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arXiv:2606.11814

Sparsified Kolmogorov-Arnold Networks for Interpretable Quantum State Tomography

Xinge Wu, Huaxin Wang, Jiajun Liu, Ruiqing He, Jiandong Shang, Hengliang Guo, Qiang Chen

quant-phcs.AIcs.LGSubmitted Jun 10, 2026• Announced Jun 11, 2026• 0 citations
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Abstract

Machine-learning approaches to quantum state tomography can achieve high reconstruction fidelity, but the physical structure used by the trained model often remains implicit. Here we ask whether a sparsified Kolmogorov-Arnold Network (KAN) can be used not only as a regressor, but also as an inspectable reconstruction rule whose internal organization can be checked against known Pauli structure. We study a controlled three-qubit GHZ-family benchmark in which all 63 non-identity Pauli expectation values are used to reconstruct three GHZ-subspace variables: the population imbalance $z$, the real off-diagonal component $c$, and the imaginary off-diagonal component $s$. Under finite-shot sampling and depolarizing noise, external ablation identifies the extended 12-channel GHZ-relevant Pauli set from the 63 measurements, with exact top-12 recovery across the tested shot counts and depolarizing-noise strengths. These support patterns remain stable across multi-seed random-initialization and noise-level analyses, and collapse under random-label controls. The dominant pruned input-hidden-output pathways organize Z-type population observables and X/Y off-diagonal observables in a pattern consistent with the analytic GHZ Pauli grouping, and sparse formula recovery recovers the canonical signed Pauli relations. The contribution of the KAN is therefore pathway-level structural interpretability within a neural reconstruction model, rather than superior sparse regression. Together with negative controls, these probes provide a consistency chain for auditing learned reconstruction rules against known physical structure.

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Hikaru Wakaura, Rahmat Mulyawan, Andriyan Bayu Suksmono

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.

arXiv:2503.21336•Mar 27, 2025•0 citations

Quantum Kolmogorov--Arnold networks are classically reproducible at their attained accuracy, and a discrete-logarithm encoding need not be: a measurement-based map of where quantum advantage can live

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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 $R^2\!\le\!0.27$) and is copied at that attained accuracy by a low-bond-dimension surrogate ($n\!\le\!10$); at fixed strict tolerance the required bond dimension grows near-exponentially. Depolarizing noise erases it (purity collapse) near $p\!\approx\!0.03$--$0.05$ 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 $\ge99\%$ accuracy while generic and classical kernels stay at chance; wired into the KAN architecture, it fits discrete-log targets to $R^2\!\approx\!1$. 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.

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Abstract Quantum steering is an important form of nonclassical correlation and plays a central role in one-sided device-independent quantum information processing. However, efficiently determining whether an arbitrary two-qubit state is steerable remains challenging, since semidefinite-programming-based certification generally depends on the choice of measurement settings and can become computationally demanding. In this work, we propose an interpretable quantum steering detection scheme based on Kolmogorov–Arnold Networks (KANs). The training labels are generated through semidefinite programming for randomly sampled two-qubit density matrices. Several physically motivated feature representations are constructed to investigate the influence of measurement and feature reduction. Owing to their learnable univariate edge functions and symbolic formula extraction capability, KANs provide not only accurate classification but also a more transparent representation of the learned steering decision boundary. Numerical experiments show that the proposed KAN-based classifiers can distinguish states with SDP-detected steering from states for which no witness is found under the sampled measurements. In particular, canonical-correlation features provide an effective representation for learning the steering boundary, and further feature compression can still retain competitive detection performance. These results indicate that KANs offer a promising, efficient, and interpretable machine-learning approach for finite-measurement steering detection of arbitrary two-qubit states.

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Enhanced Variational Quantum Kolmogorov-Arnold Network

Hikaru Wakaura, Rahmat Mulyawan, Andriyan Bayu Suksmono

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 $2^{N_q}$-dimensional KAN layer matrix by tiling controlled rotations through a sum-operator construction, using only $2^{N_q-1}$ 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 $δ\leq-0.86$ 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 ($Δ$accuracy $-0.134$, $p=0.0014$; $Δ$AUC $-0.252$, $p=0.0002$). 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.

arXiv:2503.22604•Mar 28, 2025•0 citations

Trained quantum Kolmogorov--Arnold networks can dequantize, and a discrete-logarithm encoding need not: a measurement-based map of where quantum advantage can live

Hikaru Wakaura

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.

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QKAN: quantum Kolmogorov-Arnold networks with applications in machine learning and multivariate state preparation

Petr Ivashkov, Po‐Wei Huang, Kelvin Koor +2 more

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.

arXiv:2410.04435•npj Quantum Information•Mar 11, 2026•4 citations
Machine-learning approaches to quantum state tomography can achieve high reconstruction fidelity, but the physical structure used by the trained model often remains implicit. Here we ask whether a sparsified Kolmogorov-Arnold Network (KAN) can be used not only as a regressor, but also as an inspectable reconstruction rule whose internal organization can be checked against known Pauli structure. We study a controlled three-qubit GHZ-family benchmark in which all 63 non-identity Pauli expectation values are used to reconstruct three GHZ-subspace variables: the population imbalance zzz, the real off-diagonal component ccc, and the imaginary off-diagonal component sss. Under finite-shot sampling and depolarizing noise, external ablation identifies the extended 12-channel GHZ-relevant Pauli set from the 63 measurements, with exact top-12 recovery across the tested shot counts and depolarizing-noise strengths. These support patterns remain stable across multi-seed random-initialization and noise-level analyses, and collapse under random-label controls. The dominant pruned input-hidden-output pathways organize Z-type population observables and X/Y off-diagonal observables in a pattern consistent with the analytic GHZ Pauli grouping, and sparse formula recovery recovers the canonical signed Pauli relations. The contribution of the KAN is therefore pathway-level structural interpretability within a neural reconstruction model, rather than superior sparse regression. Together with negative controls, these probes provide a consistency chain for auditing learned reconstruction rules against known physical structure.
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.
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 ⁣≤ ⁣0.27R^2\!\le\!0.27R2≤0.27) and is copied at that attained accuracy by a low-bond-dimension surrogate (n ⁣≤ ⁣10n\!\le\!10n≤10); at fixed strict tolerance the required bond dimension grows near-exponentially. Depolarizing noise erases it (purity collapse) near p ⁣≈ ⁣0.03p\!\approx\!0.03p≈0.03--0.050.050.05 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 444 to 141414 qubits while a generic IQP kernel concentrates exponentially, and a support-vector machine with it learns a classically hard concept to ≥99%\ge99\%≥99% accuracy while generic and classical kernels stay at chance; wired into the KAN architecture, it fits discrete-log targets to R2 ⁣≈ ⁣1R^2\!\approx\!1R2≈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.
Abstract Quantum steering is an important form of nonclassical correlation and plays a central role in one-sided device-independent quantum information processing. However, efficiently determining whether an arbitrary two-qubit state is steerable remains challenging, since semidefinite-programming-based certification generally depends on the choice of measurement settings and can become computationally demanding. In this work, we propose an interpretable quantum steering detection scheme based on Kolmogorov–Arnold Networks (KANs). The training labels are generated through semidefinite programming for randomly sampled two-qubit density matrices. Several physically motivated feature representations are constructed to investigate the influence of measurement and feature reduction. Owing to their learnable univariate edge functions and symbolic formula extraction capability, KANs provide not only accurate classification but also a more transparent representation of the learned steering decision boundary. Numerical experiments show that the proposed KAN-based classifiers can distinguish states with SDP-detected steering from states for which no witness is found under the sampled measurements. In particular, canonical-correlation features provide an effective representation for learning the steering boundary, and further feature compression can still retain competitive detection performance. These results indicate that KANs offer a promising, efficient, and interpretable machine-learning approach for finite-measurement steering detection of arbitrary two-qubit states.
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 2Nq2^{N_q}2Nq​-dimensional KAN layer matrix by tiling controlled rotations through a sum-operator construction, using only 2Nq−12^{N_q-1}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<0.002p<0.002p<0.002, Cliff's δ≤−0.86δ\leq-0.86δ≤−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 0.6200.6200.620, p=0.0005p=0.0005p=0.0005) but is significantly less accurate than a QNN carrying one fifth as many parameters (ΔΔΔaccuracy −0.134-0.134−0.134, p=0.0014p=0.0014p=0.0014; ΔΔΔAUC −0.252-0.252−0.252, p=0.0002p=0.0002p=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%58\%58% (Spearman p<10−3p<10^{-3}p<10−3). We also report the circuit cost in full --- three layers emit 101710171017 operations, or 411041104110 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.
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.
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.