https://github.com/mapleleaflatte03/qem-inverse-theory
Zero-noise extrapolation as a constrained quantum inverse problem.
https://github.com/mapleleaflatte03/qem-inverse-theory
Last synced: 23 days ago
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Zero-noise extrapolation as a constrained quantum inverse problem.
- Host: GitHub
- URL: https://github.com/mapleleaflatte03/qem-inverse-theory
- Owner: mapleleaflatte03
- License: mit
- Created: 2026-05-18T17:26:59.000Z (2 months ago)
- Default Branch: master
- Last Pushed: 2026-05-18T18:45:59.000Z (2 months ago)
- Last Synced: 2026-05-18T19:40:02.340Z (2 months ago)
- Language: Python
- Size: 341 KB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- Changelog: CHANGELOG.md
- License: LICENSE
- Roadmap: ROADMAP.md
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README
# qem-inverse-theory
**Zero-noise extrapolation as a constrained quantum inverse problem.**
We do not propose another extrapolation curve; we study when extrapolation is physically identifiable.
## Mission
This repository studies when quantum error mitigation is physically identifiable, statistically stable, and worth applying under finite-shot constraints.
## Motivation
Standard ZNE asks: "Which curve fits the noisy data best?"
This project asks: "Under which physical and statistical assumptions is the zero-noise limit identifiable from finite noisy observations?"
The distinction matters. Without physical constraints, ZNE is an ill-posed inverse problem: infinitely many curves pass through noisy data points but disagree at zero noise. Finite shots compound the problem by adding statistical uncertainty to already-ambiguous observations.
We study the mathematical structure of this problem: when constraints restore identifiability, when finite shots make mitigation harmful, and what structured assumptions allow escape from generic QEM lower bounds.
## How this differs from `quantum-noise-optimizer`
| | quantum-noise-optimizer | qem-inverse-theory |
|---|---|---|
| Goal | Production mitigation engine | Theoretical research |
| Output | Mitigated expectation values | Theorems, phase diagrams, risk bounds |
| Approach | AICc + bounded optimization | Inverse problem theory + Bayesian inference |
| Hardware | Wukong 180 calibration | Synthetic only |
| Claims | Empirical benchmark results | Mathematical conditions and conjectures |
## Research questions
1. **Identifiability**: Under what physical constraints is the zero-noise limit uniquely determined from finite noisy observations?
2. **Stability**: How does the estimation error amplify as a function of noise strength, model complexity, and shot budget?
3. **Help–harm boundary**: For a given circuit class and shot budget, when does ZNE reduce MSE versus amplify it?
4. **Bayesian calibration**: Can posterior credible intervals provide reliable coverage for the zero-noise estimate?
5. **Structured escape hatches**: Which problem structures (locality, symmetry, spectral gaps) allow mitigation to remain useful despite generic lower bounds?
## Mathematical framing
Let $\lambda_1 < \lambda_2 < \cdots < \lambda_n$ be noise scale factors and $y_i = f(\lambda_i) + \epsilon_i$ the observed expectations with shot noise $\epsilon_i \sim \mathcal{N}(0, \sigma_i^2)$.
ZNE seeks $f(0)$ subject to:
- **Physical admissibility**: $f(0) \in [\lambda_{\min}(O), \lambda_{\max}(O)]$ for observable $O$
- **Smoothness/structure**: $f$ belongs to some function class $\mathcal{F}$
- **Finite data**: only $n \leq 7$ observations available
This is a constrained inverse problem. We study its well-posedness in the sense of Hadamard: existence, uniqueness, and continuous dependence on data.
## Modules
- `constraints/` — Spectral bounds, probability simplex, physical admissibility
- `estimators/` — Unconstrained, constrained, regularized, Bayesian, model selection
- `theory/` — Inverse problem formulation, risk bounds, identifiability, phase diagrams
- `benchmarks/` — Synthetic response generators, shot noise, metrics
- `plots/` — Phase diagrams, risk frontiers, coverage plots
- `experiments/` — Runnable scripts demonstrating each research question
## Out of scope
- Hardware integration or SDK
- Production-ready mitigation pipeline
- MCP server or AI-agent tooling
- Claims of solving QEM lower bounds
- Peer-reviewed theorems (this is early-stage research)
## Roadmap
1. ✅ Scaffold: types, constraints, estimators, synthetic benchmarks
2. 🔜 Finite-shot phase diagrams for polynomial ZNE
3. 🔜 Bayesian GP-based ZNE with calibrated intervals
4. 🔜 Adversarial identifiability examples
5. 🔜 Theorem sketches → rigorous proofs
6. 🔜 Paper draft: "ZNE as a Constrained Quantum Inverse Problem"
## Status
Early research prototype. See [STATUS.md](STATUS.md).
## Checkpoint
Current: **v0.1-research-scaffold**
Verify with:
```bash
bash scripts/smoke_check.sh
```
Expected: 41 tests pass, all safety checks green.
## Current Paper Status
- LaTeX skeleton exists (`paper/main.tex`) with 7 sections, 2 proven propositions
- 4 figures reproducible via `python experiments/generate_figures.py`
- `refs.bib` scaffold with verified DOIs for key citations
- All results are synthetic — no hardware claims
- Not peer-reviewed; draft is work-in-progress
- Claim safety enforced by automated tests
## Install
```bash
pip install -e ".[dev]"
pytest
```
## License
MIT