Geometrically Local Quantum Kernel (GLQK)

• GLQK is a quantum machine learning kernel that exploits the exponential decay of correlations to efficiently capture local quantum information.
• It assembles local kernels derived from classical shadows into a polynomial framework, reducing the need for global data in many-body analysis.
• Empirical benchmarks show GLQK achieves near-constant sample complexity and effective phase recognition in translationally symmetric systems.

A Geometrically Local Quantum Kernel (GLQK) is a quantum machine learning kernel construction that leverages the spatial locality of quantum correlations in many-body quantum systems to achieve scalable and efficient learning, particularly in the context of quantum data generated by noncritical (gapped) systems. The GLQK framework is specifically motivated by the widespread physical phenomenon of exponentially decaying correlations and is formulated to address both the high sample complexity associated with standard quantum kernel methods and the need for methods that exploit the inherent structure of quantum many-body data (Chinzei et al., 17 Sep 2025).

1. Conceptual Foundation and Motivation

The foundational principle behind the GLQK is that, in many physically relevant quantum states—especially ground states of noncritical, gapped Hamiltonians—the connected correlation between distant subsystems decays exponentially with their separation (the exponential clustering property, ECP). Mathematically, for observables $O_A$, $O_B$ supported on disjoint regions $A, B$ and a state $\rho$,

$$|\langle O_A O_B \rangle - \langle O_A \rangle \langle O_B \rangle| \le \|O_A\|_S \|O_B\|_S \exp\left(-\frac{\operatorname{dist}(A, B)}{\xi}\right),$$

where $\xi$ is the correlation length. As a result, physically meaningful functions of quantum data, such as polynomials in local observables, can be well-approximated by restricting attention to clusters of qubits of size on the order of $\xi$.

This locality motivates the construction of kernels whose feature space is built exclusively (or predominantly) from local quantum information, in stark contrast to global kernel constructions that depend on the full many-body state or all-body Pauli correlations, which leads to exponentially increasing sample complexity.

2. Mathematical Formulation of GLQK

The GLQK framework begins with a quantum many-body state $\rho$ and a classical shadow $S_T(\rho)$ generated by a randomized measurement protocol. For a set $A(\zeta)$ of (overlapping) subsystems $A$ of size $\zeta$ (typically chosen at or slightly above the correlation length), a local kernel $k_A$ is defined for each region $A$ as

$$k_A(S_T(\rho), S_T(\tilde{\rho})) = \mathrm{function\ of\ local\ expectation\ values}.$$

The full GLQK is then assembled as a polynomial in these local kernels. A canonical form for the kernel (termed the "polynomial GLQK") is

$$k_{\mathrm{GL}}(S_T(\rho), S_T(\tilde{\rho})) = \left[ \frac{1}{|A(\zeta)|} \sum_{A \in A(\zeta)} k_A(S_T(\rho), S_T(\tilde{\rho})) \right]^h,$$

where $h$ is a parameter (often equal to the "local-cover number" of the target function; see below).

A critical result is that for any polynomial function $g(\rho)$ of local observables of bounded body size $m$ and degree $p$, and for states with exponential clustering, there exists a cluster approximation

$g_{\text{CA}}(\rho)$

which is itself a polynomial of local observables supported on spatial regions of diameter $O(\xi \log(\frac{N}{\epsilon}))$ and which approximates $g(\rho)$ to any desired error $\epsilon$ with coefficients that depend only polynomially on $n$ (the system size), $m, p, \epsilon$, and $\|g\|_1$ (the sum of absolute values of the coefficients in $g$). Thus, local quantum information is sufficient to capture the target function up to arbitrary accuracy.

3. Sample Complexity, Local-Cover/Factor Counts, and Scaling

A key metric in quantum machine learning is the number of quantum measurement samples needed for model training as a function of qubit number $n$. If the target function $g(\rho)$ can be decomposed into $\alpha_g$ clusters (the local-cover number) and $\beta_g$ local factors (the local-factor count), then the sample complexity for kernel ridge regression using GLQK is

$N = O\left(\frac{n^{\alpha_g}}{\epsilon^4}\right),$

where $\epsilon$ is the target prediction error. For "local" target polynomial functions (such as sums of local terms), $\alpha_g$ and $\beta_g$ can be constant, resulting in polynomial or even constant sample complexity in $n$.

In contrast, global kernel constructions such as the shadow kernel (Chinzei et al., 17 Sep 2025) require

$N = O\left(\frac{n^{mp}}{\epsilon^4}\right)$

with $m$ the body size and $p$ the polynomial degree, leading to a much steeper scaling when $m, p > 1$.

In the special case of translationally symmetric quantum data, GLQK achieves constant sample complexity independent of $n$, since the effective number of distinct clusters does not grow with system size.

4. Numerical Demonstrations and Comparison

The GLQK has been numerically benchmarked on two classes of tasks:

• Regression from Quantum Dynamics: In tasks involving regression of local, nonlocal, and nonlinear target functions of quantum expectations under random local Hamiltonian evolution, GLQK exhibited almost constant learning accuracy as a function of system size for translationally symmetric cases and significantly flatter scaling compared to the shadow kernel for general cases.
• Quantum Phase Recognition: When distinguishing trivial and SPT phases in bond-alternating XXZ chains, the GLQK kernelized SVM and kernel-PCA representations show clear phase separation (visible cluster separation in feature space) for large system sizes, while global kernels fail to maintain this separation as $n$ increases.

5. Generalization to Broader Settings and Metrics

The construction relies on the properties of exponential clustering but is agnostic to the detailed choice of measurement protocol, subsystem partitioning, or the base local kernel $k_A$. While the formal analysis uses classical shadows, the method extends to other quantum measurement schemes that provide access to local reduced density matrices or local expectation values.

Two central operational metrics defined are:

Measure	Definition	Role in Sample Complexity
Local-cover number $\alpha_g$	Minimal number of local subsystems whose union contains the support of all terms in $g(\rho)$	Sets exponent in sample scaling
Local-factor count $\beta_g$	Effective number of non-redundant local factors in the cluster decomposition of $g(\rho)$	Controls constant scaling with symmetry

These quantities are problem-dependent and can range from 1 (e.g. for sums of local terms in symmetric data) to $m p$ in the worst case.

6. Theoretical Guarantees and Applicability

Central rigorous results in the framework include:

• Cluster Approximation Lemma: Any degree-$p$, $m$-local polynomial can, under ECP, be approximated to error $\epsilon$ by a polynomial supported only on clusters of size $O(\xi \log(n/\epsilon))$.
• Sample Complexity Bound: Given the polynomial GLQK, if the target function’s approximation cluster numbers $(\alpha_g, \beta_g)$ are small, the number of classical shadows needed to guarantee mean-squared prediction error less than $\epsilon^2$ is $N = O(n^{\alpha_g}/\epsilon^4)$ for general data, or $N = O(1/\epsilon^4)$ for translationally symmetric data.
• Kernel Ridge Regression Guarantee: For suitable regularization and sufficient sample size, the resulting predictor achieves the desired accuracy for the class of target functions considered.

GLQK’s framework is designed for settings where the "exponentially decaying correlations" assumption is physically justified (noncritical ground states, non-delocalized quantum data, etc.). The approach may need modification or cannot guarantee improved scaling for critical or highly entangled (volume-law) states.

7. Extensions and Outlook

Several generalizations and future directions are outlined:

• The principle of extracting local features can be adapted beyond kernel methods to quantum deep learning or neural network architectures.
• Investigations into alternative measurement schemes, such as "shallow shadows," may further reduce measurement complexity.
• While the training phase is classical (once measurement data is available), realizing quantum advantage may require learning tasks where the data is classically inaccessible but GLQK can be evaluated efficiently on quantum devices.
• Generalization to higher-dimensional lattices or more exotic geometries follows directly from the definition, as long as exponential clustering holds and local patches can be identified.
• Extensions to target functions beyond polynomials, provided they can be similarly approximated by local expansions.

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GLQK provides a provably scalable framework for supervised learning on quantum many-body data by systematically exploiting geometric locality and correlation decay. It enables efficient regression and classification tasks previously hampered by the curse of dimensionality and offers rigorous theoretical guarantees, substantial empirical improvement in sample complexity, and an extensible foundation for future quantum data-driven methodologies (Chinzei et al., 17 Sep 2025).