Trained quantum Kolmogorov--Arnold networks can dequantize, and a discrete-logarithm encoding need not: a measurement-based map of where quantum advantage can live
Abstract Quantum machine-learning models are only useful if a classical computer cannot cheaply imitate them. We test this directly for quantum Kolmogorov–Arnold networks (QKANs)—neural networks whose tunable “activation functions” are small quantum circuits—by asking whether each model can be contracted as an effcient tensor network (TN), the standard tool for classical quantum simulation. We make the test practical with a task-aware yardstick: the bond dimension a classical surrogate needs to copy the model’s output at the accuracy the model itself reaches. We find a clear ladder. The amplitude-encoded Chebyshev QKAN uses only logarithmically many qubits and is exactly classical. The variational QKAN, built from all-to-all controlled rotations, looks hard at random settings but, once trained to genuinely learn a smooth task (R2 = 0.30–0.62), is copied by a constant-bond-dimension surrogate at the sizes (n ≤ 10) we test. Noise pushes it toward classicality near p ≈ 0.03–0.05, echoing noisy random circuits. Every generic attempt to force hardness—a global measurement, a deeper entangler, or an interaction-based (IQP) feature map—instead destroys trainability through exponentially small gradients (a “barren plateau”). The escape is a discrete-logarithm feature map: its kernel variance is flat (0.07±0.01) from 4 to 14 qubits while a generic IQP kernel decays as 2 −n , and a support-vector machine with it learns a classically hard concept to ≥ 99% accuracy while generic and classical kernels stay at chance—hardness placed in number theory stays both classically hard and shot-effciently learnable. The work provides a concrete, measurement-based criterion for when a quantum model is reproducible by a low-bond- dimension surrogate, and a design rule: seek advantage in structured, provably hard encodings rather than in deep entanglement.
Abstract Quantum machine-learning models are only useful if a classical computer cannot cheaply imitate them. We test this directly for quantum Kolmogorov–Arnold networks (QKANs)—neural networks whose tunable “activation functions” are small quantum circuits—by asking whether each model can be contracted as an effcient tensor network (TN), the standard tool for classical quantum simulation. We make the test practical with a task-aware yardstick: the bond dimension a classical surrogate needs to copy the model’s output at the accuracy the model itself reaches. We find a clear ladder. The amplitude-encoded Chebyshev QKAN uses only logarithmically many qubits and is exactly classical. The variational QKAN, built from all-to-all controlled rotations, looks hard at random settings but, once trained to genuinely learn a smooth task (R2 = 0.30–0.62), is copied by a constant-bond-dimension surrogate at the sizes (n ≤ 10) we test. Noise pushes it toward classicality near p ≈ 0.03–0.05, echoing noisy random circuits. Every generic attempt to force hardness—a global measurement, a deeper entangler, or an interaction-based (IQP) feature map—instead destroys trainability through exponentially small gradients (a “barren plateau”). The escape is a discrete-logarithm feature map: its kernel variance is flat (0.07±0.01) from 4 to 14 qubits while a generic IQP kernel decays as 2 −n , and a support-vector machine with it learns a classically hard concept to ≥ 99% accuracy while generic and classical kernels stay at chance—hardness placed in number theory stays both classically hard and shot-effciently learnable. The work provides a concrete, measurement-based criterion for when a quantum model is reproducible by a low-bond- dimension surrogate, and a design rule: seek advantage in structured, provably hard encodings rather than in deep entanglement.
Abstract Quantum machine-learning models are useful only if classical computers cannot cheaply imitate them. We test this directly for quantum Kolmogorov--Arnold networks (QKANs)---neural networks whose tunable ``activation functions'' are small quantum circuits---by asking whether each model can be contracted as an efficient tensor network (TN). Our task-aware yardstick is the bond dimension a classical surrogate needs to copy the model's output at the accuracy the model itself reaches. We find a clear ladder. The amplitude-encoded Chebyshev QKAN uses only logarithmically many qubits and is exactly classical. The variational QKAN, built from all-to-all controlled rotations, looks hard at random settings but generalizes only weakly once trained (held-out R2≤0.27) and is copied at that attained accuracy by a low-bond-dimension surrogate (n≤10); at fixed strict tolerance the required bond dimension grows near-exponentially. Depolarizing noise erases it (purity collapse) near p≈0.03--0.05 at the sizes we reach. Every generic attempt to force hardness---a global measurement or an interaction-based (IQP) feature map---instead destroys trainability through exponentially small gradients (a ``barren plateau''). The escape is a discrete-logarithm feature map: its kernel variance is flat from 4 to 14 qubits while a generic IQP kernel concentrates exponentially, and a support-vector machine with it learns a classically hard concept to ≥99% accuracy while generic and classical kernels stay at chance; wired into the KAN architecture, it fits discrete-log targets to R2≈1. Hardness placed in number theory stays classically hard, shot-efficient, and learnable. The work provides a measurement-based reproducibility criterion and a design rule: seek advantage in structured, provably hard encodings rather than in deep entanglement.
Variational quantum circuits (VQCs) are central to quantum machine learning, while recent progress in Kolmogorov-Arnold networks (KANs) highlights the power of learnable activation functions. We unify these directions by introducing the quantum variational activation function (QVAF), a general framework in which parameterized quantum circuits serve as learnable activation functions; in this work we study an efficient single-qubit instantiation called DatA Re-Uploading ActivatioN (DARUAN). We show that DARUAN with trainable data-preprocessing weights can realize an exponentially growing accessible frequency support with the number of re-uploading repetitions; for an explicit geometric choice of these weights, this gives a capacity-level exponential parameter reduction relative to independently parameterized Fourier activations. Embedding DARUAN into KAN yields the quantum-inspired Kolmogorov-Arnold Network (QKAN), which retains the interpretability of the KAN architecture while improving parameter efficiency, expressivity, and generalization. We further introduce layer extension and the hybrid QKAN (HQKAN) architecture to improve scalability and computational efficiency, enabling QKAN modules to act as compact replacements for multi-layer perceptrons (MLPs) in large-scale models. We provide theoretical analysis and extensive experiments on function regression, image classification, and autoregressive generative language modeling, demonstrating the efficiency and scalability of QKANs. Because the single-qubit circuits are efficiently simulable on classical quantum simulators, QKANs have quantum-inspired advantage in parameter efficiency and training stability; DARUANs and QKANs serve as present-day validation of the QVAF concept, and the trained DARUANs are directly executable and feasible on current noisy intermediate-scale quantum (NISQ) hardware for inference validation.
The Kolmogorov-Arnold Network (KAN) places the trainable functions on the synapses rather than on the neurons. Existing quantum implementations either lack accuracy (Variational Quantum KAN, VQKAN) or rely on block encoding and Quantum Signal Processing, which demand many control gates and ancillae. We propose the Enhanced Variational Quantum Kolmogorov-Arnold Network (EVQKAN), a variational ansatz that emulates a 2Nq-dimensional KAN layer matrix by tiling controlled rotations through a sum-operator construction, using only 2Nq−1 trainable spline functions per layer. On the fitting of an elementary function, EVQKAN attains a significantly lower test error than Quantum Neural Networks (QNN), VQKAN and Adaptive VQKAN (Mann-Whitney p<0.002, Cliff's δ≤−0.86 over ten attempts; EVQKAN beats VQKAN on every attempt), though classical KAN is more accurate still. On a two-dimensional classification task the ordering reverses: under a leak-free protocol introduced here, EVQKAN classifies above chance (accuracy 0.620, p=0.0005) but is significantly less accurate than a QNN carrying one fifth as many parameters (Δaccuracy −0.134, p=0.0014; ΔAUC −0.252, p=0.0002). We withdraw the classification results of an earlier version of this work: their encoding placed the target label into the circuit as a feature for EVQKAN but not for the methods it was compared against. The dominant error source is overfitting from an under-determined training set; enlarging that set closes the train-test gap by 58% (Spearman p<10−3). We also report the circuit cost in full --- three layers emit 1017 operations, or 4110 two-qubit gates once the multi-controlled gates are decomposed --- so the construction is simulator-scale and fault-tolerant-era rather than NISQ-ready, with block encoding and qubitization the route to reducing it.
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.
Kolmogorov-Arnold networks place learnable one-dimensional functions on the edges of a network rather than fixed activations on its nodes, and several quantum realisations have been proposed. Whether any of them offers a practical benefit is unclear, because the reported comparisons rest on single training runs, untuned baselines, and test sets that were also used for model selection. Here we evaluate a variational quantum Kolmogorov-Arnold network whose ansatz is grown one Pauli operator at a time, under a protocol with seed-paired comparisons, a stopping rule that never sees the test set, and a confirmatory study whose hypotheses, seeds, and analysis were committed before the runs. On four-qubit benchmarks the model is indistinguishable from a quantum neural network of the same parameter count and is outperformed by classical regressors. On a difficulty-controlled family of 12- and 16-dimensional targets learned from ten training points, the 32-parameter quantum model beats the best of four unregularised classical regressors and a tuned quantum neural network with Holm-corrected p <= 0.017; the result is unchanged with 256 measurement shots per circuit, and the trained models run on a 156-qubit IBM processor with test errors within 0.3 of the exact values. A kernel ridge regressor with hyperparameters chosen by leave-one-out cross-validation on the same ten points matches the quantum model, and a sample-size sweep shows that the quantum model's error is nearly flat in the training-set size while the classical models keep improving. The benefit of the quantum model is therefore an implicit regularisation of a low-capacity model, present only in the few-sample regime and absent at 18 dimensions, not an expressivity advantage. These results give a reproducible reference point for the resources and limits of variational quantum Kolmogorov-Arnold networks.
We introduce quantum Kolmogorov-Arnold networks (QKAN), a quantum algorithmic framework inspired by the recently proposed Kolmogorov-Arnold Networks (KAN). QKAN inherits the compositional structure of KAN and is based on block-encodings, constructed recursively from a single layer using quantum singular value transformation. We demonstrate the algorithmic utility of QKAN in two applications. First, we introduce and analyze QKAN as a quantum learning model, treating the eigenvalues of block-encoded matrices as neurons and applying parametrized activation functions on the edges of the network. We show that QKAN is a wide-and-shallow neural architecture, where shallow depth is compensated by exponentially wide layers whenever efficient block-encodings of inputs are available. We further discuss how to parametrize and train QKAN using parametrized quantum circuits and quantum linear algebra subroutines. Second, we demonstrate that QKAN can serve as a multivariate quantum state-preparation protocol for functions with shallow compositional structure. We demonstrate this by efficiently preparing a multivariate Gaussian quantum state using a two-layer QKAN. Looking forward, we anticipate that QKAN's compositional and modular design will enable new applications in quantum machine learning and quantum state preparation.