training can easily converges to a local minimum which may be far from an ideal solution.

Quantum-inspired neural networks (QiNNs) and Quantum computing-based neural networks have been shown to be more effective and efficient as compared to conventional ANNs [Sagheer2019]. In addition, QiNN models are not limited to solely those that can only be implemented on quantum computers, rather there has been renewed interest in methods that can take advantage of QiNN traits while being implemented on conventional computers [Sagheer2019].