What can we learn from quantum convolutional neural networks?

TL;DR

This work investigates why quantum convolutional neural networks (QCNNs) trained on quantum data excel at phase recognition by interpreting input states as hidden feature maps, with ground-state embeddings $|\psi_0(x)\rangle$ furnishing a basis that yields sharp, critical-point features. It contrasts ground-state embeddings with rotation-based Fourier embeddings, showing that the former enable strong generalization in finite-shot scenarios and a method to build nonlinear decision boundaries using a small observable $\hat{\mathcal{O}}$, while the latter can struggle when feature frequencies are not well-tuned. The authors demonstrate these ideas through detailed modeling of cluster-type Hamiltonians, analyzing basis functions and their transformation under QCNN pooling, and then extend the approach to regression tasks in fluid dynamics (Burgers equation) and damped oscillators, achieving notable success with ground-state embeddings. Overall, the paper provides a principled framework for understanding QCNNs with quantum data, highlighting ground-state feature maps as a design principle that can improve trainability, generalization, and cross-domain applicability of quantum data-driven models.

Abstract

Quantum machine learning (QML) shows promise for analyzing quantum data. A notable example is the use of quantum convolutional neural networks (QCNNs), implemented as specific types of quantum circuits, to recognize phases of matter. In this approach, ground states of many-body Hamiltonians are prepared to form a quantum dataset and classified in a supervised manner using only a few labeled examples. However, this type of dataset and model differs fundamentally from typical QML paradigms based on feature maps and parameterized circuits. In this study, we demonstrate how models utilizing quantum data can be interpreted through hidden feature maps, where physical features are implicitly embedded via ground-state feature maps. By analyzing selected examples previously explored with QCNNs, we show that high performance in quantum phase recognition comes from generating a highly effective basis set with sharp features at critical points. The learning process adapts the measurement to create sharp decision boundaries. Our analysis highlights improved generalization when working with quantum data, particularly in the limited-shots regime. Furthermore, translating these insights into the domain of quantum scientific machine learning, we demonstrate that ground-state feature maps can be applied to fluid dynamics problems, expressing shock wave solutions with good generalization and proven trainability.

Figures (9)

• Figure 1: Visualization for the classification process of physical phases with a quantum convolutional neural network (QCNN). We highlight that input states $\hat{\rho}(\bm{x})$ for the network come from actual physical processes, e.g. preparation of ground states for spin lattices. The analyzed states depend on underlying classical features $\bm{x}$ of the system, being the physical parameters (externally controlled like magnetic field and temperature, or internal parameters like degree of anisotropy). This can be seen as a hidden feature map (left). The goal of QCNN is then building a model based on a simple few-qubit observable $\hat{\mathcal{O}}$, with its expectation $\langle \hat{\mathcal{O}}\rangle (\bm{x})$ representing a nonlinear decision boundary with respect to the system features.
• Figure 2: Analysis of a simple $N=3$ cluster model. (a) Sketch of the spin-1/2 chain with ZXZ couplings, periodic boundary conditions, and assuming the transverse magnetic field. (b) Basis functions of the ground state embedding for the $N=3$ cluster model, shown as squared projections $|\phi_j(x)|^2$ on computational basis states $j=000,100, ..., 111$. Superscript indices indicate degeneracies for one-hot and two-hot states. (c) Products of the basis functions $\phi_j^*(x) \phi_{\bar{j}}(x)$ corresponding to antidiagonal components, and their sum that represents the expectation value of the string order operator $\langle \hat{\mathcal{O}}\rangle(x)$.
• Figure 3: Analysis of the nine-qubit quantum convolutional neural network for the cluster state Hamiltonian. (a) QCNN circuit with a fixed predefined structure, motivated by the criteria for SPT phase recognition. It comprises of convolution and pooling layers, which represent unitaries that perform a basis change from the cluster state basis into a product basis (UNPREPARE operation corresponding to pivoting unitary $\hat{\mathcal{U}}_{\mathrm{pivot}}$ and a layer of Hadamards), and conditional operations that ensure that in the lowest order of symmetry breaking terms the product state remains trivial (CORRECT operation). We also label different stages of the protocol (stage A to D) that help understanding of how the underlying basis changes during QCNN processing. (b) Visualization of the basis set for $N=9$ QCNN at stage B, where squared amplitudes projected on computational basis states are shown as a function of the Hamiltonian parameter $x$. Multiple degenerate solution of the sigmoid type are generated. (c) The resulting models $\langle \hat{\mathcal{O}} \rangle (x)$ of the fixed QCNN that are read out at different stages. Orange curve corresponds to the full QCNN operation with pooling procedure, representing a sharp decision boundary at stage D for $\tilde{N} = 1$. This coincides with string order parameter measured full width 9-qubit $\langle \hat{X}_1 \hat{X}_2 ... \hat{X}_N \rangle$ prior to QCNN action, and $\tilde{N}=3$ measurement of the order parameter $\langle \hat{\mathcal{O}}_{\tilde{N}=3}\rangle \simeq \langle \hat{X}_2 \hat{X}_5 \hat{X}_8 \rangle$ at stage C. The case of measuring $\langle \hat{\mathcal{O}}_{\tilde{N}=1}\rangle \simeq \langle \hat{X}_5 \rangle$ and $\langle \hat{\mathcal{O}}_{\tilde{N}=3}\rangle$ observables in the absence of pooling (i.e. with CORRECT layers being removed) are shown with blue and magenta dashed curves. We note that here decision boundaries can be shifted by $\pm 1$ and/or trivially scaled by $1/2$ to yield the comparison where needed.
• Figure 4: (a) Parameterized elements of the variational QCNN circuit that substitute the fixed QCNN elements for pivoting and correction layers. Layers are translationally invariant but SU(2) and ZZ unitaries have different angles. (b) Examples of the decision boundary for the 9-qubit variational QCNN during the training. We show decision boundaries, starting from random initialization (curve 1; 42% test accuracy), and increasing number of epochs to 25, 50, 100, and 200 (curves 2-5, respectively).
• Figure 5: (a) Here, we substitute the ground state embedding by the rotation-based embedding with Fourier basis. $\eta$ represents a parameter that defines the frequency range. (b) Basis sets shown for the Fourier feature map with $\eta = 2$ (top panel) and $\eta = 8$ (bottom panel), visualized as squared projections on the computational basis, $|\phi_j(x)|^2$. The state is taken after the embedding is performed, just before the variational QCNN circuit. (c) Decision boundary coming from QCNN with the Fourier feature map based on $\hat{R}_{\mathrm{Y}}(\eta x)$ rotations, shown for $\eta=2,4,6,8$. The corresponding test accuracy for each embedding is shown on the right, ranging from $65\%$ ($\eta = 8$) to $100\%$ ($\eta = 2$ and $4$).