L1 ⚛ L1-324 ⊙ Testnet

Lindblad Master Equation (Open Quantum Systems)

Quantum Mechanics · Markovian open-system dynamics

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⬇ L1-324.md

# ⚛ L1 Principle — Lindblad Master Equation (Open Quantum Systems) **ID:** `L1-324` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Quantum Mechanics — *Markovian open-system dynamics* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** density matrix trajectory > **📡 Carrier:** photon · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554081 --- ## 🧠 1. Introduction **Lindblad Master Equation (Open Quantum Systems)** is a **nonlinear inverse problem** whose unknown lives in **density matrix trajectory** space, within the **Markovian open-system dynamics** sub-domain of **Quantum Mechanics**. Measurements consist of photons collected by an optical detector via a **time resolved observable** sensing mechanism. The forward operator applies, in order: E · hamiltonian operator; E · dissipators operator; time evolution of the state; O · observables operator. Observations are corrupted by additive Gaussian noise. Trace-preserving; completely positive; well-posed for finite-dim Hilbert space. ## ⚙ 2. Forward Model Physical chain: **x** → E · hamiltonian → E · dissipators → Time derivative → O · observables → **y** (detector). ``` y = `O.observables` ∂_t `E.dissipators` `E.hamiltonian` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.hamiltonian` | E · hamiltonian operator | | `E.dissipators` | E · dissipators operator | | `D.time` | Time evolution of the state | | `O.observables` | O · observables operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Quantum Mechanics | | Sub domain | Markovian open-system dynamics | | Carrier | photon | | Problem class | nonlinear_inverse | | Solution space | density_matrix_trajectory | | Noise model | gaussian | | Integration axis | time | | Difficulty delta | 5 | | L dag | 3.5 | ## 📡 4. Measurement Model Trace-preserving; completely positive; well-posed for finite-dim Hilbert space. | Metric | Value | |---|---| | Metric | T2_relative_error | | Secondary | fidelity_error | ## 📏 5. Operating Range (Ω) **Center problem class:** `lindblad` · **Forward operator:** `lindblad_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | N dim | — | 8 | | T end ns | ns | 100 | | Gamma mhz | MHz | 1 | | N dissipators | — | 2 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N dim | — | 2 – 1000 | | T end ns | ns | 0.1 – 1000000.0 | | Gamma mhz | MHz | 0.0001 – 10000.0 | | N dissipators | — | 1 – 50 | ## 🎯 6. Tolerance (ε) **Center tolerance:** T2 rel error <= 0.03 | Metric | Range | |---|---| | T2 relative error | 0.005 – 0.3 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **T2_relative_error**, with κ = `400` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x57b1521342c0595d5f01c5f0c31e3006c082abefec5dd25e4d4594c446ef7c75` - **Chain tx hash:** `0x7656b44f0120c50f418734ac900e2c0fb211417f6188cbc015860865fd14b91c` - **Chain block:** `41554081` --- ## File Mapping This bundle consists of: `L1-324.md`, `L1-324.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-324.md` | Source of truth — edit this | Human or LLM | | `L1-324.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-324`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-324.md).
Regenerate L1-324.json so every field matches the Markdown.
Preserve the schema documented in the File Mapping footer at the bottom of the MD.
Output each file in its own fenced code block tagged with the filename.

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L2-324-001 Lindblad Master Equation (Open Quantum Systems) Mismatch-Only Spec

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📋 JSON metadata

{
  "artifact_id": "L1-324",
  "chain_block": 41554081,
  "chain_hash": "0x57b1521342c0595d5f01c5f0c31e3006c082abefec5dd25e4d4594c446ef7c75",
  "chain_tx_hash": "0x7656b44f0120c50f418734ac900e2c0fb211417f6188cbc015860865fd14b91c",
  "domain": "Quantum Mechanics",
  "hardness_fn": {
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    "kappa": 400,
    "metric": "T2_relative_error",
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    "regime": "Trace-preserving; completely positive; well-posed for finite-dim Hilbert space.",
    "secondary": "fidelity_error"
  },
  "physics_fingerprint": {
    "L_DAG": 3.5,
    "carrier": "photon",
    "difficulty_delta": 5,
    "domain": "Quantum Mechanics",
    "integration_axis": "time",
    "noise_model": "gaussian",
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      "E.dissipators",
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      "O.observables"
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    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "time_resolved_observable",
    "solution_space": "density_matrix_trajectory",
    "sub_domain": "Markovian open-system dynamics",
    "title": "Lindblad Master Equation (Open Quantum Systems)"
  },
  "size_tiers": {
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    ],
    "allowed_omega_dimensions": [
      "N_dim",
      "T_end_ns",
      "gamma_MHz",
      "N_dissipators"
    ],
    "allowed_problem_classes": [
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    "center_spec": {
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      "forward_operator": "lindblad_forward",
      "input_format": "measurement_only",
      "omega": {
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        "T_end_ns": 100,
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    "epsilon_bounds": {
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        0.3
      ]
    },
    "omega_bounds": {
      "N_dim": [
        2,
        1000
      ],
      "N_dissipators": [
        1,
        50
      ],
      "T_end_ns": [
        0.1,
        1000000.0
      ],
      "gamma_MHz": [
        0.0001,
        10000.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Markovian open-system dynamics",
  "title": "Lindblad Master Equation (Open Quantum Systems)"
}

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