Training deep quantum neural networks

Abstract

Neural networks enjoy widespread success in both research and industry and, with the advent of quantum technology, it is a crucial challenge to design quantum neural networks for fully quantum learning tasks. Here we propose a truly quantum analogue of classical neurons, which form quantum feedforward neural networks capable of universal quantum computation. We describe the efficient training of these networks using the fidelity as a cost function, providing both classical and efficient quantum implementations. Our method allows for fast optimisation with reduced memory requirements: the number of qudits required scales with only the width, allowing deep-network optimisation. We benchmark our proposal for the quantum task of learning an unknown unitary and find remarkable generalisation behaviour and a striking robustness to noisy training data.

Fig. 1. A general quantum feedforward neural network.

A quantum neural network has an input, output, and L hidden layers. We apply the perceptron unitaries layerwise from top to bottom (indicated with colours for the first layer): first the violet unitary is applied, followed by the orange one, and finally the yellow one.

Fig. 2. Numerical results.

In both plots, the insets show the behaviour of the quantum neural network under approximate depolarizing noise. The colours indicate the strength t of the noise: black t = 0, violet t = 0.0033, orange t = 0.0066, yellow t = 0.01. For a more detailed discussion of the noise model see Supplementary Note 3 and Supplementary Fig. 3. Panel (a) shows the ability of the network to generalize. We trained a 3-3-3 network with ϵ = 0.1, η = 2∕3 for 1000 rounds with n = 1, 2, …, 8 training pairs and evaluated the cost function for a set of 10 test pairs afterwards. We averaged this over 20 rounds (orange points) and compared the result with the estimated value of the optimal achievable cost function (violet points). Panel (b) shows the robustness of the QNN to noisy data. We trained a 2-3-2 network with ϵ = 0.1, η = 1 for 300 rounds with 100 training pairs. In the plot, the number on the x-axis indicates how many of these pairs were replaced by a pair of noisy (i.e. random) pairs and the cost function is evaluated for all “good” test pairs.