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Theory of Zulfia Quantum Kernel: Beyond Classical Kernel

DR. ZULFIQAR ALI KHAN

Published Oct 2, 2026• 0 citations
Publisher

Abstract

For centuries, only the classical kernel CK = {x | f (x) = 0} and its directquantum analogue CKQ = {ψ | Hψ = 0} have been known.Using the Zulfia Deviation Measure, we develop the Zulfia Quantum KernelKQZ .Let H be classical Hamiltonian and HZ = H + DZ be its Zulfia deformation.For any quantum state ψ, ZDMQ(ψ) = Hψ − HZ ψ = −DZ (ψ).We define:KQZ := {ψ ∈ H | HZ (ψ) = 0} = {ψ | Hψ = ZDMQ(ψ), ∥ψ∥ = 1}In this framework, the classical kernel emerges as a limiting case:limDZ →0 KQZ = CKQ → CKproving KQZ is fundamental.The purpose of this paper is to utilize this newly developed Zulfia QuantumKernel to characterize Zulfia-balanced states where quantum evolution is exactlycompensated by its own deviation, opening a new window to study structuredstability, non-trivial zeros, and hidden symmetries beyond classical zero.

DOI
10.5281/zenodo.23092245

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For centuries, only the classical kernel CK = {x | f (x) = 0} and its directquantum analogue CKQ = {ψ | Hψ = 0} have been known.Using the Zulfia Deviation Measure, we develop the Zulfia Quantum KernelKQZ .Let H be classical Hamiltonian and HZ = H + DZ be its Zulfia deformation.For any quantum state ψ, ZDMQ(ψ) = Hψ − HZ ψ = −DZ (ψ).We define:KQZ := {ψ ∈ H | HZ (ψ) = 0} = {ψ | Hψ = ZDMQ(ψ), ∥ψ∥ = 1}In this framework, the classical kernel emerges as a limiting case:limDZ →0 KQZ = CKQ → CKproving KQZ is fundamental.The purpose of this paper is to utilize this newly developed Zulfia QuantumKernel to characterize Zulfia-balanced states where quantum evolution is exactlycompensated by its own deviation, opening a new window to study structuredstability, non-trivial zeros, and hidden symmetries beyond classical zero.
Abstract Quantum machine learning, an emerging field at the intersection of quantum computing and classical machine learning, has shown great promise in enhancing computational capabilities beyond classical bounds. A key element in this area of research is the utilization of Quantum Kernel Estimators, traditionally grounded in the symmetries of SU(2) groups associated with qubits. This study extends the conceptual framework of Quantum Kernel Estimators to incorporate a broader spectrum of symmetry groups. By harnessing the various structures of Lie groups, we develop novel quantum-inspired feature maps that offer more flexible and potentially powerful ways to encode and compress classical data into quantum states. We present a comprehensive theoretical introduction for this approach, followed by a methodology that integrates the developed feature maps into kernel-based quantum classifiers. Our results, derived from a series of computational experiments across various datasets, demonstrate the efficacy of this approach in comparison to traditional quantum and classical machine learning models. The findings not only underline the versatility of Lie-group theory in enhancing quantum machine learning algorithms but also open new avenues for exploring complex symmetries in quantum information processing. This research bridges a gap between the study of symmetries and machine learning, paving the way for more sophisticated quantum algorithms capable of tackling complex, high-dimensional data in ways previously unattainable.
A bstract The construction of exactly-solvable models has recently been advanced by considering integrable TT‾ T\overline{T} TT T T ¯ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise in the description of energy diffusion that emerges in quantum systems from time-keeping errors in a real clock used to track time evolution. We show that the latter can be related to an inverse TT‾ T\overline{T} TT T T ¯ deformation with a purely imaginary deformation parameter. In this case, the integral transforms take a particularly simple form when the initial state is a coherent Gibbs state or a thermofield double state, as we illustrate by characterizing the purity, Rényi entropies, logarithmic negativity, and the spectral form factor. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we further study the spectral properties of the Liouvillians, i.e., the dynamical generators associated with the deformed theories. As an application, we discuss the interplay between decoherence and quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and the Sachdev-Ye-Kitaev model.
Quantum kernel functions built using quantum-mechanical principles and have emerged as a centerpiece of quantum machine learning. The initial enthusiasm for quantum kernel machines has been tempered by recent studies suggesting that quantum kernels could not offer significant computational or statistical advantages when learning from classical data. However, most of the research in this area has been devoted to scalar-valued kernels in standard classification or regression settings for which classical kernel methods are efficient and effective, leaving very little room for improvement with quantum kernels. In this position paper, we argue that progress in this field requires moving beyond scalar-valued kernels toward more expressive kernel frameworks. Scalar-valued kernels lack the degrees of freedom necessary to fully exploit intrinsically quantum resources such as entanglement and are not rich enough to deal with complex learning tasks where classical learning methods struggle. Building on recent advances in operator-valued kernel learning and C∗C^*C∗-algebraic kernel representations, we propose a roadmap for designing quantum kernels capable of leveraging entanglement and non-commutative structures to tackle complex structured prediction problems. To support this viewpoint, we present an initial proof-of-concept illustrating how quantum operator-valued kernel formulations can reveal structural dependencies that remain difficult to access for scalar-valued kernel methods. This shift in focus could open a pathway toward a new generation of quantum kernel machines and a more faithful exploration of their potential advantages.
For more than a century since Schr¨odinger, the quantum wave equation has been as-sumed to hold on a fully accessible domain Ω without certified origin. No extension in theavailable literature has revisited its foundational probability assumption.We replace this assumption with a deviation origin. Using Zulfia Decomposition inProbability (ZDM) Z(t) = MZ [η(t0)] + ∆(t) within Zulfia Bi-Face Probability Theory(ZBPT) [1], the information field after collision is restricted to deviation domain Ω \ ΩC =∆−1(̸= 0) where ∆(t)̸ = 0 where PZ does not exist and C(ω) = 0, with certified domainΩC = ∆−1(0) excluded where ∆(t) = 0.iℏZ CZ (ω) ∂∆(t)∂t = − ℏ2Z2mZ∇ · (CZ (ω)∇∆(t)) + VZ (ω)∆(t) on Ω \ ΩC (1)The conventional equationiℏ∂tψ = − ℏ22m ∇2ψ + V ψ (2)is the special case for Ω \ ΩC ≈ Ω, CZ = 1.After Collision is strictly informational, mass-free on Ω \ ΩC .
Abstract Since the entry of kernel theory in the field of quantum machine learning, quantum kernel methods (QKMs) have gained increasing attention with regard to both probing promising applications and delivering intriguing research insights. Benchmarking these methods is crucial to gain robust insights and to understand their practical utility. In this work, we present a comprehensive large-scale study examining QKMs based on fidelity quantum kernels (FQKs) and projected quantum kernels (PQKs) across a manifold of design choices. Our investigation encompasses both classification and regression tasks for five dataset families and 64 datasets, systematically comparing the use of FQKs and PQKs quantum support vector machines and kernel ridge regression. This resulted in over 20,000 models that were trained and optimized using a state-of-the-art hyperparameter search to ensure robust and comprehensive insights. We delve into the importance of hyperparameters on model performance scores and support our findings through rigorous correlation analyses. Additionally, we provide an in-depth analysis addressing the design freedom of PQKs and explore the underlying principles responsible for learning. Our goal is not to identify the best-performing model for a specific task but to uncover the mechanisms that lead to effective QKMs and reveal universal patterns.
We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton–Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton–Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure. Quanta 2022; 11: 42–52.