# ⚛  L1 Principle — Quantum Monte Carlo (VMC + DMC)

**ID:** `L1-321` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Quantum Mechanics — *Stochastic many-body electron structure*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** many body wavefunction
> **📡 Carrier:** electron · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554080

---

## 🧠 1. Introduction

**Quantum Monte Carlo (VMC + DMC)** is a **nonlinear inverse problem** whose unknown lives in **many body wavefunction** space, within the **Stochastic many-body electron structure** sub-domain of **Quantum Mechanics**.

Measurements consist of electrons collected by an electron detector via a **energy benchmark** sensing mechanism.

The forward operator applies, in order: E · trial wavefunction operator; int · stochastic operator; O · energies operator.

Observations are corrupted by Poisson counting noise. Stochastic error ~ 1/sqrt(N_walkers * N_steps); fixed-node systematic error.

## ⚙ 2. Forward Model

Physical chain: **x** → E · trial wavefunction → int · stochastic → O · energies → **y** (detector).

```
y = `O.energies` `int.stochastic` `E.trial_wavefunction` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.trial_wavefunction` | E · trial wavefunction operator |
| `int.stochastic` | Int · stochastic operator |
| `O.energies` | O · energies operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Quantum Mechanics |
| Sub domain | Stochastic many-body electron structure |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | many_body_wavefunction |
| Noise model | poisson |
| Integration axis | imaginary_time |
| Difficulty delta | 10 |
| L dag | 3.8 |

## 📡 4. Measurement Model

Stochastic error ~ 1/sqrt(N_walkers * N_steps); fixed-node systematic error.

| Metric | Value |
|---|---|
| Metric | energy_error_mHa |
| Secondary | density_L2_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `quantum_monte_carlo` · **Forward operator:** `qmc_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Method | — | DMC |
| N steps | — | 100000 |
| N walkers | — | 1000 |
| N electrons | — | 20 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Method | — | VMC, DMC, AFQMC |
| N steps | — | 1000 – 100000000.0 |
| N walkers | — | 100 – 1000000 |
| N electrons | — | 2 – 1000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** E error <= 0.5 mHa

| Metric | Range |
|---|---|
| Energy error mha | 0.1 – 50 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **energy_error_mHa**, with κ = `1500` and δ = `10`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xaa13a8fdf1754fc484d9ba3afe9e0466a5bd877eb82dc17fdd2024344c6f2d80`
- **Chain tx hash:** `0xc44ddf8bae284e89c21786da1c8448328f8d46ea0aee9be4ee173b6af3d43c8a`
- **Chain block:** `41554080`

---

## File Mapping

This bundle consists of: `L1-321.md`, `L1-321.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-321.md` | Source of truth — edit this | Human or LLM |
| `L1-321.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 `L1-321`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._