Merged amplitude encoding for Chebyshev quantum Kolmogorov--Arnold networks: trading qubits for circuit executions

TL;DR

This paper introduces merged amplitude encoding for Chebyshev quantum networks, significantly reducing circuit executions with minimal additional qubits, and demonstrates empirically that trainability is preserved.

Contribution

The authors propose merged amplitude encoding, which packs multiple input vectors into one amplitude state, reducing circuit executions while maintaining trainability.

Findings

Merged amplitude encoding reduces circuit executions by a factor of n.

No significant difference in trainability between original and merged circuits under tested conditions.

Comparable classification accuracy achieved with merged encoding on MNIST dataset.

Abstract

Quantum Kolmogorov--Arnold networks based on Chebyshev polynomials (CCQKAN) evaluate each edge activation function as a quantum inner product, creating a trade-off between qubit count and the number of circuit executions per forward pass. We introduce merged amplitude encoding, a technique that packs the element-wise products of all $n$ input-edge vectors for a given output node into a single amplitude state, reducing circuit executions by a factor of $n$ at a cost of only 1--2 additional qubits relative to the sequential baseline. The merged and original circuits compute the same mathematical quantity exactly; the open question is whether they remain equally trainable within a gradient-based optimization loop. We address this question through numerical experiments on 10 network configurations under ideal, finite-shot, and noisy simulation conditions, comparing original, parameter-transferred, and independently initialized merged circuits over 16 random seeds. Wilcoxon signed-rank tests show no significant difference between the independently initialized merged circuit and the original ($p > 0.05$ in 28 of 30 comparisons), while parameter transfer yields significantly lower loss under ideal conditions ($p < 0.001$ in 9 of 10 configurations). On 10-class digit classification with the $8\times8$ MNIST dataset using a one-vs-all strategy, original and merged circuits achieve comparable test accuracies of 53--78\% with no significant difference in any configuration. These results provide empirical evidence that merged amplitude encoding preserves trainability under the simulation conditions tested.