What question did this study set out to answer?

This research aims to develop a method for protecting modular quantum states from thermal decoherence using a Parent Lindbladian.

June 22, 2026Open Access

Dissipative Protection of Modular Quantum States, A Parent Lindbladian for Z/6Z Superselection

Key Points

This research aims to develop a method for protecting modular quantum states from thermal decoherence using a Parent Lindbladian.
Constructed a Parent Lindbladian to maintain unique steady states for quantum states.
Executed numerical experiments via a Colab notebook for systems with up to 6 qubits, focusing on exact diagonalization and spectral gap computations.
Analyzed dissipative phase transitions and entanglement entropy within varied leakage parameters.
Demonstrated a crossover at a critical leakage parameter (δ_c ≈ 0.05) indicating shifts in entropy behavior.
Proved the Area Law in a macroscopic limit, showing entropy saturation below ergodic limits.
Confirmed the stability of the topological state as an attractor through spectral gap analysis.

Abstract

This repository contains the full computational framework accompanying the article "Dissipative Protection of Modular Quantum States: A Parent Lindbladian for Z/6Z Superselection". The modular ring Z/6Z partitions a quantum register into two resonant channels (C₁, C₅) and four sterile ones (C₀, C₂, C₃, C₄). While the companion work establishes how to prepare a topological state confined to the resonant subspace, this article addresses the subsequent challenge: protecting that state against thermal decoherence on NISQ hardware. We construct a Parent Lindbladian—a frustration-free dissipative generator whose unique steady state is exactly the topologically superselected density matrix. The jump operators, derived from the modular deterministic finite automaton, continuously pump amplitude from the sterile channels back to the resonant ones, acting as a passive error-correction mechanism. Repository contents: Dissipative Protection of Modular Quantum States. pdf — The complete article (LaTeX source available in the accompanying GitHub repository). DissipativeProtectionₒfModularQuantumStates. ipynb — Reproducible Colab notebook implementing all numerical experiments: Exact diagonalisation of the Lindblad superoperator for spin chains up to N=6 qubits. Dissipative phase transition: strict mask (δ=0) enforces an Area Law; breaking the mask (δ>0) restores the Volume Law (ETH). Leakage parameter sweep (δ=0 to 0. 5) showing a sharp crossover at δc ≈ 0. 05. Liouvillian spectral gap computation confirming the topological state as a stable attractor. Macroscopic MPDO tensor network contraction (N=10 to 60 qubits) under depolarising noise, demonstrating S₂ entropy saturation at ~1. 65 bits—far below the ergodic limit. AreaLawProof. png — Thermodynamic extrapolation of the bipartite Rényi entropy, proving the Area Law in the macroscopic limit. TransitionETHAreaLaw. png — Dissipative phase transition: von Neumann entropy vs. system size for strict and broken topological masks. LiouvillianGap. png — Evolution of the spectral gap as a function of the leakage parameter. DeltaSweep. png — Entanglement entropy crossover at fixed N=4, identifying the critical noise threshold. All experiments are fully deterministic and reproducible on any standard Google Colab instance using QuTiP, NumPy, SciPy, and Matplotlib.

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