Completely quantum neural networks

Artificial neural networks are at the heart of modern deep learning algorithms. We describe how to embed and train a general neural network in a quantum annealer without introducing any classical element in training. To implement the network on a state-of-the-art quantum annealer, we develop three crucial ingredients: binary encoding the free parameters of the network; polynomial approximation of the activation function; and reduction of binary higher-order polynomials into quadratic ones. Together, these ideas allow encoding the loss function as an Ising model Hamiltonian. The quantum annealer then trains the network by finding the ground state. We implement this for an elementary network and illustrate the advantages of quantum training: its consistency in finding the global minimum of the loss function and the fact that the network training converges in a single annealing step, which leads to short training times while maintaining a high classification performance. After training the network using a quantum annealer, one can then use the quantum network weights in a classical network algorithm of identical design for inference. Our approach opens an avenue for the quantum training of general machine learning models.

Figure 1

Schematic representation of a NN with two inputs and three layers $L^{\left(1\right)}$, $L^{\left(2\right)}$, and $L^{\left(3\right)}$, with four, five, and three units, respectively. The inputs and units are shown as blue circles. The arrows represent the action of the layers, which are affine transformations followed by the element-wise application of the activation function, as displayed in Eq. (3). The outputs of layers $L^{\left(1\right)}$ and $L^{\left(2\right)}$ are denoted $z^{\left(1\right)}$ and $z^{\left(2\right)}$

Figure 2

Polynomial approximations to the popular ReLU (top) and sigmoid (middle) activation functions, and to the logistic loss function $L$ (bottom). The logistic loss is defined here as $L(y,Y)=-ylogp\left(Y\right)-(1-y)log[1-p(Y\left)\right]$, where $p\left(x\right)=1/(1+e^{-10x})$.

Figure 3

Left: Number of physical qubits needed to embed the quantum NN as a function of the number of spins in its abstract encoding, for varying values of the number of features $N_f=2$–4, the number of units in the hidden layer $N_h=2$–4 and the number of spins per parameter $N_b=1$–3. The graph shows that the number of physical qubits only depends on these parameters through the number of abstract spins, which is a function of them. Right: Number of qubits as function of $N_h$, for each value of $N_f$ and fixed $N_b=1$.

Figure 4

Decision boundary obtained with the quantum NN for each dataset, together with the points in the dataset. Points labeled as signal and background are shown in blue and red, respectively. The decision boundary, displayed as a black line, is the set of points for which the network prediction is $Y_{v,w}\left(x\right)=0$.

Figure 5

Schematic representation of the loss function. Depending on the starting point and training method, classical training may end up in any local minima, such as the red one. Quantum training can tunnel from any of these minima to the global one, shown in green.

Figure 6

Comparison of the results of the quantum NN to its classical analog. Each bar gives the median of ten training runs. The error bars show the percentiles 20 and 80.