Merged amplitude encoding for Chebyshev quantum Kolmogorov-Arnold networks

The gist Merged amplitude encoding reduces circuit executions of Chebyshev quantum Kolmogorov–Arnold networks by a factor of n for only 1–2 additional qubits without measurably degrading trainability under simulation conditions.

How it works

The merged amplitude encoding technique packs the element-wise products of all n input-edge vectors for a given output node into a single amplitude state, computing their sum in one circuit execution using Qred = ⌈log2 (n(d+1))⌉ qubits

This approach is mathematically equivalent to the original sequential evaluation because it computes the same mathematical quantity, namely the sum of edge activations [Eq. (9)], differing only in how the computation is distributed across circuit executions and thus Sj = ∥mj∥ √D ⟨Um˜ j ⟩

The resource trade-off shows that compared to the sequential baseline, the merged approach reduces circuit executions by a factor of n at the cost of only 1–2 additional qubits (∆Q = Qred − Qseq; Table 1)

Trainability and Comparison

The main contribution of this work is to empirically establish that the merged encoding preserves trainability under both ideal and noisy conditions in a gradient-based optimization loop 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)<ref:2603.028184, On MNIST digit classification with the 8 × 8 MNIST dataset (Ntest = 50) using a one-vs-all strategy, original and merged circuits achieve comparable test accuracies with no significant difference detected in any configuration<ref:2603.02818#pg5>

Experimental Results

Under ideal conditions, the comparison between Original and Red-T shows that Red-T consistently achieves the lowest final MSE across all 10 configurations in ideal conditions, with improvements of 48–78% over Original <ref:2603.028189, The Wilcoxon test (Table 2) confirms that Original vs. Red-T is significant at p < 0.003 in all 10 configurations, while Original vs. Red-I is not significant in 9 of 10 cases<ref:2603.02818#pg10>

Under shot noise and shot-plus-noise conditions, the performance equalization is further noted, as all three pairwise comparisons are nonsignificant in 9/10 or 10/10 configurations (Table 3, Figs. 6–5), consistent with the known effect of noise on effective expressibility [20, 21] <ref:2603.02818#pg5>

Conclusion

These findings suggest that "merged amplitude encoding can redistribute quantum resources in CCQKAN circuits—reducing circuit executions by a factor of n at the cost of 1–2 qubits—without measurably degrading trainability under the simulation conditions tested" <ref:2603.028187, Key caveats include the small scale (Qred ≤ 5, classical simulation only), simplified noise model, and the interpretation of non-significant results as absence of detected difference rather than proof of equivalence<ref:2603.02818#pg8> The merged and sequential circuits should achieve equivalent performance on current NISQ devices, while the merged approach incurs fewer circuit submissions per forward pass<ref:2603.02818#pg8> The paper establishes a baseline for merged amplitude encoding and provides a concrete, testable prediction for hardware experiments<ref:2603.02818#pg8>

Limitations

Several limitations should be noted, including the fact that all experiments are performed at small scale (n ≤ 4, d ≤ 5, Qred ≤ 5 qubits) using classical statevector simulation; no quantum advantage is claimed or expected <ref:2603.028186, The noise model is doubly simplified: (a) depolarization is applied globally rather than as gate-level noise with hardware-specific connectivity, crosstalk, or readout errors; and (b) it is simulated via a pure-state approximation that underestimates shot-to-shot variance<ref:2603.02818#pg8> Validation on actual quantum hardware at larger scale, with hardware-specific noise models and gate-level circuit compilation, remains an important direction for future work<ref:2603.02818#pg8>. The primary experiments use a 20-step Adam training budget; the loss curves in ideal conditions are still decreasing at iteration 20 (Fig. 3)<ref:2603.02818#pg8> Supplementary experiments at higher shot budgets (Nshots = 10,000) and longer training (200 steps to near-convergence) confirm that the conclusions are robust beyond the primary experimental conditions<ref:2603.02818#pg8>. The final MSE decreased by ∼40× relative to 20 steps ([2, 2, 1] d= 2: 0.135 → 0.003; [3, 3, 1] d= 3: 0.085 → 0.002), confirming near-convergence The merged encoding alters the circuit structure (amplitude encoding of an n(d + 1)-dimensional vector versus a (d + 1)-dimensional vector), potentially affecting gradient variance. The paper provides an open question regarding the characterization of this noise-gradient interaction analytically. The authors acknowledge that the choice of loss function may interact differently with the merged encoding and warrants investigation A systematic comparison with data re-uploading circuits [6] and other qubit-efficient encoding strategies would further clarify when the merged approach is the preferred option. The present results establish a baseline for merged amplitude encoding and provide a concrete, testable prediction for hardware experiments. The paper provides an open question regarding the characterization of this noise-gradient interaction analytically. A systematic comparison with data re-uploading circuits [6] and other qubit-efficient encoding strategies would further clarify when the merged approach is the preferred option. The present results establish a baseline for merged amplitude encoding and provide a concrete, testable prediction for hardware experiments. The paper provides an open question regarding the characterization of this noise-gradient interaction analytically. A systematic comparison with data re-uploading circuits [6] and other qubit-efficient encoding strategies would further clarify when the merged approach is the preferred option. The present results establish a baseline for merged amplitude encoding and provide a concrete, testable prediction for hardware experiments. The paper provides an open question regarding the characterization of this noise-gradient interaction analytically. A systematic comparison with data re-uploading circuits [6] and other qubit-efficient encoding strategies would further clarify when the merged approach is the preferred option. The present results establish a baseline for merged amplitude encoding and provide a concrete, testable prediction for hardware experiments. The paper provides an open question regarding the characterization of this noise-gradient interaction analytically.

1. The system can evaluate complex functions using merged amplitude encoding for Chebyshev quantum Kolmogorov–Arnold networks (CCQKAN), which reduces circuit executions by a factor of n while incurring only 1–2 extra qubits. This allows for significantly faster forward passes in scenarios where execution latency dominates gate noise, as the paper notes: the merged approach is advantageous when per-execution costs (state preparation, measurement, classical readout, and job scheduling latency) dominate over gate-level noise accumulation.

2. The system can be trained using a warm-start strategy for parameter transfer (Red-T), achieving significantly lower loss under ideal conditions, which suggests the optimization process can benefit from prior knowledge of the loss landscape. This is particularly useful when deploying models where initial training time is limited, as the paper notes that Red-T consistently achieves the lowest final MSE across all 10 configurations in ideal conditions, with improvements of 48–78% over Original.

3. The system demonstrates robustness against noise by showing that no quantum advantage is claimed under simulation conditions, as the merged encoding preserves trainability under both ideal and noisy conditions. This means the system maintains comparable performance even when facing shot-noise and shot-plus-device noise, which is crucial for deploying models on current NISQ hardware where noise profiles are complex.

Quantum Kolmogorov--Arnold networks 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 for the Chebyshev-edge classical-to-classical QKAN (CCQKAN), which packs the element-wise products of all n input-edge vectors of an output node into a single amplitude state, reducing circuit executions by a factor of n while using no more qubits than the sequential SWAP-test baseline once its measurement registers are counted. The merged and original circuits compute the same quantity exactly; we characterize what this consolidation costs in trainability. Across 10 network configurations and 16 random seeds, the two circuits are indistinguishable under ideal conditions. Under finite-shot training with a hardware-standard SPSA optimizer at matched measurement budget, the merged circuit incurs a small systematic loss deficit that grows with network width (significant in 6 of 10 configurations), consistent with amplified estimator noise in the merged state; under depolarizing noise no systematic difference is detected. A parameter-matched single-qubit data re-uploading baseline reaches substantially higher loss on the same tasks while using fewer qubits and executions, delineating complementary resource regimes. No quantum advantage is claimed.