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

TL;DR

Empirical evidence is provided that merged amplitude encoding preserves trainability under the simulation conditions tested, and introduces merged amplitude encoding, a technique that packs the element-wise products of all input-edge vectors for a given output node into a single amplitude state.

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.

Figures (13)

• Figure 1: Resource trade-off for all 10 configurations. Each configuration maps to three points: parallel (circle), sequential (square), merged (triangle). The merged approach reduces circuit executions $C$ by a factor of $n$ relative to sequential, at a cost of 1--2 additional qubits.
• Figure 2: Final MSE loss distributions under ideal conditions (16 seeds) for four representative configurations. Box: interquartile range; line: median; dots: individual seeds. Blue: Original, orange: Red-T, green: Red-I. The overlap between Original and Red-I distributions is consistent with the non-significant Wilcoxon results.
• Figure 3: Training loss curves under ideal conditions for all 10 configurations ($[n,n,1]$, degrees as labeled). Solid lines: mean over 10 seeds; shaded regions: $\pm 1$ standard deviation. Blue: Original, orange: Red-T (parameter transfer), green: Red-I (independent initialization).
• Figure 4: Training loss curves under 1000-shot measurement noise. Colors as in Fig. \ref{['fig:ideal_loss']}. Shot noise dominates the loss, compressing the differences between models.
• Figure 5: Final MSE loss distributions under shot noise plus depolarizing noise ($p=0.01$) for four representative configurations (16 seeds). Colors as in Fig. \ref{['fig:boxplot_ideal']}. All three model distributions overlap substantially, consistent with the non-significant Wilcoxon results.