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arxiv: 2410.04435 · v2 · pith:MIOCJBQ2new · submitted 2024-10-06 · 🪐 quant-ph

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

Petr Ivashkov , Po-Wei Huang , Kelvin Koor , Lirand\"e Pira , Patrick Rebentrost This is my paper

Pith reviewed 2026-05-23 20:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum Kolmogorov-Arnold networksblock-encodingquantum singular value transformationquantum machine learningquantum state preparationmultivariate functionswide-and-shallow architecture
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0 comments X

The pith

QKAN constructs wide-and-shallow quantum networks from recursive block-encodings to support both learning models and multivariate state preparation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces quantum Kolmogorov-Arnold networks as a framework that inherits the compositional structure of classical KANs. It builds these networks recursively from a single layer via block-encodings and quantum singular value transformation. The resulting architecture treats eigenvalues as neurons with parametrized activations for machine learning tasks and enables efficient preparation of states such as multivariate Gaussians using only two layers. A sympathetic reader would care because the design claims to offset shallow circuit depth with exponential layer width whenever efficient block-encodings exist, opening a route for quantum resources to handle compositional functions without deep circuits.

Core claim

QKAN is a quantum algorithmic framework based on block-encodings constructed recursively from a single layer using quantum singular value transformation. It functions as a learning model by taking eigenvalues of block-encoded matrices as neurons and placing parametrized activation functions on network edges. The same structure serves as a multivariate quantum state-preparation protocol for functions with shallow compositional structure, demonstrated by preparing a multivariate Gaussian state in two layers. The architecture is wide-and-shallow, with shallow depth compensated by exponentially wide layers when efficient block-encodings of inputs are available. Parametrization and training occur

What carries the argument

Block-encodings constructed recursively from a single layer using quantum singular value transformation, which carries the compositional structure and enables the wide-and-shallow scaling.

If this is right

  • QKAN serves as a quantum learning model whose training uses parametrized quantum circuits together with quantum linear algebra subroutines.
  • A two-layer QKAN prepares a multivariate Gaussian quantum state with resources set by the compositional depth rather than the input dimension.
  • Shallow depth is offset by exponentially wide layers precisely when efficient block-encodings exist for the inputs.
  • The modular, compositional design opens new applications in quantum machine learning and quantum state preparation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If block-encodings prove efficient for broad classes of functions, QKAN could reduce the circuit depth required for approximation tasks compared with standard variational approaches.
  • The recursive block-encoding construction may combine with other quantum algorithms that already rely on block-encodings for linear-algebra primitives.
  • Hybrid training loops could alternate between classical optimization of the activation parameters and quantum evaluation of the block-encoded layers.

Load-bearing premise

Efficient block-encodings of the input matrices or functions are available so that exponential width can compensate for shallow depth.

What would settle it

Implement the two-layer QKAN for multivariate Gaussian state preparation on a quantum device and check whether the output state fidelity reaches the target with circuit depth and width scaling as claimed.

Figures

Figures reproduced from arXiv: 2410.04435 by Kelvin Koor, Lirand\"e Pira, Patrick Rebentrost, Petr Ivashkov, Po-Wei Huang.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces quantum Kolmogorov-Arnold networks (QKAN), a framework that inherits the compositional structure of classical KANs and constructs them recursively from block-encodings via quantum singular value transformation (QSVT). It presents QKAN as a quantum learning model in which eigenvalues of block-encoded matrices act as neurons with parametrized activation functions on the edges, and demonstrates its use for multivariate quantum state preparation via a two-layer construction that prepares a Gaussian state. The central claim is that QKAN yields a wide-and-shallow architecture in which exponentially wide layers compensate for shallow depth whenever efficient block-encodings of the inputs are available; the manuscript discusses parametrization via parametrized quantum circuits and quantum linear-algebra subroutines.

Significance. If the claims hold, QKAN would supply a modular, compositional quantum architecture that directly transplants the edge-function flexibility of KANs into the quantum setting, potentially enabling new quantum-ML models and state-preparation protocols that exploit shallow compositional structure. The explicit two-layer Gaussian preparation constitutes a concrete, falsifiable example that can be checked for resource scaling; the recursive QSVT construction itself is a reusable primitive.

major comments (2)
  1. [Abstract and §4] Abstract and §4 (quantum learning model): the headline claim that 'shallow depth is compensated by exponentially wide layers whenever efficient block-encodings of inputs are available' is load-bearing for the wide-and-shallow characterization, yet the text invokes the premise for both the eigenvalue-neuron model and the state-preparation protocol without supplying a general construction or poly(log N) resource bound that would establish when such block-encodings exist for arbitrary input matrices or compositional functions.
  2. [§5.2] §5.2 (multivariate Gaussian state preparation): the two-layer QKAN is asserted to prepare the state 'efficiently,' but no explicit query-complexity or gate-count analysis is given that would confirm the exponential-width compensation actually materializes beyond the assumption of an efficient block-encoding; this leaves the concrete demonstration without a verifiable resource comparison to standard state-preparation methods.
minor comments (2)
  1. [§3] Notation for the recursive QSVT construction (likely §3) would benefit from an explicit diagram or pseudocode showing how a single-layer block-encoding is lifted to the multi-layer network; the current prose description leaves the width scaling implicit.
  2. [Introduction] The manuscript should cite the original KAN paper (Liu et al., 2024) and recent block-encoding/QSVT surveys in the introduction to situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and for highlighting the importance of clarifying the assumptions underlying our claims. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (quantum learning model): the headline claim that 'shallow depth is compensated by exponentially wide layers whenever efficient block-encodings of inputs are available' is load-bearing for the wide-and-shallow characterization, yet the text invokes the premise for both the eigenvalue-neuron model and the state-preparation protocol without supplying a general construction or poly(log N) resource bound that would establish when such block-encodings exist for arbitrary input matrices or compositional functions.

    Authors: We agree that no general construction or poly(log N) bound is supplied for arbitrary input matrices, as the existence of efficient block-encodings is inherently problem-dependent and relies on specific structure (e.g., sparsity or oracle access). The manuscript's claim is explicitly conditional on such encodings being available, consistent with standard assumptions in QSVT and quantum linear algebra. We will revise the abstract and §4 to emphasize this conditionality, note that a universal construction lies outside the paper's scope, and clarify that the exponential width arises from the block-encoding dimension when the premise holds. revision: partial

  2. Referee: [§5.2] §5.2 (multivariate Gaussian state preparation): the two-layer QKAN is asserted to prepare the state 'efficiently,' but no explicit query-complexity or gate-count analysis is given that would confirm the exponential-width compensation actually materializes beyond the assumption of an efficient block-encoding; this leaves the concrete demonstration without a verifiable resource comparison to standard state-preparation methods.

    Authors: The efficiency claim for the Gaussian example rests on the assumption of an efficient block-encoding, without a detailed query or gate-count analysis in the current text. We will add an explicit discussion in §5.2 outlining the resource assumptions (including how the block-encoding dimension contributes to width) and a high-level comparison to standard state-preparation approaches, while noting that concrete numbers depend on the specific block-encoding implementation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external block-encoding premise and known QSVT.

full rationale

The paper defines QKAN via recursive construction from block-encodings using QSVT, inheriting KAN's compositional structure without reducing any target quantity (e.g., width compensation or state-preparation efficiency) to a fitted parameter or self-citation by definition. The 'wide-and-shallow' claim is explicitly conditioned on the external availability of efficient block-encodings rather than derived tautologically. No self-citations appear load-bearing, no ansatz is smuggled, and the Gaussian demonstration is presented as an explicit two-layer protocol without renaming or circular prediction. The chain is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; full technical details unavailable. The central claims rest on the domain assumption that efficient block-encodings exist for the inputs of interest.

axioms (1)
  • domain assumption Efficient block-encodings of inputs are available
    Invoked to justify the wide-and-shallow architecture claim in the learning and state-preparation sections.

pith-pipeline@v0.9.0 · 5760 in / 1195 out tokens · 31366 ms · 2026-05-23T20:11:57.538167+00:00 · methodology

discussion (0)

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Forward citations

Cited by 8 Pith papers

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  3. Merged amplitude encoding for Chebyshev quantum Kolmogorov--Arnold networks: trading qubits for circuit executions

    quant-ph 2026-03 conditional novelty 7.0

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