Shot-frugal and Robust quantum kernel classifiers

TL;DR

This work tackles the resource bottleneck of quantum kernel classifiers by shifting focus from exact kernel evaluation to reliable classification under shot noise. It introduces a reliability-centric framework (ShofaR) that uses subgaussian tail bounds and chance constraints to derive bounds on the number of measurements $N$ required to reproduce the ideal classifier with high probability, rather than achieving precise kernel entries. A key finding is that a margin-driven bound with $N$ scaling like $m_{sv}\log M/\gamma^2$ enables dramatic reductions in measurements, and a robust primal formulation further reduces the shots needed (by factors up to $64$ in reported cases) while maintaining or improving reliability, even under depolarizing noise. The approach includes a practical estimation program for the training kernel and norm-relaxations that yield sparser support vectors, making the method viable on near-term quantum hardware. Overall, the paper provides a principled, resource-efficient route to robust quantum-kernel classification with broad applicability to unbiased noise models and realistic quantum devices.

Abstract

Quantum kernel methods are a candidate for quantum speed-ups in supervised machine learning. The number of quantum measurements N required for a reasonable kernel estimate is a critical resource, both from complexity considerations and because of the constraints of near-term quantum hardware. We emphasize that for classification tasks, the aim is reliable classification and not precise kernel evaluation, and demonstrate that the former is far more resource efficient. Furthermore, it is shown that the accuracy of classification is not a suitable performance metric in the presence of noise and we motivate a new metric that characterizes the reliability of classification. We then obtain a bound for N which ensures, with high probability, that classification errors over a dataset are bounded by the margin errors of an idealized quantum kernel classifier. Using chance constraint programming and the subgaussian bounds of quantum kernel distributions, we derive several Shot-frugal and Robust (ShofaR) programs starting from the primal formulation of the Support Vector Machine. This significantly reduces the number of quantum measurements needed and is robust to noise by construction. Our strategy is applicable to uncertainty in quantum kernels arising from any source of unbiased noise.

Figures (8)

• Figure 1: Reliability and Accuracy. Performance of \ref{['eq:skc*']} is shown as a function of $N$, calculated over $N_{\mathrm {trials}}=200$ independent trials. \ref{['eq:ekc*']} is thus The observed accuracies $\mathrm {Acc}\left(f^{(N)}\right)$ of \ref{['eq:skc*']} can show large variations over the different trials (red shaded area shows the range). Reliability $\widehat{\mathcal{R}}(f^{(N)})$ measures the fraction of data points which are reliably classified (definition \ref{['def:emp_rel']}). Notice that even when the average accuracy is high ($>85\%$ at $N=512$), the reliability can be very low ($0\%$ at $N=512$).
• Figure 2: (a) GATES circuit measures the probability of obtaining the final state $\ket{0}^{\otimes n}$. The resulting distribution for $K({\mathbf x},{\mathbf x}')$ is given by (\ref{['gates_dist']}). (b) SWAP circuit measures the probability of obtaining the ancillary qubit (top line) in the $\ket{0}$ state. The resulting kernel estimate is given by (\ref{['swap_dist']}).
• Figure 3: (a) Practical and theoretical bounds on $N$ as a function of the margin $\gamma$. The green shading represents the region $\epsilon_{\gamma}\left(f^{*}\right)=0$ where the stochastic kernel function \ref{['eq:skc*']} is expected to have perfect accuracy, with $\delta_{\mathrm {target}}=0.01$. The red shading represents the region with large margin error, $\epsilon_{\gamma}\left(f^{*}\right)>0.5$, and low accuracy. The theoretical bound is smooth as a function of $\gamma$, but the practical bound shows a rapid change in $N$ around $\gamma=1$. The positions of the vertical green and red boundaries are expected to be problem dependent. The theoretical bound on $N$ to obtain an accurate classifier \ref{['eq:skc*']} is optimal when we choose the largest $\gamma$ in the green region. (b) The $\gamma$-margin error of the Exact Kernel Classifier. $\epsilon_{\gamma}(f^{*})=0$ for $\gamma>0$ implies that \ref{['eq:ekc*']} makes no errors over the dataset, i.e., $\epsilon_{0}(f^{*})=0$. It can be seen that $\gamma^{*} = \mathop{\mathrm{argmax}}\limits_{\gamma} \{\epsilon_{\gamma}(f^{*})=0\}$ lies in the interval $[0.9,1]$.
• Figure 4: (a) Comparison of the reliabilities of classifiers found using \ref{['opt:p']} and the robust formulation \ref{['eq:rob_primal']}, for the training data $\mathcal{D}_{\mathrm {train}}$ itself. (b) The relative accuracy $\textrm{RA}(h^{*},f^{*})$ for the same dataset. Looking at (a) and (b) together, (i) the relative accuracy of $h^{*}$, and (ii) the reliability $\widehat{\mathcal{R}}(h^{(N)},h^{*})$, become 1 much before the reliability $\widehat{\mathcal{R}}(f^{(N)},f^{*})$ reaches 1.
• Figure 5: Comparison of the \ref{['eq:rob_primal']} (top panel) and \ref{['eq:J_rob_l2_est']} (bottom panel) Programs over a test set $\mathcal{D}_{\mathrm {test}}$. The top panel compares the (a) reliability, and (b) relative accuracies, of the stochastic kernel classifiers arising from the \ref{['eq:rob_primal']} and \ref{['opt:p']} Programs. The bottom panel compares the (c) reliability, and (d) relative accuracies, of the stochastic kernel classifiers arising from the \ref{['eq:J_rob_l2_est']} and \ref{['opt:p']} Programs. It is seen that both \ref{['eq:rob_primal']} and \ref{['eq:J_rob_l2_est']} lead to significant savings in $N$ over the nominal SVM obtained by solving \ref{['opt:p']}. The measurement requirements of \ref{['eq:J_rob_l2_est']} are marginally higher than \ref{['eq:rob_primal']} to reproduce \ref{['eq:ekc*']}. \ref{['eq:J_rob_l2_est']} uses an estimated kernel matrix but yet significantly outperforms the nominal \ref{['opt:p']}, even though the latter has access to the exact kernel matrix over the training data.

Theorems & Definitions (38)

• Definition 1: Subgaussian random variable
• Lemma 1
• Definition 2: Exact Kernel Classifier
• Definition 3: Stochastic Kernel Classifier
• Definition 4: Accuracy
• Definition 5: Reliability
• Definition 6: Empirical Reliability
• Definition 7: Circuit Factor
• Remark 1
• proof