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

TL;DR

QKAN is a novel quantum neural network framework inspired by classical KAN, enabling efficient quantum machine learning and multivariate state preparation through recursive block-encodings and parametrized circuits.

Contribution

Introduces QKAN, a quantum neural network architecture with a recursive, block-encoded structure, and demonstrates its applications in quantum learning and state preparation.

Findings

QKAN can be wide-and-shallow with exponential width for input encoding.

QKAN can be trained using parametrized quantum circuits.

Efficiently prepares multivariate Gaussian quantum states.

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.