Variational Quantum Kolmogorov-Arnold Network

Abstract We implemented a quantum version of the Kolmogorov-Arnold Network (KAN) on a quantum circuit. It optimizes the weight of each synaptic connection to enhance the overall performance of the neurons. This approach offers a significant boost in efficiency over traditional neural networks, as it can be optimized more accurately and requires only a few neurons to solve complex problems. The KAN is a multi-layer network, similar to a traditional neural network, with each layer represented as a matrix containing parameters and feedback. This structure is linear and mirrors the form of Variational Quantum Algorithms. Thus, we propose the Variational Quantum Kolmogorov-Arnold Network (VQKAN) as a quantum version of the Kolmogorov-Arnold Network. We demonstrated the optimization of VQKAN for fitting the given function in various ways, such as by using ansatzes and comparisons.Abstract We implemented a quantum version of the Kolmogorov-Arnold Network (KAN) on a quantum circuit. It optimizes the weight of each synaptic connection to enhance the overall performance of the neurons. This approach offers a significant boost in efficiency over traditional neural networks, as it can be optimized more accurately and requires only a few neurons to solve complex problems. The KAN is a multi-layer network, similar to a traditional neural network, with each layer represented as a matrix containing parameters and feedback. This structure is linear and mirrors the form of Variational Quantum Algorithms. Thus, we propose the Variational Quantum Kolmogorov-Arnold Network (VQKAN) as a quantum version of the Kolmogorov-Arnold Network. We demonstrated the optimization of VQKAN for fitting the given function in various ways, such as by using ansatzes and comparisons.

Dec 26, 20242 citations

Variational quantum Kolmogorov–Arnold network

Hikaru Wakaura, Rahmat Mulyawan, Andriyan Bayu Suksmono

Abstract We implemented a quantum version of the Kolmogorov-Arnold Network (KAN) on a quantum circuit. It optimizes the weight of each synaptic connection to enhance the overall performance of the neurons. This approach offers a significant boost in efficiency over traditional neural networks, as it can be optimized more accurately and requires only a few neurons to solve complex problems. The KAN is a multi-layer network, similar to a traditional neural network, with each layer represented as a matrix containing parameters and feedback. This structure is linear and mirrors the form of Variational Quantum Algorithms. Thus, we propose the Variational Quantum Kolmogorov-Arnold Network (VQKAN) as a quantum version of the Kolmogorov-Arnold Network. We demonstrated the optimization of VQKAN for the fitting of the given function in various ways, such as by using ansatzes and comparisons.Abstract We implemented a quantum version of the Kolmogorov-Arnold Network (KAN) on a quantum circuit. It optimizes the weight of each synaptic connection to enhance the overall performance of the neurons. This approach offers a significant boost in efficiency over traditional neural networks, as it can be optimized more accurately and requires only a few neurons to solve complex problems. The KAN is a multi-layer network, similar to a traditional neural network, with each layer represented as a matrix containing parameters and feedback. This structure is linear and mirrors the form of Variational Quantum Algorithms. Thus, we propose the Variational Quantum Kolmogorov-Arnold Network (VQKAN) as a quantum version of the Kolmogorov-Arnold Network. We demonstrated the optimization of VQKAN for the fitting of the given function in various ways, such as by using ansatzes and comparisons.

Quantum Information ProcessingAug 10, 20260 citations

The quantum version of Kolmogorov-Arnold Network ( KAN ) : Variational Quantum Kolmogorov-Arnold Network.

Hikaru Wakaura, Andriyan Bayu Suksmono

Abstract We implemented a quantum version of the Kolmogorov-Arnold Network (KAN) on a quantum circuit. It optimizes the weight of each synaptic connection to enhance the overall performance of the neurons. This approach offers a significant boost in efficiency over traditional neural networks, as it can be optimized more accurately and requires only a few neurons to solve complex problems. The KAN is a multi-layer network, similar to a traditional neural network, with each layer represented as a matrix containing parameters and feedback. This structure is linear and mirrors the form of Variational Quantum Algorithms. Thus, we propose the Variational Quantum Kolmogorov-Arnold Network (VQKAN) as a quantum version of the Kolmogorov-Arnold Network. We demonstrated the optimization of VQKAN for the fitting of the given function in some ways, such as by using ansatzes and comparisons.Abstract We implemented a quantum version of the Kolmogorov-Arnold Network (KAN) on a quantum circuit. It optimizes the weight of each synaptic connection to enhance the overall performance of the neurons. This approach offers a significant boost in efficiency over traditional neural networks, as it can be optimized more accurately and requires only a few neurons to solve complex problems. The KAN is a multi-layer network, similar to a traditional neural network, with each layer represented as a matrix containing parameters and feedback. This structure is linear and mirrors the form of Variational Quantum Algorithms. Thus, we propose the Variational Quantum Kolmogorov-Arnold Network (VQKAN) as a quantum version of the Kolmogorov-Arnold Network. We demonstrated the optimization of VQKAN for the fitting of the given function in some ways, such as by using ansatzes and comparisons.

Jun 3, 20241 citation

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.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:2503.22604Mar 28, 20250 citations

QuKAN: A Quantum Circuit Born Machine Approach to Quantum Kolmogorov Arnold Networks

Yannick Werner, Akash Malemath, Mengxi Liu +4 more

Kolmogorov Arnold Networks (KANs), built upon the Kolmogorov Arnold representation theorem (KAR), have demonstrated promising capabilities in expressing complex functions with fewer neurons. This is achieved by implementing learnable parameters on the edges instead of on the nodes, unlike traditional networks such as Multi-Layer Perceptrons (MLPs). However, KANs potential in quantum machine learning has not yet been well explored. In this work, we present an implementation of these KAN architectures in both hybrid and fully quantum forms using a Quantum Circuit Born Machine (QCBM). We adapt the KAN transfer using pre-trained residual functions, thereby exploiting the representational power of parametrized quantum circuits. In the hybrid model we combine classical KAN components with quantum subroutines, while the fully quantum version the entire architecture of the residual function is translated to a quantum model. We demonstrate the feasibility, interpretability and performance of the proposed Quantum KAN (QuKAN) architecture.Kolmogorov Arnold Networks (KANs), built upon the Kolmogorov Arnold representation theorem (KAR), have demonstrated promising capabilities in expressing complex functions with fewer neurons. This is achieved by implementing learnable parameters on the edges instead of on the nodes, unlike traditional networks such as Multi-Layer Perceptrons (MLPs). However, KANs potential in quantum machine learning has not yet been well explored. In this work, we present an implementation of these KAN architectures in both hybrid and fully quantum forms using a Quantum Circuit Born Machine (QCBM). We adapt the KAN transfer using pre-trained residual functions, thereby exploiting the representational power of parametrized quantum circuits. In the hybrid model we combine classical KAN components with quantum subroutines, while the fully quantum version the entire architecture of the residual function is translated to a quantum model. We demonstrate the feasibility, interpretability and performance of the proposed Quantum KAN (QuKAN) architecture.

arXiv:2506.22340Scientific ReportsOct 9, 20252 citations

Generative Adversarial Variational Quantum Kolmogorov-Arnold Network

Hikaru Wakaura

Kolmogorov Arnold Networks is a novel multilayer neuromorphic network that can exhibit higher accuracy than a neural network. It can learn and predict more accurately than neural networks with a smaller number of parameters, and many research groups worldwide have adopted it. As a result, many types of applications have been proposed. This network can be used as a generator solely or with a Generative Adversarial Network; however, KAN has a slower speed of learning than neural networks for the number of parameters. Hence,it has not been researched as a generator. Therefore, we propose a novel Generative Adversarial Network called Generative Adversarial Variational Quantum KAN that uses Variational Quantum KAN as a generator. This method enables efficient learning with significantly fewer parameters by leveraging the computational advantages of quantum circuits and their output distributions. We performed the training and generation task on MNIST and CIFAR10, and revealed that our method can achieve higher accuracy than neural networks and Quantum Generative Adversarial Network with less data.Kolmogorov Arnold Networks is a novel multilayer neuromorphic network that can exhibit higher accuracy than a neural network. It can learn and predict more accurately than neural networks with a smaller number of parameters, and many research groups worldwide have adopted it. As a result, many types of applications have been proposed. This network can be used as a generator solely or with a Generative Adversarial Network; however, KAN has a slower speed of learning than neural networks for the number of parameters. Hence,it has not been researched as a generator. Therefore, we propose a novel Generative Adversarial Network called Generative Adversarial Variational Quantum KAN that uses Variational Quantum KAN as a generator. This method enables efficient learning with significantly fewer parameters by leveraging the computational advantages of quantum circuits and their output distributions. We performed the training and generation task on MNIST and CIFAR10, and revealed that our method can achieve higher accuracy than neural networks and Quantum Generative Adversarial Network with less data.

quant-pharXiv:2512.11014Dec 12, 20250 citations

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.