Iterative Quantum Feature Maps

TL;DR

This paper addresses the practical deployment of quantum feature maps by introducing Iterative Quantum Feature Maps (IQFMs), a hybrid quantum-classical framework that connects shallow QFM blocks with trainable classical augmentation to form a deep representation while fixing quantum circuit parameters. Through multi-basis quantum feature extraction and layer-wise contrastive learning, IQFMs reduce quantum runtime and mitigate noise, yielding robustness on noisy quantum data and competitive performance on classical image data. Empirical results show IQFMs outperform a QCNN in noisy quantum phase recognition tasks and achieve near-parallel performance to width-matched classical networks on Fashion-MNIST, highlighting their versatility and practicality for NISQ devices. The work also outlines potential extensions, including one-step learning, direct feedback alignment as an alternative training signal, and quantum transfer learning, offering a promising pathway to scalable quantum-enhanced learning on NISQ devices and pointing to extensions such as one-step learning, DFA-based training, and quantum transfer learning.

Abstract

Quantum machine learning models that leverage quantum circuits as quantum feature maps (QFMs) are recognized for their enhanced expressive power in learning tasks. Such models have demonstrated rigorous end-to-end quantum speedups for specific families of classification problems. However, deploying deep QFMs on real quantum hardware remains challenging due to circuit noise and hardware constraints. Additionally, variational quantum algorithms often suffer from computational bottlenecks, particularly in accurate gradient estimation, which significantly increases quantum resource demands during training. We propose Iterative Quantum Feature Maps (IQFMs), a hybrid quantum-classical framework that constructs a deep architecture by iteratively connecting shallow QFMs with classically computed augmentation weights. By incorporating contrastive learning and a layer-wise training mechanism, the IQFMs framework effectively reduces quantum runtime and mitigates noise-induced degradation. In tasks involving noisy quantum data, numerical experiments show that the IQFMs framework outperforms quantum convolutional neural networks, without requiring the optimization of variational quantum parameters. Even for a typical classical image classification benchmark, a carefully designed IQFMs framework achieves performance comparable to that of classical neural networks. This framework presents a promising path to address current limitations and harness the full potential of quantum-enhanced machine learning.

Figures (12)

• Figure 1: IQFMs and the representation learning in processing classical input $\boldsymbol{x}$ and quantum input $\ket{\phi}$. (a) The $l$-th QFM starts with a quantum state $\ket{\psi}_{l-1}$ and uses an embedding circuit $\mathcal{U}_{\Psi(\boldsymbol{h}_{l-1})}$ to map classical features $\boldsymbol{h}_{l-1}$ into a quantum state, followed by a preprocessing circuit $P_l$, then a parameterized circuit $\Omega_l(\boldsymbol{\theta}_l)$ to adapt the measurement basis. Extracted features $\boldsymbol{g}_l$ from the $l$-th QFM are passed to the $(l+1)$-th QFM through classical augmentation with trainable weights. The outputs $\boldsymbol{h}_l$ from all augmentation layers are aggregated into a classical feature vector, which is fed into multiple prediction networks to predict properties of $\boldsymbol{x}$ and $\ket{\phi}$. (b) The IQFMs framework employs contrastive learning to train the classical augmentation weights sequentially, leaving quantum circuit parameters fixed. Here, positive representations $(\boldsymbol{h}_1^{+}\ldots,\boldsymbol{h}_L^{+})$ are derived from data similar to the input (e.g., perturbed versions or same-class samples), while negative representations $(\boldsymbol{h}_1^{-}\ldots,\boldsymbol{h}_L^{-})$ come from dissimilar data (e.g., different-class samples). For each layer $l$, the weight $\boldsymbol{W}_l$ is trained to minimize the cost function $\mathcal{C}(\boldsymbol{h}_1^{+},\boldsymbol{h}_1^{-})$, which pulls positive representations closer and pushes negative representations away. This process begins with $\boldsymbol{W}_1$, fixes it, then trains $\boldsymbol{W}_2$, and continues layer by layer until the final layer, then repeating until termination conditions are met.
• Figure 2: The modular IQFMs. The classical data $\boldsymbol{h}$ is partitioned and processed in parallel by multiple QFMs within each module. The resulting feature vectors of QFMs in each module are then combined and forwarded to the next layer.
• Figure 3: The quantum feature extraction involves multi-basis measurement. The initial feature vector $\boldsymbol{g}^{(0)}$ is derived from all qubits measurement in the Z-basis. To expand the feature set, measurements are performed on other $B$ distinct bases to acquire $\boldsymbol{g}^{(1)}, \ldots, \boldsymbol{g}^{(B)}$. Here, $\boldsymbol{g}^{(i)}$ is obtained by applying a random single-qubit rotation $RX(\theta^i)$ to all qubits. The total extracted feature vector $\boldsymbol{g}$ is a concatenation of $\boldsymbol{g}^{(0)}, \ldots, \boldsymbol{g}^{(B)}$.
• Figure 4: Ground-state phase diagrams of Tasks A and B.
• Figure 5: For quantum data classification tasks, the IQFMs framework incorporates a re-input structure: $\ket{\psi}_0 = \ket{\phi}$ and $\ket{\psi}_{l-1} = \ket{\psi}_0$ for $l=1,2,\ldots$, where $\ket{\phi}$ is the input quantum state. The $l$-th preprocessing circuit $P_l$ starts with Hadamard gates applied to all qubits, followed by several layers of randomly parameterised RX, RZ, and RZZ gates. Starting from the second QFM layer, an embedding circuit $\mathcal{U}_{\Psi(\boldsymbol{h}_{l-1})}$ maps classical data $\boldsymbol{h}_{l-1}$ into a quantum state. This circuit comprises $d$ layers, each consisting of Hadamard gates on all qubits, followed by RZ and RZZ gates parameterized by $\boldsymbol{h}_{l-1}$. In both the preprocessing and embedding circuits, RX and RZ gates are applied to each qubit individually, and RZZ gates are applied to all adjacent qubit pairs arranged in a circular topology.