hamming-quantum-kernel-svm

name	hamming-quantum-kernel-svm
description	Hamming quantum kernel for SVMs that avoids exponential concentration problem of fidelity quantum kernel. Uses full measurement statistics rather than single fidelity value. Outperforms fidelity kernel at 15+ qubits and classical Gaussian kernel on synthetic quantum data. Scales to 27 qubits without additional quantum resources. Activation: hamming quantum kernel, quantum SVM, exponential concentration, quantum kernel scalability, fidelity kernel alternative, scalable quantum kernel
metadata	{"arxiv_id":"2605.31449","published":"2026-05-29","authors":"Anant Agnihotri, Michael Krebsbach, Florentin Reiter, Thomas Wellens","tags":["quantum","svm","kernel-methods","scalability","qml","classification"]}

Core Problem: Exponential Concentration in Fidelity Quantum Kernels

The fidelity quantum kernel $K(x, x') = |\langle \psi(x) | \psi(x') \rangle|^2$ suffers from exponential concentration as system size increases:

• Kernel values concentrate to a constant as $n_q$ grows
• Prevents efficient scaling beyond few-qubit systems
• Makes quantum SVM useless for large-scale problems

Hamming Quantum Kernel Solution

Instead of using a single fidelity value, the Hamming quantum kernel uses the full measurement statistics:

$$K_H(x, x') = \text{Hamming similarity of measurement outcome distributions}$$

Key properties:

• Classical post-processing only: No additional quantum resources needed
• Same measurement outcomes: Uses data already collected for fidelity kernel
• Avoids exponential concentration: Full statistics preserve information that single fidelity value loses
• Scalable: Demonstrated up to 27 qubits in simulation

Performance Results

Metric	Fidelity Kernel	Hamming Kernel	Classical Gaussian
≤14 qubits	Competitive	Competitive	Competitive
≥15 qubits	Degrades	Outperforms	Varies
Synthetic quantum data	Poor	Outperforms	Outperformed
MNIST	Competitive	Competitive	Competitive

Implementation Pattern

# Pseudocode for Hamming quantum kernel
def hamming_quantum_kernel(measurements_x, measurements_x_prime):
    """
    Compute Hamming kernel from measurement bitstrings.
    
    measurements_x: array of bitstrings from circuit with input x
    measurements_x_prime: array of bitstrings from circuit with input x'
    """
    # Count matching bitstrings across measurement distributions
    matching = count_hamming_similar(measurements_x, measurements_x_prime)
    total = len(measurements_x) * len(measurements_x_prime)
    return matching / total

# Use with sklearn SVM
from sklearn.svm import SVC
K_train = compute_hamming_kernel_all_pairs(train_measurements)
svm = SVC(kernel='precomputed')
svm.fit(K_train, train_labels)

When to Use

• Quantum SVM problems with ≥15 qubits where fidelity kernel fails
• Classical post-processing is acceptable (no quantum overhead)
• Synthetic quantum data classification
• Scaling quantum kernel methods beyond NISQ limitations

Key Advantage

Zero additional quantum cost: The Hamming kernel uses the same measurement data as the fidelity kernel. The improvement comes entirely from better classical processing of existing quantum data.

Pitfalls

• Requires sufficient measurement shots for reliable statistics
• Classical computation cost grows with number of shots (not qubits)
• Still needs quantum circuit execution for data encoding