Building quantum neural networks based on a swap test

An artificial neural network, consisting of many neurons in different layers, is an important method to simulate the human brain. Usually, one neuron has two operations: one is linear, the other is nonlinear. The linear operation is the inner product and the nonlinear operation is represented by an activation function. In this work, we introduce a kind of quantum neuron whose inputs and outputs are quantum states. The inner product and activation operator of the quantum neurons can be realized by quantum circuits. Based on the quantum neuron, we propose a model of a quantum neural network in which the weights between neurons are all quantum states. We also construct a quantum circuit to realize this quantum neural network model. A learning algorithm is proposed meanwhile. We show the validity of the learning algorithm theoretically and demonstrate the potential of the quantum neural network numerically.

Figure 1

Quantum circuit to prepare $|{\varphi }_r\rangle$.

Figure 2

Quantum circuit to implement $G_r$.

Figure 3

Quantum circuit to run $I^{\otimes (n+1)}-{2|0\rangle }^{\otimes (n+1)}{\langle 0|}^{\otimes (n+1)}$.

Figure 4

Quantum phase estimation to estimate $\theta$.

Figure 5

Structure of a quantum neuron, where $|d\rangle =f(\langle x|w\rangle )$ is the output state.

Figure 6

The quantum neuron to generate $R_Z(\pi y_i/2^{t-1})|d_r\rangle$, where $|d_r\rangle =R_Y(\pi y_r/2^{t-1})|0\rangle$.

Figure 7

The quantum neuron in the ideal case, where $|d\rangle =f(\langle x|w\rangle )$.

Figure 8

The quantum neuron in the general case.

Figure 9

The quantum feedforward neural network.

Figure 10

Construction of the quantum feedforward neural network. Here the input states are $|x_1\rangle$ and $|x_2\rangle$, and the output state is $|z_1^{\left(2\right)}\rangle$. (a) Quantum neural network model with three neurons. (b) The quantum neural network represented by a circuit. The transformations $G_{r{1}^{\left(2\right)}}$ and $G_{i1}^{\left(2\right)}$ are controlled by $4t$ qubits compared to before (see Fig. 11).

Figure 11

The transformation $G_{r1}^{\left(2\right)}$.

Figure 12

The learning curve.

Figure 13

The testing result: The correct predictions are represented with dots, and the incorrect predictions are labeled with crosses.