[Paper Review] Covariant quantum kernels for data with group structure

Jennifer R. Glick, Tanvi P. Gujarati+5 more|arXiv (Cornell University)|May 7, 2021

Quantum Computing Algorithms and Architecture53 references33 citations

TL;DR

The paper introduces covariant quantum kernels for data with group structure, defining kernels via unitary group representations and a tunable fiducial state, and demonstrates learning on coset-structured data with a 27-qubit superconducting device using kernel alignment.

ABSTRACT

The use of kernel functions is a common technique to extract important features from data sets. A quantum computer can be used to estimate kernel entries as transition amplitudes of unitary circuits. Quantum kernels exist that, subject to computational hardness assumptions, cannot be computed classically. It is an important challenge to find quantum kernels that provide an advantage in the classification of real-world data. We introduce a class of quantum kernels that can be used for data with a group structure. The kernel is defined in terms of a unitary representation of the group and a fiducial state that can be optimized using a technique called kernel alignment. We apply this method to a learning problem on a coset-space that embodies the structure of many essential learning problems on groups. We implement the learning algorithm with $27$ qubits on a superconducting processor.

Motivation & Objective

• Motivate and address the challenge of achieving quantum advantage in kernel-based learning for real-world data with group structure.
• Define covariant quantum kernels using unitary group representations and fiducial states that can be efficiently prepared on quantum hardware.
• Show how fiducial-state optimization (kernel alignment) improves classification performance on coset-structured data.
• Demonstrate a hardware experiment implementing covariant kernels on a 27-qubit superconducting processor.
• Highlight the role of error mitigation in enhancing kernel quality and learning outcomes.

Proposed method

• Define covariant feature maps Phi(x)=D_x |psi><psi| D_x^† using a unitary group representation D_x and a fiducial state |psi> prepared by an efficient circuit V.
• Express the kernel as K(x, x̃)=|<psi| D_x^† D_x̃ |psi>|^2, equivalent to a transition amplitude between feature states.
• Implement the quantum kernel estimation (QKE) by estimating the overlap via measurement of the all-zero outcome after applying D_x^† D_x̃ to the fiducial state.
• Optimize the fiducial state |psi> through kernel alignment, solving min_lambda max_alpha F(alpha, lambda) with F given by the SVM-related upper bound on generalization error.
• Use a stochastic SPSA-based procedure to update fiducial-state parameters lambda based on kernel matrices evaluated on a quantum processor.
• Experimentally validate with a coset-space learning problem (LCE) and compare mitigated vs unmitigated hardware runs.

Figure 1: Labeling cosets . (a), (b) Two covariant feature maps for a single-qubit example of the labeling cosets learning problem introduced in the text. We take $S=\{\mathds{1},A,A^{2}\}$ as subgroup of $G=SU(2)$ , where $A=\exp(i(2\pi/3)X)$ . Choosing two elements ${\bm{c}}_{+},{\bm{c}}_{-}\in SU

Experimental results

Research questions

• RQ1 Can covariant quantum kernels exploit group structure to achieve advantages over classical learners for coset-based data?
• RQ2 How does the choice of fiducial state influence kernel expressivity and learning performance?
• RQ3 What is the impact of error mitigation on the quality of quantum kernel estimates and downstream classification accuracy?
• RQ4 How does kernel alignment guide fiducial-state optimization in a practical quantum hardware setting?

Key findings

• A covariant quantum kernel defined via a group representation and fiducial state provides a learnable, left-invariant kernel suitable for group-structured data.
• Kernel alignment with a variational fiducial state improves classification performance and can reach ideal parameter values faster under error mitigation.
• Hardware demonstration on a 27-qubit superconducting processor shows successful kernel estimation and high-accuracy classification for the LCE coset problem.
• Error mitigation significantly improves the kernel cost landscape and accelerates convergence of the fiducial-state optimization.
• The experimental setup uses SU(2)^{⊗27} with a graph-stabilizer subgroup, demonstrating the practicality of covariant kernels on near-term devices.
• The results align with theoretical indications that quantum kernels can outperform classical learners on specific group-structured tasks.

Figure 2: Device layout and circuit mapping. (a) The connectivity of the 27-qubit device ibmq_kolkata . (b) The quantum circuit used to evaluate the kernel matrix elements for the learning problem labeling cosets with errors . Here, we define the single-qubit rotations as $R_{P}(\phi)=\exp(-i(\phi/2