In Search of Quantum Advantage: Estimating the Number of Shots in Quantum Kernel Methods

TL;DR

The paper tackles the practicality of quantum kernel estimation (QKE) by formalizing shot-budget rules that account for two key effects: the spread of kernel values within a Gram matrix and exponential concentration toward a fixed value as quantum device size grows. It introduces per-entry shot estimates $\tilde{N}_{spread}$ and $\tilde{N}_{CA}$ and aggregates them into a dataset-wide budget $\bar{N}$, while extending the framework to noisy circuits with $N^f_{spread}$ and $\tilde{N}_{CA}^{FQ/PQ}$. Theoretical groundwork covers data embedding, fidelity and projected kernels, and the Bernoulli-measurement model, all tied to explicit bounds and practical formulas. Through a case study on ZZ-feature maps and real datasets, the authors show that exponential concentration can drive shot requirements up rapidly, challenging the practical advantage of QKE for many problems, though they acknowledge that certain kernel families or data structures might still offer feasible pathways. Overall, the work provides a concrete, quantitative planning toolkit for quantum kernel experiments and highlights substantial resource costs that currently limit the real-world utility of QKE.

Abstract

Quantum Machine Learning (QML) has gathered significant attention through approaches like Quantum Kernel Machines. While these methods hold considerable promise, their quantum nature presents inherent challenges. One major challenge is the limited resolution of estimated kernel values caused by the finite number of circuit runs performed on a quantum device. In this study, we propose a comprehensive system of rules and heuristics for estimating the required number of circuit runs in quantum kernel methods. We introduce two critical effects that necessitate an increased measurement precision through additional circuit runs: the spread effect and the concentration effect. The effects are analyzed in the context of fidelity and projected quantum kernels. To address these phenomena, we develop an approach for estimating desired precision of kernel values, which, in turn, is translated into the number of circuit runs. Our methodology is validated through extensive numerical simulations, focusing on the problem of exponential value concentration. We stress that quantum kernel methods should not only be considered from the machine learning performance perspective, but also from the context of the resource consumption. The results provide insights into the possible benefits of quantum kernel methods, offering a guidance for their application in quantum machine learning tasks.

Figures (7)

• Figure 1: The general idea behind the quantum kernel machines. First, the classical or quantum data is encoded into the quantum circuits. The circuits are run, the kernel matrix entries or density matrices are estimated. For different quantum kernel families the subsequent classical post-processing might be necessary. Then, the results of a quantum kernel estimation serve as an input to classical routines.
• Figure 2: Schematic quantum circuits used for quantum fidelity and projected kernel estimation. Left: Fidelity quantum kernel circuit for the concatenation method of the kernel value estimation. Two data points $x,y$ are encoded-decoded with the embedding map unitary transformation $U(\cdot)$. The outcome of the measurement is used for gathering the final vacuum/non-vacuum state statistics. Right: Projected quantum kernel circuit for the one-qubit quantum state tomography. First, the data point $x$ is encoded with the feature map unitary transformation $U(x)$, producing full $n$-qubit embedded state. Then, for each of the qubit registers,the one-qubit reduced density matrix is measured in different bases. After performing quantum tomography of the reduced states, the estimated density matrices are processed further classically in order to obtain projected kernel values.
• Figure 3: Schematic representation of the two effects taken into account during the number of shots, $N$, estimation. Left: The comparison between the distribution of single quantum kernel measurement with the histogram of the independent kernel values in the kernel matrix. The uncertainty of the kernel value estimation $\Delta_{kernel\ value}$ is similar to the spread of the independent kernel values in the kernel matrix $\Delta_{ensemble}$. Increasing $N$, reduces the $\Delta_{kernel\ value}$ allowing for distinguishing the single kernel values in the ensemble. Right: The uncertainty related to the measurement is big enough to face challenges with distuingishing whether measured value $\hat{\mathcal{M}}$ is less or greater than the concentration value $\mu_{\mathcal{M}}$. Increasing $N$ allows to boost the probability of measuring the $\hat{\mathcal{M}}$ value being smaller than the concentration value $\mu_{\mathcal{M}}$. The plot is created for simulated data, the distributions are rescaled to improve the clarity of the reception of the figure.
• Figure 4: Exponential concentration for the Connectionist Bench dataset, embedded into the quantum state with ZZ-feature map and for the fidelity (Eq. \ref{['eq:fidelity_kernel']}) (upper panel) and projected kernel (Eq. \ref{['eq:projected_kernel']}) (lower panel). The dashed lines indicate extrapolations. Left: Exponential concentration for the mean of independent kernel entries. Right: Exponential concentration for the standard deviation of independent kernel entries.
• Figure 5: Number of shots $N$ estimation for Connectionist Bench dataset, embedded into the quantum state with the ZZ-feature map (Eq. \ref{['eq:ZZmap']}), for the fidelity (left) and projected (right) quantum kernel families as a function of number of qubits $n$. The full entanglement strategy was employed in both cases, the results are presented with different feature map repetitions $r$ and are divided into two bound for $N$: spread (Eq. \ref{['eq:Spread']}) and concentration avoidance (Eq. \ref{['eq:sr']}).

Theorems & Definitions (1)

• Definition 2.1: Exponential kernel value concentration