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.
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.
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.
Implementing polynomial functions of Hermitian matrices on quantum hardware is a foundational task in quantum computing, critical for accurate Hamiltonian simulation, quantum linear system solving, high-fidelity state preparation, machine learning kernels, and other advanced quantum algorithms. Existing state-of-the-art techniques, including Qubitization, Quantum Singular Value Transformation (QSVT), and Quantum Signal Processing (QSP), rely heavily on block-encoding the Hermitian matrix. These methods are often constrained by the complexity of preparing the block-encoded state, the overhead associated with the required ancillary qubits, or the challenging problem of angle synthesis for the polynomial's phase factors, which limits the achievable circuit depth and overall efficiency. In this work, we propose a novel and resource-efficient approach to implement arbitrary polynomials of a Hermitian matrix by leveraging the Generalized Quantum Signal Processing (GQSP) framework. Our method circumvents the need for block-encoding and avoids the compounding post-selection overheads characteristic of LCU-based constructions, achieving a stable, degree-independent success probability. We derive closed-form expressions for symmetric polynomial expansions and demonstrate how linear combinations of GQSP circuits can realize the desired transformation. This approach reduces resource overhead and opens new pathways for quantum algorithm design for functions of Hermitian matrices, particularly in settings where the Hermitian operator arises naturally from symmetric combinations of unitaries.
Quantum signal processing (QSP) is a framework which was proven to unify and simplify a large number of known quantum algorithms, as well as discovering new ones. QSP allows one to transform a signal embedded in a given unitary using polynomials. Characterizing which polynomials can be achieved with QSP protocols is an important part of the power of this technique, and while such a characterization is well-understood in the case of univariate signals, it is unclear which multivariate polynomials can be constructed when the signal is a vector, rather than a scalar. This work uses a slightly different formalism than what is found in the literature, and uses it to find simpler necessary conditions for decomposability, as well as a sufficient condition – the first, to the best of our knowledge, proven for a (generally inhomogeneous) multivariate polynomial in the context of quantum signal processing.
In this paper, we study quantum generating functions (QGFs) as algorithmic primitives for quantum computation with a significant focus on machine learning. We develop a systematic formulation for encoding sequences and functions into quantum states, enabling efficient quantum algorithms for evaluation, transformation, and analysis.We demonstrate the utility of QGFs across three key areas of machine learning: i) quantum activation functions with efficient, explicit circuit implementations; ii) quantum kernel methods via generating function representations of feature maps; iii) quantum Kolmogorov-Arnold networks (KANs) for function approximation. For each area we provide proof-of-concept numerical simulations of the proposed circuits. We analyze circuit complexities and applications, offering a foundation for quantum machine learning. The quantum implementations of generating functions for structured function classes consist of simple quantum rotations. Therefore, they can provide efficient algorithmic primitives to design machine learning methods. At the same time, we are explicit about the scope of this efficiency: the simplest primitives are product-state circuits with classically computable angles, which are classically simulable, so a quantum advantage can only materialize when the constructions are embedded in coherent pipelines with entangling processing; we state these limitations alongside the corresponding results. Throughout the paper, complexity statements concern the core quantum computation; the costs of initial data loading and final classical readout are stated separately where relevant.
Quantum Signal Processing (QSP), a framework for implementing matrix-valued polynomials, is a fundamental primitive in various quantum algorithms. Despite its versatility, a potentially underappreciated challenge is that all systematic protocols for implementing QSP rely on post-selection. This can impose prohibitive costs for tasks when amplitude amplification cannot sufficiently improve the success probability. For example, in the context of ground-state preparation, this occurs when using a too poor initial state. In this work, we introduce a new formula for implementing QSP transformations of Hermitian matrices, which requires neither auxiliary qubits nor post-selection. Rather, using approximation to the exact unitary synthesis, we leverage the theory of the double-bracket quantum algorithms to provide a new quantum algorithm for QSP, termed Double-Bracket QSP (DB-QSP). The algorithm requires the energy and energetic variance of the state to be measured at each step and has a recursive structure, which leads to circuit depths that can grow super exponentially with the degree of the polynomial. With these strengths and caveats in mind, DB-QSP should be viewed as complementing the established QSP toolkit. In particular, DB-QSP can deterministically implement low-degree polynomials to ``warm start" QSP methods involving post-selection.
Abstract Quantum architecture Search (QAS) is a promising direction for optimization and automated design of quantum circuits towards quantum advantage. Recent techniques in QAS emphasize Multi-Layer Perceptron (MLP)-based deep Q-networks. However, their interpretability remains challenging due to the large number of learnable parameters and the complexities involved in selecting appropriate activation functions. In this work, to overcome these challenges, we utilize the Kolmogorov-Arnold Network (KAN) in the QAS algorithm, analyzing their efficiency in the task of quantum state preparation and quantum chemistry. In quantum state preparation, our results show that in a noiseless scenario, the probability of success is 2× to 5× higher than MLPs. In noisy environments, KAN outperforms MLPs in fidelity when approximating these states, showcasing its robustness against noise. In tackling quantum chemistry problems, we enhance the recently proposed QAS algorithm by integrating curriculum reinforcement learning with a KAN structure. This facilitates a more efficient design of parameterized quantum circuits by reducing the number of required 2-qubit gates and circuit depth. Further investigation reveals that KAN requires a significantly smaller number of learnable parameters compared to MLPs; however, the average time of executing each episode for KAN is higher.