Nonlinear quantum neuron: A fundamental building block for quantum neural networks

Quantum computing enables quantum neural networks (QNNs) to have great potential to surpass artificial neural networks. The powerful generalization of neural networks is attributed to nonlinear activation functions. Although various models related to QNNs have been developed, they are facing the challenge of merging the nonlinear, dissipative dynamics of neural computing into the linear, unitary quantum system. In this paper, we establish different quantum circuits to approximate nonlinear functions and then propose a generalizable framework to realize any nonlinear quantum neuron. We present two quantum neuron examples based on the proposed framework. The quantum resources required to construct a single quantum neuron are polynomial in function of the input size. Finally, both IBM Quantum Experience results and numerical simulations illustrate the effectiveness of the proposed framework.

Figure 1

Panels (a) and (b) show two quantum circuits for implementing $U_f$ in Eq. (6).

Figure 2

A schematic of a classical neuron.

Figure 3

The proposed framework of implementing a nonlinear quantum neuron. $V_1$ represents the initialization of auxiliary qubits. $V_2$ refers to the preparation of quantum dataset. $U\left({\theta }_1\right)$ denotes the calculation of the inner product of the input vector and the weight vector. $U\left({\theta }_2\right)$ denotes the approximation of nonlinear functions. ${\theta }_1$ and ${\theta }_2$ are the circuit parameters.

Figure 4

Panels (a) and (b) show nonlinear quantum neuron examples based on basis encoding and amplitude encoding, respectively.

Figure 5

Implementing discrete ReLU with different quantum circuits. All circuits are executed 8192 times (the maximum allowed) on IBM's quantum simulator and quantum computers. The accuracy indicates the proportion of the correct outputs of all outputs for given inputs. (a) The results of using four qubits to implement discrete ReLU with the circuit in Fig. 1. (b) The results of using four qubits to implement discrete ReLU with the circuit in Fig. 1. (c) An example of a typical quantum circuit for a nonlinear quantum neuron based on the Fig. 4 with four qubits, where the sign bit of the activation output is ignored due to ReLU's non-negative output. In this example, the input data is $x=1$ and the $U_w=R_z(w/4),w\in \{-2,-1,0,1\}$. (d) The running results of the circuit in Fig. 5.

Figure 6

Different types of activation functions: (a) Sigmoid, (b) ReLU, (c) Tanh, and (d) GELU.

Figure 7

Construction of a simple quantum feedforward neural network with the proposed nonlinear quantum neurons. Here the input states are $|x_1\rangle$ and $|x_2\rangle$, and the output state is $|z\rangle$. (a) Quantum feedforward neural network model. (b) The quantum feedforward neural network represented by a circuit, where ${|x\rangle }_b=|x_1\rangle \otimes |x_2\rangle , U_{w_i^{\left(1\right)}}=R_z\left(w_{i1}^{\left(1\right)}\right)\otimes R_z\left(w_{i2}^{\left(1\right)}\right), U_{w_i^{\left(2\right)}}=R_z\left(w_{i1}^{\left(2\right)}\right)\otimes R_z\left(w_{i2}^{\left(2\right)}\right)\otimes \cdots \otimes R_z\left(w_{im}^{\left(2\right)}\right)$.

Figure 8

The learning curve.