Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.09584

Algebraic Kolmogorov--Arnold representation theorem for quantum measurement

Sviatoslav V. Dzhenzher

quant-phmath.FASubmitted Jun 8, 2026• Announced Jun 9, 2026• 0 citations• Revised Sep 29, 2026 (v3)
v1 · Jun 8, 2026v2 · Sep 12, 2026v3 · Sep 29, 2026v4 · Oct 10, 2026
arXivHTMLPDF

Abstract

We establish an operational framework connecting the classical Kolmogorov--Arnold (KA) representation theorem to quantum information theory. By introducing and proving an algebraic, bounded-degree polynomial version of the theorem, we demonstrate that any target physical property of an unentangled multi-qubit product state can be exactly decomposed using a finite, fixed set of local ``inner'' observables and a shallow architecture of univariate polynomials. We further analyse the stability of this Quantum Kolmogorov--Arnold (QKA) representation under adversarial perturbations. In stark contrast to the pathological instabilities and severe reparameterisation sensitivities inherent to the classical Kolmogorov--Arnold representation theorem, our algebraic quantum framework exhibits remarkable resilience. We prove that the representation remains stable against bounded physical perturbations acting on the inner measurement operators, and show via the Heisenberg picture that it is inherently immune to adversarial quantum channel attacks acting on the input states.

Comments
9 pages, no figures; major revision: proofs are corrected, the `presumably trivial' version is replaced with the `incorrect' version

Related papers

Quantum Bernstein--Kolmogorov--Arnold approximation

Sviatoslav V. Dzhenzher

This paper addresses the expressivity and representation gap in Quantum Neural Networks by establishing a quantum analogue of the classical Kolmogorov--Arnold representation theorem. We propose a constructive approximation framework over fixed input coherent states, where the dependence of the function argument is embedded entirely in coordinate qubits, while the target function dictates a single outer unitary evolution. By applying a Sprecher-type representation to the rotation angle of the state, we demonstrate that the multidimensional operator factorises into \(2d+1\) commuting universal inner observables. We prove that the resulting measurement probability precisely reproduces the multidimensional Bernstein polynomial of the target function. For any continuous multivariate function, the joint quantum state converges uniformly to the target output register, linking classical superposition theory with probabilistic and coherent quantum circuits.

quant-pharXiv:2610.41608•Nov 2, 2026

Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps

Sviatoslav V. Dzhenzher

The classical Kolmogorov--Arnold representation theorem states that any continuous multivariate function can be exactly decomposed into a finite composition of univariate continuous functions and addition operations. This foundational result has recently inspired the development of Kolmogorov--Arnold Networks (KANs) in classical machine learning, as well as their extensions into the quantum domain (QKANs). In this paper, we establish two quantum analogues of the Kolmogorov--Arnold representation theorem for continuous unitary-valued maps of several variables within an open $1$-neighbourhood of the identity matrix \(O_1(\mathbf{I}) \subset \mathcal{U}(n)\). First, we prove a representation theorem that yields an exact additive decomposition inside the matrix exponent of anti-Hermitian-valued maps. Second, due to the non-commutative nature of quantum operators, we derive a factorised version expressing the target unitary map as a finite sequential product of univariate matrix exponentials. Finally, we provide a concrete topological counterexample based on the lifting property of \(\mathcal{SU}(2)\) to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group \(\mathcal{U}(n)\) without encountering fundamental structural obstructions.

quant-pharXiv:2607.03187•Jul 6, 2026•0 citations

An algebraic approach to quantum information theory with applications in quantum cryptography

Pim Tuyls

We present an algebraic approach to quantum information theory and its application to quantum cryptography. Therefore, we model the measurement process in terms of an algebraic object: a partition of unity. The POVM associated to such a partition describes the measurement results and the completely positive map associated to it the disturbance of the state. This approach leads to a compact description of quantum cryptography in which general measurements, noisy channels as well as measurement apparatus of the receiver that are out of sync with those of the sender can be described.

IEEE International Symposium on Information Theory (ISIT)•Jun 25, 2003•1 citation

Quantum Measurements and Information-Theoretical Restrictions in Observable Algebras Formalism

Sergey Nikolaevich Mayburov

Information-Theoretical restrictions on the information transfer are applied to Quantum Measurements. For the measurement of quantum object S by information system O this restrictions are calculated in Algebraic QM formalism as the inference map to the (sub)algebra of O observables.

arXiv:quant-ph/0506065•Jun 8, 2005•0 citations

Weighted Quantum Signal Processing: Low-Depth Polynomial Approximation with Applications to Kolmogorov-Arnold Networks

Rohit Sarma Sarkar, Rupayan Bhattacharjee, Elias F. Combarro +5 more

Quantum Signal Processing is a powerful quantum framework for generating and approximating univariate polynomials. However, QSP is often limited by circuit-depth bottlenecks and parity constraints on the class of realizable polynomials. In this work, we introduce Weighted Quantum Signal Processing, an extension of QSP in which a weight function is assigned to the central rotation operator. This formulation provides a deeper understanding of QSP, which emerges as the special case of WQSP with unit weights. The choice of weights determines the structure and expressive capabilities of WQSP circuits. When the weights are natural numbers greater than one, WQSP reduces to a pruned version of QSP, revealing parameter redundancies in the standard framework. Through appropriate selection of integer weights, WQSP achieves linear-to-exponential reductions in the number of parameters required to realize arbitrary bounded univariate polynomials while preserving approximation quality. For generic weights, we establish corresponding approximation error bounds and show that, in many cases, the approximation is exact. We analyze WQSP from both a deterministic perspective, where polynomial generation is formulated as the solution of a linear system, and a quantum machine learning perspective, where WQSP serves as a structured and expressive quantum learning model. We further employ this learning framework to parameterize learnable activation functions in Kolmogorov--Arnold Networks for multivariate function approximation. Our results show that WQSP provides a compact, flexible, and theoretically grounded framework for realizing arbitrary univariate polynomials while requiring significantly fewer trainable parameters than conventional QSP. This yields expressive and parameter-efficient neural architectures, highlighting the potential of WQSP as a scalable primitive for quantum-enhanced machine learning.

quant-pharXiv:2609.21567•Sep 21, 2026•0 citations

Efficient Representation of multicategorical local Hilbert spaces: nonlinear Restricted Boltzmann Machines to Kolmogorov-Arnold Networks

Abhiroop Lahiri

Neural network quantum states (NQS) for representing spin-$\frac{1}{2}$ systems are typically built from Multi-Layer Perceptrons (MLPs) with binary visible variables, in which the input spins are first combined through linear affine maps before more expressive nonlinear transformations are applied. The standard way for representing multi-categorical systems, namely spin-1 and spin-2, or the q-state quantum Potts systems with more than two local degrees of freedom, is a unary or one-hot encoded MLP architecture. Following a few similar studies, I observe that while the one-hot construction is the mathematically faithful encoding for systems with more than two local states such as spin-$S$ models with $S > 1/2$ or $q$-state Potts models, its parameter count grows with the number of local states, and the resulting optimization landscape can hinder convergence. I find numerically that allowing $\log Ψ$ to be a nonlinear function of the raw multi-valued spin variable is a natural categorical generalization: it preserves the labelling freedom of the local basis and reproduces the one-hot model with strictly fewer parameters, often with improved trainability. I first benchmark this idea on shallow Restricted Boltzmann Machines (RBMs), equipping them with several distinct nonlinear connections, for spin-1, 2, and 3 Heisenberg chains. I then turn to Kolmogorov-Arnold Networks (KANs), where each edge carries a learnable univariate nonlinearity, and show that they provide a strictly more expressive realization of the same principle. Finally, I demonstrate that this framework captures the critical behaviour of the quantum Potts Hamiltonian, recovering its phase transition.

quant-pharXiv:2609.06672•Sep 8, 2026•0 citations
We establish an operational framework connecting the classical Kolmogorov--Arnold (KA) representation theorem to quantum information theory. By introducing and proving an algebraic, bounded-degree polynomial version of the theorem, we demonstrate that any target physical property of an unentangled multi-qubit product state can be exactly decomposed using a finite, fixed set of local ``inner'' observables and a shallow architecture of univariate polynomials.
We further analyse the stability of this Quantum Kolmogorov--Arnold (QKA) representation under adversarial perturbations. In stark contrast to the pathological instabilities and severe reparameterisation sensitivities inherent to the classical Kolmogorov--Arnold representation theorem, our algebraic quantum framework exhibits remarkable resilience. We prove that the representation remains stable against bounded physical perturbations acting on the inner measurement operators, and show via the Heisenberg picture that it is inherently immune to adversarial quantum channel attacks acting on the input states.
This paper addresses the expressivity and representation gap in Quantum Neural Networks by establishing a quantum analogue of the classical Kolmogorov--Arnold representation theorem. We propose a constructive approximation framework over fixed input coherent states, where the dependence of the function argument is embedded entirely in coordinate qubits, while the target function dictates a single outer unitary evolution. By applying a Sprecher-type representation to the rotation angle of the state, we demonstrate that the multidimensional operator factorises into 2d+12d+12d+1 commuting universal inner observables. We prove that the resulting measurement probability precisely reproduces the multidimensional Bernstein polynomial of the target function. For any continuous multivariate function, the joint quantum state converges uniformly to the target output register, linking classical superposition theory with probabilistic and coherent quantum circuits.
The classical Kolmogorov--Arnold representation theorem states that any continuous multivariate function can be exactly decomposed into a finite composition of univariate continuous functions and addition operations. This foundational result has recently inspired the development of Kolmogorov--Arnold Networks (KANs) in classical machine learning, as well as their extensions into the quantum domain (QKANs). In this paper, we establish two quantum analogues of the Kolmogorov--Arnold representation theorem for continuous unitary-valued maps of several variables within an open 111-neighbourhood of the identity matrix O1(I)⊂U(n)O_1(\mathbf{I}) \subset \mathcal{U}(n)O1​(I)⊂U(n). First, we prove a representation theorem that yields an exact additive decomposition inside the matrix exponent of anti-Hermitian-valued maps. Second, due to the non-commutative nature of quantum operators, we derive a factorised version expressing the target unitary map as a finite sequential product of univariate matrix exponentials. Finally, we provide a concrete topological counterexample based on the lifting property of SU(2)\mathcal{SU}(2)SU(2) to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group U(n)\mathcal{U}(n)U(n) without encountering fundamental structural obstructions.
We present an algebraic approach to quantum information theory and its application to quantum cryptography. Therefore, we model the measurement process in terms of an algebraic object: a partition of unity. The POVM associated to such a partition describes the measurement results and the completely positive map associated to it the disturbance of the state. This approach leads to a compact description of quantum cryptography in which general measurements, noisy channels as well as measurement apparatus of the receiver that are out of sync with those of the sender can be described.
Quantum Signal Processing is a powerful quantum framework for generating and approximating univariate polynomials. However, QSP is often limited by circuit-depth bottlenecks and parity constraints on the class of realizable polynomials. In this work, we introduce Weighted Quantum Signal Processing, an extension of QSP in which a weight function is assigned to the central rotation operator. This formulation provides a deeper understanding of QSP, which emerges as the special case of WQSP with unit weights. The choice of weights determines the structure and expressive capabilities of WQSP circuits. When the weights are natural numbers greater than one, WQSP reduces to a pruned version of QSP, revealing parameter redundancies in the standard framework. Through appropriate selection of integer weights, WQSP achieves linear-to-exponential reductions in the number of parameters required to realize arbitrary bounded univariate polynomials while preserving approximation quality. For generic weights, we establish corresponding approximation error bounds and show that, in many cases, the approximation is exact. We analyze WQSP from both a deterministic perspective, where polynomial generation is formulated as the solution of a linear system, and a quantum machine learning perspective, where WQSP serves as a structured and expressive quantum learning model. We further employ this learning framework to parameterize learnable activation functions in Kolmogorov--Arnold Networks for multivariate function approximation. Our results show that WQSP provides a compact, flexible, and theoretically grounded framework for realizing arbitrary univariate polynomials while requiring significantly fewer trainable parameters than conventional QSP. This yields expressive and parameter-efficient neural architectures, highlighting the potential of WQSP as a scalable primitive for quantum-enhanced machine learning.
Neural network quantum states (NQS) for representing spin-12\frac{1}{2}21​ systems are typically built from Multi-Layer Perceptrons (MLPs) with binary visible variables, in which the input spins are first combined through linear affine maps before more expressive nonlinear transformations are applied. The standard way for representing multi-categorical systems, namely spin-1 and spin-2, or the q-state quantum Potts systems with more than two local degrees of freedom, is a unary or one-hot encoded MLP architecture. Following a few similar studies, I observe that while the one-hot construction is the mathematically faithful encoding for systems with more than two local states such as spin-SSS models with S>1/2S > 1/2S>1/2 or qqq-state Potts models, its parameter count grows with the number of local states, and the resulting optimization landscape can hinder convergence. I find numerically that allowing log⁡Ψ\log ΨlogΨ to be a nonlinear function of the raw multi-valued spin variable is a natural categorical generalization: it preserves the labelling freedom of the local basis and reproduces the one-hot model with strictly fewer parameters, often with improved trainability. I first benchmark this idea on shallow Restricted Boltzmann Machines (RBMs), equipping them with several distinct nonlinear connections, for spin-1, 2, and 3 Heisenberg chains. I then turn to Kolmogorov-Arnold Networks (KANs), where each edge carries a learnable univariate nonlinearity, and show that they provide a strictly more expressive realization of the same principle. Finally, I demonstrate that this framework captures the critical behaviour of the quantum Potts Hamiltonian, recovering its phase transition.