[Paper Review] Quantum Fisher kernel for mitigating the vanishing similarity issue

Yudai Suzuki, Hideaki Kawaguchi+1 more|arXiv (Cornell University)|Oct 29, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes the quantum Fisher kernel (QFK), a novel quantum kernel method that mitigates the vanishing similarity issue—where fidelity-based quantum kernels lose discriminative power with increasing qubits—by leveraging anti-symmetric logarithmic derivatives (ALD) in an alternating layered ansatz (ALA). The ALDQFK maintains high expressivity comparable to fidelity-based kernels while avoiding exponential decay in kernel values, enabling practical quantum advantage in machine learning.

ABSTRACT

Quantum kernel method is a machine learning model exploiting quantum computers to calculate the quantum kernels (QKs) that measure the similarity between data. Despite the potential quantum advantage of the method, the commonly used fidelity-based QK suffers from a detrimental issue, which we call the vanishing similarity issue; detecting the difference between data becomes hard with the increase of the number of qubits, due to the exponential decrease of the expectation and the variance of the QK. This implies the need to design QKs alternative to the fidelity-based one. In this work, we propose a new class of QKs called the quantum Fisher kernels (QFKs) that take into account the geometric structure of the data source. We analytically and numerically demonstrate that the QFK based on the anti-symmetric logarithmic derivatives (ALDQFK) can avoid the issue when the alternating layered ansatzs (ALAs) are used, while the fidelity-based QK cannot even with the ALAs. Moreover, the Fourier analysis numerically elucidates that the ALDQFK can have expressivity comparable to that of the fidelity-based QK. These results indicate that the QFK paves the way for practical applications of quantum machine learning with possible quantum advantages.

Motivation & Objective

• Address the vanishing similarity issue in fidelity-based quantum kernels, where kernel values decay exponentially with qubit count, undermining quantum advantage.
• Develop a quantum kernel design that incorporates the geometric structure of data sources to improve stability and generalization.
• Propose the quantum Fisher kernel (QFK) as a quantum extension of the classical Fisher kernel, embedding data structure into kernel computation.
• Demonstrate that the ALDQFK with alternating layered ansatz (ALA) avoids the vanishing similarity issue where fidelity-based kernels fail.
• Establish that the ALDQFK maintains expressivity comparable to fidelity-based kernels through Fourier analysis and classification benchmarks.

Proposed method

• Introduce the quantum Fisher kernel (QFK) as a quantum extension of the classical Fisher kernel, using the quantum Fisher information matrix to encode data structure.
• Define the ALDQFK using anti-symmetric logarithmic derivatives (ALD) of the density matrix ρx,θ, expressed as Tr[ρx,θ B ρx',θ B] with B = ∂ρx,θ/∂θ.
• Implement the ALDQFK using an alternating layered ansatz (ALA) for parameterized quantum circuits, ensuring structural robustness against vanishing kernel values.
• Apply Fourier analysis to decompose the QK into frequency components, enabling comparison of expressivity between ALDQFK and fidelity-based QK.
• Use support vector machines (SVC) with synthetic one-dimensional datasets to evaluate classification performance and validate kernel expressivity.
• Truncate Fourier frequency sets to {−12,…,12} and fit QKs to their Fourier representations using curve-fitting algorithms for quantitative comparison.

Figure 1: Summary of this work. (a) The vanishing similarity issue for the fidelity-based QK, where the expectation and the variance of the QK decrease exponentially as the number of qubits is increased. Hence, estimating each entry of the Gram matrix on a real quantum device requires an exponential

Experimental results

Research questions

• RQ1 Can a quantum kernel design that incorporates data source geometry avoid the vanishing similarity issue in high-dimensional Hilbert spaces?
• RQ2 Does the ALDQFK with ALA maintain expressivity comparable to the fidelity-based quantum kernel despite avoiding exponential decay?
• RQ3 How does the Fourier spectrum of the ALDQFK compare to that of the fidelity-based QK in terms of frequency content and amplitude?
• RQ4 Can the ALDQFK achieve low misclassification rates on synthetic datasets requiring non-trivial frequency components?
• RQ5 What is the relationship between the ALDQFK structure and known quantum dynamical quantities like Loschmidt echo or out-of-time-ordered correlators?

Key findings

• The ALDQFK with ALA avoids the vanishing similarity issue, as the expectation and variance of kernel values do not decay exponentially with qubit count, unlike fidelity-based QK.
• The ALDQFK exhibits Fourier coefficients with amplitudes and frequencies nearly identical to those of the fidelity-based QK, indicating comparable expressivity.
• Classification tasks on synthetic sine-based datasets show that the ALDQFK achieves low misclassification rates, with performance strongly correlated to the amplitude of Fourier coefficient cω,−ω.
• For datasets with frequency w=12, the ALDQFK maintains non-zero cω,−ω components, while fidelity-based QK shows near-zero amplitudes due to vanishing kernel values.
• The ALDQFK structure Tr[ρx,θ B ρx',θ B] is isomorphic to quantum dynamical measures such as the Loschmidt echo and out-of-time-ordered correlators, suggesting deeper connections to quantum chaos and information scrambling.
• The ALDQFK enables practical quantum advantage by maintaining stable kernel values and generalization performance even for large-scale quantum systems.

Figure 2: The expectation and the variance of the QK in Eq. ( 1 ), where each element of input data $\bm{x}$ and parameters in PQC $\bm{\theta}$ is randomly chosen from the range $[-\pi,\pi)$ . The tensor-product and the IQP-type quantum circuits with depth $L=2$ are used as $U(\bm{x},\bm{\theta})$