Problem-Dependent Power of Quantum Neural Networks on Multiclass Classification

Quantum neural networks (QNNs) have become an important tool for understanding the physical world, but their advantages and limitations are not fully understood. Some QNNs with specific encoding methods can be efficiently simulated by classical surrogates, while others with quantum memory may perform better than classical classifiers. Here we systematically investigate the problem-dependent power of quantum neural classifiers (QCs) on multiclass classification tasks. Through the analysis of expected risk, a measure that weighs the training loss and the generalization error of a classifier jointly, we identify two key findings: first, the training loss dominates the power rather than the generalization ability; second, QCs undergo a U-shaped risk curve, in contrast to the double-descent risk curve of deep neural classifiers. We also reveal the intrinsic connection between optimal QCs and the Helstrom bound and the equiangular tight frame. Using these findings, we propose a method that exploits loss dynamics of QCs to estimate the optimal hyperparameter settings yielding the minimal risk. Numerical results demonstrate the effectiveness of our approach to explain the superiority of QCs over multilayer Perceptron on parity datasets and their limitations over convolutional neural networks on image datasets. Our work sheds light on the problem-dependent power of QNNs and offers a practical tool for evaluating their potential merit.

Figure 1

Risk curve and geometry of the unified QCs. (a) The left plot shows the relation between the zero classification error (yellow star) and zero loss (blue star) during the training process. The middle and right plots depict the risk curve of CCs and QCs (labeled by “C-$\mathcal{R}$” and “Q-$\mathcal{R}$“). Unlike CCs with a double-descent tendency, the risk curve of QCs yields a “U” shape whose power is dominated by the empirical risk (“Q-${\mathcal{R}}_{\mathrm{ERM}}$”), dominated by the expected risk. (b) The unified QC consists of two parts, the feature state $\rho$ and the measure operator $\mathit{o}$. (c) Geometric interpretation of $\{{\rho }^{(i,k)}\}$ and $\mathit{o}$ of QCs with (near) zero training loss: (i) the feature states associated with train samples belonging to the same class concentrate around their class-feature mean, i.e., ${\overline{\rho }}^{*(k)}≔{\rho }^{*(1,k)}=\dots ={\rho }^{*(n_c,k)}$ for $\forall k\in [K]$; (ii) the class-feature means are maximally distant with each other, i.e., $\mathrm{Tr}({\overline{\rho }}^{*(k)}{\overline{\rho }}^{*(k^{\prime })})\sim {\delta }_{k,k^{\prime }}$; (iii) the measure operator should align with class-feature means, i.e., $\mathrm{Tr}({\overline{\rho }}^{*(k)}{o}^{*(k^{\prime })})\sim {\delta }_{k,k^{\prime }}$.

Figure 2

Binary classification on the parity dataset. (a) The learning dynamic of QC. The $x$ axis denotes the epoch numbers. Shaded region represents variance. The Bloch spheres display the quantum feature states at different epochs. (b) The fitted risk curve of QC and MLP. The $x$ axis denotes the number of trainable parameters. The label “QC risk” (“MLP risk”) refers to the fitted risk curve of QC and MLP. The label “QC res“ (“MLP res”) refers to the collected results used for fitting the curves.

Figure 3

Multiclass classification on the image dataset with $K=9$. (a) The learning performance of QC when the layer number is 50. (b) The fitted risk curve of QC and CNN. All labels have the same meaning as those used in Fig. 2.