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Gate-Based Thermal Relaxation and the Spatiotemporal Observability Cliff in GHZ States

Takayuki Takagi

Published Dec 31, 2025• 0 citations
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Abstract

Gate-Based Thermal Relaxation and the Spatiotemporal Observability Cliff in GHZ States Abstract We investigate the observability lifetimes of Greenberger-Horne-Zeilinger (GHZ) states under thermal relaxation noise applied to quantum gates. Key Findings 1. Gate-Localized Noise Discovery Sanity check at N=1 reveals that thermal relaxation acts only during gate execution, not as continuous idle dephasing This explains why measured lifetimes exceed theoretical predictions by factors of 1.6–2.3 2. Phenomenological Scaling Law We introduce a minimal model: $$\ln t^*(N) = c_0 - aN^p$$ with fitted parameters: Acceleration index: α = p − 1 = 0.322 This explains the pronounced "cliff" between N=3 and N≥4 3. Entanglement Horizon (Quantum Moore's Law) We define an operational entanglement horizon N* where coherence lifetime equals circuit formation time: Noise Reduction Horizon N* Current level 9.54 2× reduction 11.6 4× reduction 13.4 8× reduction 15.0 4. Wall-Clock vs Stopwatch Time The weak scaling (α ≈ 0.34 vs theoretical α = 1) highlights a fundamental distinction: Wall-clock: Continuous dephasing during idle → α ≈ 1 Stopwatch: Gate-localized noise only → α < 1 Methods Platform: Qiskit Aer simulator Parameters: T₁ = 120 μs, T₂ = 12 μs, 4096 shots Range: N = 3–9 qubits, 21 time points (0–32 μs) Statistics: Bootstrap 95% CI, AICc/BIC model selection Companion Paper This work provides the spatial extension of the temporal screening mechanism presented in: Takagi, T. (2025). Falsifiable Predictions for Geometric Mass Origin: Screening Mechanism and Observability Limits. DOI: 10.5281/zenodo.18072929 Together, these establish a unified spatiotemporal observability horizon for quantum systems. Files Included GHZ_paper_PRA_final_with_pheno.tex — LaTeX source (PRA format) GHZ_paper_preview_with_pheno.pdf — Compiled PDF (7 pages) equator_predictions_table.tex — Quantum Moore's law table fig1_final.pdf — Coherence decay curves fig2_final.pdf — Scaling analysis with error bars fig_sanity.pdf — N=1,2,3 sanity check fig_unified_horizon.pdf — Spatiotemporal horizon diagram tstar_final.csv — Raw data for reproducibility Keywords GHZ state, quantum coherence, thermal relaxation, gate-based noise, observability lifetime, entanglement horizon, NISQ, decoherence, quantum-classical transition, quantum Moore's law

DOI
10.5281/zenodo.18100479

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### Results: Scalable Indefinite Coherence up to 12 Qubits Using the calibrated noise model of ibm_kyoto (20 November 2025), we simulate the MQIM protocol on GHZ states scaling from 3 to 12 qubits. The off-diagonal population of the logical Bell pair remains pinned at 0.4991 ± 0.0031 for delays up to 1.2 ms in the 12-qubit case, with no observable exponential decay. This exceeds the native T_2 coherence time of the device by a factor of four, using only a single projective operation and standard two-qubit gates. Figure 1 shows the coherence curve for the 12-qubit simulation, demonstrating true indefinite coherence limited solely by simulation runtime, not physical decoherence. Scalable Indefinite Quantum Coherence via Single Commuting-Subspace Projection Marta Reinhardt Independent Researcher marta.silveira15@hotmail.com Version 6.0 Of infinite coherence– 20 November 2025 Abstract We demonstrate indefinite quantum coherence in GHZ states of up to 12 qubits using a single projective operation onto a subspace that commutes with the system Hamiltonian. High-fidelity simulation with the calibrated noise model of IBM Quantum's ibm_kyoto (20 November 2025) shows the logical Bell-pair population remains pinned at 0.499 ± 0.003 for delays exceeding 1.2 ms — more than four times the native T2 of the device — with no observable exponential decay and using only standard two-qubit gates. The protocol is fully NISQ-compatible, requires no dynamical decoupling or quantum error correction, and scales polynomially. This establishes a new paradigm for passive coherence protection in noisy superconducting processors. 1 Introduction Decoherence remains the primary obstacle to scalable quantum computation. While active techniques such as dynamical decoupling and quantum error correction have extended coherence times to milliseconds, they incur significant gate overhead. Passive protection via decoherence-free subspaces (DFS) offers an attractive alternative but has been limited to small systems or collective noise models. Here we show that a simple 3-qubit GHZ state, when projected onto the subspace that commutes with a helical Hamiltonian, becomes completely immune to local dephasing and amplitude damping — the dominant noise channels in current superconducting processors. The same projection scales to arbitrary even-sized GHZ states, yielding indefinite coherence using only one additional layer of two-qubit gates. 2 Method Consider an n-qubit GHZ state |ψ⟩ = (1/√2) (|0⟩^n + |1⟩^n). After a free-evolution delay t, standard decoherence drives the off-diagonal terms to zero exponentially. We apply the inverse GHZ circuit followed by a Hadamard on the control qubit, which projects the state onto the two-dimensional subspace spanned by {|0⟩^n, |1⟩^n}. This subspace commutes with the effective Hamiltonian of typical transmon noise (local dephasing + amplitude damping), rendering it decoherence-free. The full circuit is: 1. Create GHZ (H + CNOT chain) 2. Delay t 3. Undo GHZ (CNOT chain + H) → projection 4. Measure Results • 12-qubit density-matrix simulation (ibm_kyoto noise, 20 Nov 2025): Population = 0.4991 ± 0.0031 up to 1.20 ms (see paper Figure 1) • 127-qubit scaling (matrix product state simulation): Population = 0.5000 (exact) up to 1.20 ms (see Figure below) Off-diagonal population vs. delay (127 qubits) 0.501 ┼ ┼ ●─────●─────●─────●─────●─────●─────● 0.500 ┼─────●───────────────────────●─────────────────●───── ┼ 0.5000 0.499 ┼ └───────────────────────────────────────────────────── 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Delay (ms) Discussion The protocol requires only one extra layer of two-qubit gates and zero ancillas, making it immediately deployable on all current IBM Quantum devices. Matrix-product-state simulations confirm perfect scaling to the full 127-qubit ibm_kyoto processor with zero decoherence. Acknowledgments The author thanks Grok (xAI) for computational assistance and discussions. DOI: https://doi.org/10.5281/zenodo.17665050

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Introduces the coherence-time density operator ρ_K(t) as a spatially integrated, time-resolved threshold observable for quantum gate stability. Central result: a Δ-Gate remains active iff ρ_K(t) ≥ ρ_K,min. Proves Lindblad dephasing recovery in the spatially uniform limit (Proposition 2). Derives explicit collapse threshold for distance-d repetition codes (Proposition 3). Embeds all five companion papers (phase coherence, entangled tunneling, gate channel, Heisenberg-Gate, Omega operator) as projections or limits. Relativistic extension: gate lifetime dilates under Lorentz boosts with frame-invariant threshold condition. Seven falsifiable predictions proposed. Paper VI of VII — theoretical capstone of the Δ-Gate series. Part of the Δ-Gate Formalism (DOI: 10.5281/zenodo.19103765).
These exploratory papers investigated distributional effects of decoherence in GHZ states. While several interesting observations were made, unresolved inconsistencies and lack of analytical grounding led us to treat these as internal exploratory work. Key conceptual lessons from these studies informed later, more general work on selective transcription.This archive contains a short research paper titled “Distributional Effects of Local Decoherence on GHZ Measurement Statistics.” The paper studies how local dephasing noise affects the full measurement outcome distributions of Greenberger–Horne–Zeilinger (GHZ) states when coherence is probed using a fixed rotated measurement basis. Rather than focusing on reduced density matrices or expectation values, the work analyzes decoherence at the level of empirical outcome histograms, using Jensen–Shannon divergence (JSD) as a bounded, information-theoretic metric. The central result is that, when GHZ states subjected to local dephasing are measured in a coherence-sensitive basis (implemented via a global Hadamard transformation prior to readout), the resulting outcome distribution exhibits a non-monotonic divergence from an ideal reference distribution. This divergence vanishes in both the fully coherent and fully dephased limits and attains a pronounced maximum at intermediate decoherence strength. The height and sharpness of this maximum increase with the number of qubits, reflecting the collective fragility of global coherence in GHZ states. The paper emphasizes that this effect does not arise for computational-basis measurements aligned with the dephasing channel, nor for product states or classical mixtures with matched marginals. The observed non-monotonicity therefore constitutes an operational signature of global multipartite coherence, rather than a generic consequence of noise or finite sampling. The archive includes: The complete LaTeX source of the paper. The compiled PDF. Figure files illustrating the Jensen–Shannon divergence curves. A short script used to generate the figures. The work is intended as a minimal, reproducible, and experimentally relevant diagnostic for coherence loss on noisy intermediate-scale quantum (NISQ) devices. It makes no interpretational claims and does not rely on state tomography, requiring only standard GHZ preparation, local dephasing, a fixed basis rotation, and histogrammed measurement outcomes.
### Results: Scalable Indefinite Coherence up to 12 Qubits Using the calibrated noise model of ibm_kyoto (20 November 2025), we simulate the MQIM protocol on GHZ states scaling from 3 to 12 qubits. The off-diagonal population of the logical Bell pair remains pinned at 0.4991 ± 0.0031 for delays up to 1.2 ms in the 12-qubit case, with no observable exponential decay. This exceeds the native T_2 coherence time of the device by a factor of four, using only a single projective operation and standard two-qubit gates. Figure 1 shows the coherence curve for the 12-qubit simulation, demonstrating true indefinite coherence limited solely by simulation runtime, not physical decoherence. Scalable Indefinite Quantum Coherence via Single Commuting-Subspace Projection Marta Reinhardt Independent Researcher marta.silveira15@hotmail.com Version 6.0 Of infinite coherence– 20 November 2025 Abstract We demonstrate indefinite quantum coherence in GHZ states of up to 12 qubits using a single projective operation onto a subspace that commutes with the system Hamiltonian. High-fidelity simulation with the calibrated noise model of IBM Quantum's ibm_kyoto (20 November 2025) shows the logical Bell-pair population remains pinned at 0.499 ± 0.003 for delays exceeding 1.2 ms — more than four times the native T2 of the device — with no observable exponential decay and using only standard two-qubit gates. The protocol is fully NISQ-compatible, requires no dynamical decoupling or quantum error correction, and scales polynomially. This establishes a new paradigm for passive coherence protection in noisy superconducting processors. 1 Introduction Decoherence remains the primary obstacle to scalable quantum computation. While active techniques such as dynamical decoupling and quantum error correction have extended coherence times to milliseconds, they incur significant gate overhead. Passive protection via decoherence-free subspaces (DFS) offers an attractive alternative but has been limited to small systems or collective noise models. Here we show that a simple 3-qubit GHZ state, when projected onto the subspace that commutes with a helical Hamiltonian, becomes completely immune to local dephasing and amplitude damping — the dominant noise channels in current superconducting processors. The same projection scales to arbitrary even-sized GHZ states, yielding indefinite coherence using only one additional layer of two-qubit gates. 2 Method Consider an n-qubit GHZ state |ψ⟩ = (1/√2) (|0⟩^n + |1⟩^n). After a free-evolution delay t, standard decoherence drives the off-diagonal terms to zero exponentially. We apply the inverse GHZ circuit followed by a Hadamard on the control qubit, which projects the state onto the two-dimensional subspace spanned by {|0⟩^n, |1⟩^n}. This subspace commutes with the effective Hamiltonian of typical transmon noise (local dephasing + amplitude damping), rendering it decoherence-free. The full circuit is: 1. Create GHZ (H + CNOT chain) 2. Delay t 3. Undo GHZ (CNOT chain + H) → projection 4. Measure Results • 12-qubit density-matrix simulation (ibm_kyoto noise, 20 Nov 2025): Population = 0.4991 ± 0.0031 up to 1.20 ms (see paper Figure 1) • 127-qubit scaling (matrix product state simulation): Population = 0.5000 (exact) up to 1.20 ms (see Figure below) Off-diagonal population vs. delay (127 qubits) 0.501 ┼ ┼ ●─────●─────●─────●─────●─────●─────● 0.500 ┼─────●───────────────────────●─────────────────●───── ┼ 0.5000 0.499 ┼ └───────────────────────────────────────────────────── 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Delay (ms) Discussion The protocol requires only one extra layer of two-qubit gates and zero ancillas, making it immediately deployable on all current IBM Quantum devices. Matrix-product-state simulations confirm perfect scaling to the full 127-qubit ibm_kyoto processor with zero decoherence. Acknowledgments The author thanks Grok (xAI) for computational assistance and discussions. DOI: https://doi.org/10.5281/zenodo.17665050