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Fractional-Time Qubits: Leveraging Memory Kernels to Combat Decoherence in Quantum Computing

Erika Barker

Published Mar 10, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.15000394

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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.
Building upon our prior work [1], we present a unified stochastic drift model (SdM) for superconducting charge qubits based on memory multi-fractional Brownian motion (mmFBM). The classical sector employs a time-dependent Hurst exponent H(t) and adaptive memory kernel K(t,s), capturing non-stationary 1/f^beta noise and long-range temporal correlations inaccessible to conventional models. The quantum extension is formulated via a time-dependent Caldeira--Leggett environment with spectral density J(omega;t) = eta(t) omega_c^{1-s(t)} omega^{s(t)} exp(-omega/omega_c), where s(t) = 2H(t)-1, consistently reproducing beta(t) = 2H(t)-1. Four central results emerge: (1) relaxation and noise amplitudes act independently on energy decay; (2) time-varying H(t) matches experimental 1/f spectra more accurately than any constant exponent; (3) adaptive kernel dynamics preserve correlations without artificial damping; and (4) simulations predict coherence times (T1 ~ 5.00 x 10^6 ns, T2 ~ 4.18 x 10^5 ns) consistent with theory when charge noise dominates. The qubit exhibits stretched-exponential Ramsey and echo decay, non-Markovian dephasing, and a temperature-driven quantum-to-classical crossover. We derive the effective time-local Lindblad master equation, establish the classical mmFBM limit at high temperatures, and provide experimentally testable scaling relations. The non-exponential decay patterns reveal fundamental limitations of Markovian approaches, and the framework guides the design of noise-resilient qubit architectures.
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.
In this paper, we investigate the memory effects introduced by the time-fractional Schrödinger equation proposed by Naber on quantum entanglement and quantum dense coding under amplitude damping noise. Two formulations are analyzed: one with fractional operations applied to the imaginary unit and one without. Numerical results show that the formulation without fractional operations on the imaginary unit may be more suitable for describing non-Markovian (power-law) behavior in dissipative environments. This finding provides a more physically meaningful interpretation of the memory effects in time-fractional quantum dynamics and indirectly addresses fundamental concerns regarding the violation of unitarity and probability conservation in such frameworks. Our work offers a new perspective for the application of fractional quantum mechanics to realistic open quantum systems and shows promise in supporting the theoretical modeling of decoherence and information degradation.
A significant obstacle impeding the advancement of the time fractional Schrödinger equation lies in the challenge of determining its precise mathematical formulation. In order to address this, we undertake an exploration of the time fractional Schrödinger equation within the context of a non-Markovian environment. By leveraging a two-level atom as an illustrative case, we find that the choice to raise i to the order of the time derivative is inappropriate. In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment, the time fractional Schrödinger equation, when devoid of fractional-order operations on the imaginary unit i, emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects. Meanwhile, we also prove that it is meaningless to study the memory of time fractional Schrödinger equation with time derivative 1 < α ≤ 2. It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrödinger equation. This will be the focus of future research. Our study might provide a new perspective on the role of time fractional Schrödinger equation.
This paper proposes a novel quantum computing error correction code predicated on the principles of quantum decoherence. The core concept leverages the inherent fragility of quantum states to create a robust error detection and correction mechanism. Specifically, the code aims to control and manipulate the decoherence time of individual qubits, utilizing this temporal constraint within a structured error correction scheme. The theoretical framework details the encoding, decoding, and error correction processes, incorporating mathematical formulations to quantify the redundancy introduced and the resulting improvement in qubit fidelity. The proposed code's effectiveness is demonstrated through simulations, showcasing its potential to significantly enhance the reliability of quantum computations, a critical step towards achieving scalable quantum computing. The key contributions lie in the direct integration of decoherence dynamics into the error correction strategy, offering a fundamentally different approach compared to traditional methods. The paper concludes with a discussion of potential optimizations and future research directions.
Quantum computing promises revolutionary advancements across various fields, but its practical realization is fundamentally limited by the phenomenon of quantum decoherence. Decoherence refers to the loss of quantum superposition and entanglement due to interactions with the surrounding environment, leading to computational errors. This paper presents a comprehensive modeling and prediction framework for quantum decoherence, aiming to provide a deeper understanding of this critical process. The framework utilizes a mathematically rigorous approach based on the principles of quantum mechanics, specifically focusing on the master equation formulation of decoherence. We derive a set of equations that accurately describe the evolution of the quantum state under the influence of various environmental noise sources. Furthermore, we explore methods for estimating decoherence rates and predicting the decoherence time, which is a crucial parameter for assessing the viability of quantum computations. The model incorporates key concepts such as the Bloch sphere representation, the Lindblad master equation, and the concept of decoherence channels. We demonstrate the applicability of the model to various quantum systems and discuss the potential for developing effective decoherence mitigation strategies. The ultimate goal is to provide a tool that can guide the design and optimization of quantum computers, ultimately paving the way for robust and reliable quantum computation.