Fractional calculus provides a natural framework for incorporating memory, dissipation, and temporal nonlocality into quantum dynamics. In this chapter, fractional time evolution is employed to extend conventional quantum gates and entangling operations while preserving Hilbert-space structure, locality, and energetic consistency. Fractional quantum gates are shown to arise from standard local Hamiltonians acting through Mittag Leffler propagators, leading to non-unitary yet physically meaningful operations characterized by Lévy-flight–like temporal statistics. This approach reshapes single- and two-qubit dynamics, producing gradual population transfer, algebraic dephasing, and history-dependent entanglement generation. The framework is further applied to optical and photonic systems, where fractional dynamics naturally emerge through dispersion, mode coupling, and structured propagation, offering insight into phenomena such as PSF broadening and coherence loss. Finally, implications for quantum communication and quantum key distribution are discussed, highlighting how channel memory modifies standard security assumptions and motivates memory-aware protocol design. Overall, fractional quantum dynamics are presented as a unifying theoretical lens for understanding realistic quantum information operations processing in and complex environments.
Fractional calculus provides a natural framework for incorporating memory, dissipation, and temporal nonlocality into quantum dynamics. In this chapter, fractional time evolution is employed to extend conventional quantum gates and entangling operations while preserving Hilbert-space structure, locality, and energetic consistency. Fractional quantum gates are shown to arise from standard local Hamiltonians acting through Mittag Leffler propagators, leading to non-unitary yet physically meaningful operations characterized by Lévy-flight–like temporal statistics. This approach reshapes single- and two-qubit dynamics, producing gradual population transfer, algebraic dephasing, and history-dependent entanglement generation. The framework is further applied to optical and photonic systems, where fractional dynamics naturally emerge through dispersion, mode coupling, and structured propagation, offering insight into phenomena such as PSF broadening and coherence loss. Finally, implications for quantum communication and quantum key distribution are discussed, highlighting how channel memory modifies standard security assumptions and motivates memory-aware protocol design. Overall, fractional quantum dynamics are presented as a unifying theoretical lens for understanding realistic quantum information operations processing in and complex environments.
Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian CPTP evolution, many physical platforms display non-Markovian features such as algebraic relaxation and coherence backflow. Fractional calculus provides a natural way to model such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we develop a unified framework that embeds fractional master equations within the broader hierarchy of open-system formalisms. The fractional equation forms a structured subclass of memory-kernel models, reduces to the Lindblad form at unit order, and, through Bochner--Phillips subordination, admits a CPTP representation as an average over Lindblad semigroups. Its resolvent structure further connects fractional dynamics to established non-Markovian approaches, including Nakajima--Zwanzig kernels and hierarchical equations of motion, providing a compact surrogate for long-memory effects. This formulation positions fractional calculus as a rigorous and practical language for quantum dynamics with intrinsic memory, supporting both analytical insight and efficient quantum simulation.
Abstract Our work explores the time evolution of entanglement, local quantum uncertainty, and correlated coherence within a system modeled by two coupled double quantum dots. The dynamics is described using a time-fractional Schrödinger equation, which is treated here as a phenomenological effective framework introducing temporal nonlocality and memory-like effects, rather than as a first-principles open-system reduction. We vary the fractional parameter τ , the tunneling amplitudes δ A and δ B , and the inter-dot Coulomb interaction strength V , in order to investigate how these key parameters govern the generation, stabilization, and decay of quantum resources within the system. The obtained results reveal that, for both initial states (separable and partially entangled), fractional dynamics with a small τ rapidly generates entanglement reaching maximal values LN ≈ 1 together with strong non-classical correlations quantified by the local quantum uncertainty. Conversely, when τ is closer to unity, entanglement builds up more slowly but the memory-like effects allow quantum resources to remain significant for a longer time, with the negativity remaining above ≈ 0.6 . We also find that higher Coulomb-interaction frequencies V accelerate the build-up of correlations and stabilize coherence, while a strong tunneling asymmetry degrades entanglement and coherence despite the initial benefits of increasing quantum resources. We find that the standard unitary Schrödinger evolution is recovered analytically at τ = 1 , where the Mittag–Leffler functions reduce to ordinary exponentials, while decreasing τ progressively introduces fractional-time nonlocal (memory-like) effects.
Classical Maxwell–Bloch models are often inadequate for describing electromagnetic wave propagation in complex optical media where memory effects, anomalous relaxation processes, and long-range electromagnetic interactions are important. To address these limitations, this work develops a fractional nonlocal Maxwell–Bloch framework that simultaneously incorporates Caputo fractional temporal memory and nonlocal electromagnetic coupling. The principal novelty lies in the unified treatment of temporal memory and spatial nonlocality within the Maxwell–Bloch framework, together with the corresponding analysis of their effects on optical-wave dynamics. The proposed model captures hereditary polarization dynamics and spatially distributed electromagnetic interactions that are not represented in the standard local integer-order formulation. The analytical part establishes the well-posedness structure of the governing system, including existence, boundedness, and stability properties of solutions in appropriate functional spaces. A spectral stability analysis is further employed to characterize fractional instability thresholds and transitions in the resulting wave propagation. Numerical simulations demonstrate the emergence of localized fractional optical structures, memory-dependent pulse propagation, nonlinear wave interactions, and complex dynamical regimes. A systematic parametric study further shows that variations in the fractional order significantly modify the electric-field profile, energy evolution, and pulse amplitude. The results demonstrate that the combined effects of fractional temporal memory and spatial nonlocality can substantially alter the propagation and stability of electromagnetic waves compared with the classical Maxwell–Bloch limit. The framework therefore provides a unified mathematical and computational basis for investigating memory-dependent and spatially nonlocal optical dynamics.
This study investigates the time-fractional Schrödinger equation (TFSE) for a free particle using the Caputo fractional derivative. The model extends the classical Schrödinger equation by incorporating memory effects through a non-integer order time derivative. By applying the Fourier transform method, an explicit solution is derived in terms of the Mittag–Leffler function, which generalizes the classical exponential evolution. The analysis reveals that the resulting dynamics are non-unitary, leading to a time-dependent total probability that deviates from the conservation property of standard quantum mechanics. Numerical illustrations demonstrate the influence of the fractional order ν on wave packet evolution and probability behavior. The results highlight the role of fractional calculus in modeling non-Markovian quantum systems and provide insight into the transition from fractional to classical dynamics as v→1. .
This is a novel approach to mitigating decoherence in quantum computing with "Fractional-Time Qubits: Leveraging Memory Kernels to Combat Decoherence in Quantum Computing." In this work, Erika Barker introduces fractional quantum dynamics by replacing the conventional time derivative in Schrödinger’s equation with a fractional derivative of order β. This modification incorporates long-range memory kernels—crucial for modeling 1/f noise and other non-Markovian effects—that fundamentally alter decoherence profiles and extend qubit lifetimes. Key highlights include: Expanded Hilbert Space Formalism: A novel framework that embeds fractional dynamics into a larger, unitary space, ensuring probability conservation and a generalized energy conservation law reminiscent of pseudo-Hermitian quantum mechanics. Noise Mitigation Strategies: Detailed analysis and numerical simulations demonstrate how fractional dynamics produce stretched-exponential and power-law decays, effectively countering decoherence in 1/f, Ohmic, and sub-Ohmic noise environments. Resource Efficiency: A comparative discussion on resource overhead reveals potential advantages over traditional quantum error correction, offering a promising pathway for near-term quantum devices with limited qubit counts. Interdisciplinary Impact: The paper also connects these techniques with broader theoretical frameworks in quantum gravity and fractal time, suggesting far-reaching implications for both practical quantum engineering and fundamental physics. This comprehensive study provides essential insights for researchers focused on quantum decoherence, advanced error mitigation techniques, and the development of robust quantum computing architectures. Keywords: quantum computing, decoherence, fractional dynamics, memory kernels, 1/f noise, non-Markovian, Hilbert space, quantum error correction.
In this study, we extend the concept of the quantum Gamma function (q-Gamma) by introducing a new (q,\tau)-deformed Gamma function. This generalization allows us to construct an enriched family of Hadamard-type fractional operators, which we then apply to the analysis of memory and decoherence in open quantum systems. The inclusion of the deformation parameter q together with the delay-like scaling parameter \tau makes the proposed (q,\tau)-Hadamard framework particularly suited to capture nonlocal and non-Markovian effects, thereby offering a flexible tool for describing structured reservoirs and anomalous dissipation. To investigate the analytical consequences, we employ a (q,\tau)-Mittag-Leffler function, through which explicit solutions are obtained for a class of fractional differential equations, with particular emphasis on population dynamics models. These solutions reveal a variety of memory-driven features, including sub-exponential decay of coherence and the occurrence of revival phenomena. Both the fractional order alpha and the deformation parameters play a decisive role in shaping the temporal behavior. Beyond population models, the framework also provides insights into fractional quantum master equations, quantum walks with fractional memory, and noise effects in quantum information processing.