Coherent Feed Forward Quantum Neural Network

TL;DR

The paper tackles scalability and resource inefficiencies in existing quantum neural networks by introducing a coherent feed-forward quantum neural network (CFFQNN) that preserves a feed-forward neural network structure while maintaining quantum coherence across hidden layers. The model encodes data with Ry rotations and connects layers via entangling CR_y gates, enabling a qubit-count that does not scale with the number of data features. Two variants are explored: the standard CFFQNN and FixedCFFQNN, the latter freezing initial-layer weights to reduce training load. Through simulations on credit card fraud and Wisconsin breast cancer datasets, the CFFQNN demonstrates improved accuracy and reduced quantum resources compared to traditional QNNs and competitive classical baselines, highlighting a practical path toward scalable quantum neural networks on near-term devices.

Abstract

Quantum machine learning, focusing on quantum neural networks (QNNs), remains a vastly uncharted field of study. Current QNN models primarily employ variational circuits on an ansatz or a quantum feature map, often requiring multiple entanglement layers. This methodology not only increases the computational cost of the circuit beyond what is practical on near-term quantum devices but also misleadingly labels these models as neural networks, given their divergence from the structure of a typical feed-forward neural network (FFNN). Moreover, the circuit depth and qubit needs of these models scale poorly with the number of data features, resulting in an efficiency challenge for real-world machine-learning tasks. We introduce a bona fide QNN model, which seamlessly aligns with the versatility of a traditional FFNN in terms of its adaptable intermediate layers and nodes, absent from intermediate measurements such that our entire model is coherent. This model stands out with its reduced circuit depth and number of requisite C-NOT gates to outperform prevailing QNN models. Furthermore, the qubit count in our model remains unaffected by the data's feature quantity. We test our proposed model on various benchmarking datasets such as the diagnostic breast cancer (Wisconsin) and credit card fraud detection datasets. We compare the outcomes of our model with the existing QNN methods to showcase the advantageous efficacy of our approach, even with a reduced requirement on quantum resources. Our model paves the way for application of quantum neural networks to real relevant machine learning problems.

Figures (7)

• Figure 1: A perceptron, inspired by neural networks in the brain. Inputs $\{X_i\}$ are combined with weights $\{W_i\}$ including a bias $W_0$ that are processed nonlinearly to produce a binary output.
• Figure 2: (a) Architecture of a QNN acting on $n$ qubits; the data $\mathbf{X}$ are loaded with a feature map $U(\mathbf{X})$ and the data are processed using a parametrized circuit $U_{var}(\theta)$. Subsequent measurement allows the parameters $\theta$ to be optimized and updated. These steps may be repeated. (b) A 3-qubit feature map circuit administering the commonly used ZZFeatureMap. Here $H$ represents the Hadamard gate, $P$ represents the phase gate, $\tilde{X}_{i} = 2X_{i}$, and $X_{ij} = 2(\pi - X_{i})(\pi - X_{j})$. (c) A 3-qubit variational circuit with weight parameters $\{\theta_j\}$ explicit.
• Figure 3: The depiction of data encoding stage. Rotation gates with angles $X_0W_0$ act on a single qubit in analogy with inputs acting on a single neuron. An extra rotation gate with $X_0=1$ and $W_0=b$ is added for flexibility to bias the initial qubit.
• Figure 4: An illustration of a single intermediate node in CFFQNN and how the weight values are applied from one node to the next using controlled rotations by angle $W_i$, with a possible single-qubit rotation by bias angle $W_0$.
• Figure 5: (a) Architecture of an artificial neural network with two hidden layers. Here $W$ represents the weight parameters, $\mathbf{X}$ are data points, $\sigma$ is a non-linear activation function, and $h_{ij}=W_{ij}X_{i}$. (b) Architecture of a CFFQNN with two hidden layers where $\mathbf{X}$ are data points, and $\mathbf{W}$ and $\boldsymbol{\theta}$ represent the weight parameters. The number of modes in a layer of the ANN correspond to the number of qubits in a layer of the CFFQNN. In contrast to earlier QNN models such as those in Fig. \ref{['fig1']}(c), the parameters of the CFFQNN change the controlled operations such that the CFFQNN circuits are not solely parametrized by their single-qubit gates; this is what allows the CFFQNN to resemble an ANN.