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Topological Model of Quantum Decoherence Based on Graph Algebraic Connectivity: From Markovian Hitting Time to a Unified Analytical Equation

Ethan Feng

Published Apr 6, 2026• 0 citations
Publisher

Abstract

Accurately predicting quantum decoherence time is a core bottleneck in scaling up noisy intermediate-scale quantum (NISQ) devices. Traditional frameworks relying on the Lindblad master equation suffer from the curse of dimensionality and obscure the intuitive physical mechanisms of decoherence behind complex system-environment coupling assumptions. Based on graph algebraic connectivity, this paper proposes a deterministic topological model of quantum decoherence. We rigorously map multi-particle quantum entanglement to the algebraic connectivity (Fiedler value, ) of an underlying structural graph and redefine decoherence as a Poissonian random edge-deletion process. By modeling this topological evolution as a continuous-time Markov chain (CTMC), we derive a unified and exact recursive analytical equation for the expected hitting time of complete decoherence for arbitrary asymmetric topological states. The analytical solutions strictly prove the relative robustness of prototypical entangled states . Furthermore, the model demonstrates unprecedented computational advantages by instantly yielding the exact analytical decoherence time for a 10-body asymmetric complex network, which is perfectly validated by 50,000 Monte Carlo simulations with a relative error of . We calibrate the core parameter (single-edge fracture rate) using the latest public experimental data of superconducting quantum chips from IBM and Google, and the theoretical predictions are in high agreement with the experimental results within the measurement error range . This topological downgrade mechanism reduces the computational complexity of decoherence calculations to , providing a powerful theoretical and algorithmic tool for the optimization of quantum processing unit (QPU) underlying architectures and the design of fault-tolerant multi-particle quantum networks.

DOI
10.6084/m9.figshare.31941237

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Accurately predicting quantum decoherence time is a core bottleneck in scaling up noisy intermediate-scale quantum (NISQ) devices. Traditional frameworks relying on the Lindblad master equation suffer from the curse of dimensionality and obscure the intuitive physical mechanisms of decoherence behind complex system-environment coupling assumptions. Based on graph algebraic connectivity, this paper proposes a deterministic topological model of quantum decoherence. We rigorously map multi-particle quantum entanglement to the algebraic connectivity (Fiedler value, ) of an underlying structural graph and redefine decoherence as a Poissonian random edge-deletion process. By modeling this topological evolution as a continuous-time Markov chain (CTMC), we derive a unified and exact recursive analytical equation for the expected hitting time of complete decoherence for arbitrary asymmetric topological states. The analytical solutions strictly prove the relative robustness of prototypical entangled states . Furthermore, the model demonstrates unprecedented computational advantages by instantly yielding the exact analytical decoherence time for a 10-body asymmetric complex network, which is perfectly validated by 50,000 Monte Carlo simulations with a relative error of . We calibrate the core parameter (single-edge fracture rate) using the latest public experimental data of superconducting quantum chips from IBM and Google, and the theoretical predictions are in high agreement with the experimental results within the measurement error range . This topological downgrade mechanism reduces the computational complexity of decoherence calculations to , providing a powerful theoretical and algorithmic tool for the optimization of quantum processing unit (QPU) underlying architectures and the design of fault-tolerant multi-particle quantum networks.
Abstract Accurately predicting quantum decoherence time is a core bottleneck in scaling up noisy intermediate-scale quantum (NISQ) devices. Traditional frameworks relying on the Lindblad master equation suffer from the curse of dimensionality (O(4^n )) and obscure the intuitive physical mechanisms of decoherence behind complex system-environment coupling assumptions. Based on graph algebraic connectivity, this paper proposes a deterministic topological model of quantum decoherence. We rigorously map multi-particle quantum entanglement to the algebraic connectivity (Fiedler value, λ_2) of an underlying structural graph and redefine decoherence as a Poissonian random edge-deletion process. By modeling this topological evolution as a continuous-time Markov chain (CTMC), we derive a unified and exact recursive analytical equation for the expected hitting time of complete decoherence for arbitrary asymmetric topological states. The analytical solutions strictly prove the relative robustness of prototypical entangled states (T_"Bell" >T_W≫T_"GHZ" ). Furthermore, the model demonstrates unprecedented computational advantages by instantly yielding the exact analytical decoherence time for a 10-body asymmetric complex network, which is perfectly validated by 50,000 Monte Carlo simulations with a relative error of <0.5%. We calibrate the core parameter γ (single-edge fracture rate) using the latest public experimental data of superconducting quantum chips from IBM and Google, and the theoretical predictions are in high agreement with the experimental results within the measurement error range (<3%). This topological downgrade mechanism reduces the computational complexity of decoherence calculations to O(n^3 ), providing a powerful theoretical and algorithmic tool for the optimization of quantum processing unit (QPU) underlying architectures and the design of fault-tolerant multi-particle quantum networks.
We present a comprehensive analytical and numerical study of time-dependent decoherence in open quantum systems, focusing on a single-qubit model with time-varying Lindblad dissipation rates. Employing a master equation framework, we derive exact solutions for the evolution of off-diagonal density matrix elements and characterize the emergence of robust pointer states. Our findings reveal an initial rapid coherence decay followed by a gradual slowdown, mirroring experimental trends observed in superconducting qubits influenced by quasiparticle dynamics and in photonic cavities subject to fluctuating environments. By incorporating real experimental parameters and validating the model through quantum simulations using QuTiP, we provide a robust quantitative tool for analyzing decoherence in quantum technologies. This extended model addresses limitations in static-rate approximations, offering insights into non-Markovian effects without resolving the measurement problem, and suggests extensions to multi-qubit systems.
Topological entanglement, a fundamental concept in quantum mechanics, exhibits unique properties that challenge classical intuition. This research explores a novel framework for quantifying and controlling the dynamic evolution of topological entanglement, predicting and mitigating decoherence in complex quantum systems. We propose a dynamically evolving, multi-scale topology based on a 'self-organizing' entanglement graph, utilizing adaptive feedback loops to minimize dissipation. This approach significantly expands the scope of entanglement analysis, offering predictive capabilities exceeding existing methods. The core mechanism involves modeling entanglement structure through a graph-based approach, incorporating adaptive control to ensure stability and coherence. This work presents a theoretical foundation and preliminary experimental validation demonstrating the framework's ability to predict and counteract decoherence patterns, thereby advancing our understanding of topological entanglement dynamics.
Quantum processors in the Noisy Intermediate-Scale Quantum (NISQ) era are severely constrained by spatial hardware asymmetries, temporal drift, and multi-qubit crosstalk. Traditional error suppression protocols apply static decoupling or phenomenological calibrations that struggle under continuous decoherence stress. Building upon the Entropic-Resistance framework—where spacetime is represented as a discrete, algorithmically expanding topological network and time is governed by recursive relaxation—we extend deterministic Causal-AI Taylor sensitivity operators (j1, j2) to multi-qubit topologies. We introduce a Just-In-Time (JIT) compensation engine that queries real-time QPU telemetry (qubit-specific dephasing T2 times and native two-qubit gate error rates) to dynamically compute non-commutative phase compensations. By decomposing physical microwave rotations inside logic gates, microscopic counter-rotations are embedded directly into hardware instructions. We validate this protocol on IBM Quantum's Heron processor (ibm_fez) by subjecting 3-qubit and 5-qubit Greenberger-Horne-Zeilinger (GHZ) states to deliberate decoherence stress via calibrated temporal delays. On 3-qubit states, the Lasky-Tensor engine achieves an absolute fidelity improvement of +3.81% (32.5% relative error reduction). On fragile 5-qubit linear networks, it delivers a +3.42% absolute gain (17.2% relative error reduction). These results establish that recursive network stress models scale effectively, providing a deterministic, white-box quantum error mitigation paradigm without physical qubit overhead. Methodology & Hardware: Hardware: IBM Quantum (ibm_fez, Heron architecture) Framework: Qiskit Runtime SamplerV2 Underlying Theory: Entropic-Resistance Paradigm / Causal-AI What-If Engine
It is known that under some assumptions the hitting time in quantum Markov chains is quadratically smaller than the hitting time in classical Markov chains. This work extends this result for decoherent quantum Markov chains. The decoherence is introduced using a percolation-like graph model, which allows us to define a decoherent quantum hitting time and to establish a decoherent-intensity range for which the decoherent quantum hitting time is quadratically smaller than the the classical hitting time. The detection problem under decoherence is also solved with quadratic speedup in this range.
This paper presents a novel approach to modeling quantum decoherence using probabilistic models. Traditional quantum decoherence models often rely on simplified assumptions, leading to inaccuracies in predicting decoherence times. This work introduces a more rigorous framework based on statistical physics principles, explicitly incorporating probabilistic elements to capture the complex dynamics of quantum systems. The core of the model lies in formulating decoherence as a probabilistic process, allowing for a more accurate quantification of the rate at which quantum information is lost. We derive key equations that describe the evolution of the quantum state under the influence of various decoherence mechanisms. The model's parameters are determined through statistical analysis of the underlying quantum system, providing a predictive capability for decoherence times. Simulation results demonstrate the improved accuracy of this probabilistic approach compared to conventional models. This work represents a significant advancement in quantum decoherence modeling, offering a more reliable tool for designing and analyzing quantum computing architectures and quantum information processing protocols.