Trainability of Dissipative Perceptron-Based Quantum Neural Networks

Several architectures have been proposed for quantum neural networks (QNNs), with the goal of efficiently performing machine learning tasks on quantum data. Rigorous scaling results are urgently needed for specific QNN constructions to understand which, if any, will be trainable at a large scale. Here, we analyze the gradient scaling (and hence the trainability) for a recently proposed architecture that we call dissipative QNNs (DQNNs), where the input qubits of each layer are discarded at the layer’s output. We find that DQNNs can exhibit barren plateaus, i.e., gradients that vanish exponentially in the number of qubits. Moreover, we provide quantitative bounds on the scaling of the gradient for DQNNs under different conditions, such as different cost functions and circuit depths, and show that trainability is not always guaranteed. Our work represents the first rigorous analysis of the scalability of a perceptron-based QNN.

Figure 1

Schematic diagram of a dissipative perceptron-based quantum neural network (DQNN). Top: The DQNN is composed of input, hidden, and output layers. Each node in the network corresponds to a qubit, which can be connected to qubits in adjacent layers via perceptrons (depicted as lines). The input and output of the DQNN are quantum states denoted as ${\rho }^{\mathrm{in}}$ and ${\rho }^{\mathrm{out}}$, respectively. Bottom: Quantum circuit description of the DQNN. The $j$th qubit of the $l$th layer is denoted $q_j^l$. Each perceptron corresponds to a unitary operation on the qubits it connects, with $V_j^l$ denoting the $j$th perceptron in the $l$th layer.

Figure 2

Global and local perceptrons. (a) The global perceptron acts nontrivially on all input qubits, i.e., $m=n$. (b) The local perceptron acts nontrivially only on a small number of input qubits. For the case shown, $m=3$.

Figure 3

Shallow local perceptrons ansatzes. (a) Here $m=1$ so that each perceptron acts on a single input and output qubit. Moreover, for all $j$ and $l$ we have $V_j^l=V$. The unitaries $V$ are simply given by a SWAP operator followed by a single qubit rotation around the $y$ axis. (b) Local perceptrons $V_j^l$ with $m=2$. The local perceptrons are given by the unitaries $V_1$, or $V_2$. Specifically, for $l$ odd on $j$ odd (even) $V_j^l=V_1(V_2)$, while for $l$ even and $j$ odd (even) we have $V_j^l=V_2(V_1)$. Here we also show the order in which the perceptrons are applied so that we first implement the unitaries with $j$ odd, followed by the unitaries with $j$ even. The $W$ gate in $V_1$ forms a local two-design on two qubits.