docs: rewrite getting_started.md for presentation-ready format (#74)

* docs: polish documentation and expand API reference

- Update getting_started.md: fix "Next Steps" to reflect BP+OSD is available
- Expand api_reference.md with comprehensive API documentation:
  - Add DEM functions (extract_dem, build_parity_check_matrix, build_decoding_uai)
  - Add BatchBPDecoder and BatchOSDDecoder documentation
  - Add Syndrome Database functions
  - Add Circuit Generation functions
  - Organize into logical sections

* docs: rewrite getting_started.md for presentation-ready format

Restructure the getting started guide into 5 clear sections designed
for a ~15 minute presentation:

1. The Decoding Problem - circuit-level noise, detection events
2. Pipeline Overview - Mermaid flowchart, DEM as rulebook
3. BP Algorithm & Why It Fails - factor graph, degeneracy problem
4. OSD Post-Processing - OSD-0 algorithm, soft-weighted cost
5. Live Demo - minimal code example, threshold results

Key changes:
- Reduced from 493 to 280 lines
- Added 3 Mermaid flowcharts for visual understanding
- Condensed code examples to essential snippets
- Corrected threshold values (BP+OSD ~0.7%, not ~10%)
- Moved detailed CLI options to usage_guide.md
- Removed troubleshooting and advanced usage sections

* docs: update poster and lecture notes

* compile poster

Author: ChanceSiyuan

docs/api_reference.md

@@ -1,6 +1,76 @@
-## PyTorch BP API Reference
+## API Reference
 
-This reference documents the public API exported from `bpdecoderplus.pytorch_bp`.
+This reference documents the public API for BPDecoderPlus.
+
+---
+
+## Detector Error Model (DEM)
+
+Functions for working with Stim detector error models.
+
+### DEM Extraction
+
+- `extract_dem(circuit, decompose_errors=True) -> stim.DetectorErrorModel`
+  Extract DEM from a Stim circuit.
+
+- `load_dem(path) -> stim.DetectorErrorModel`
+  Load a DEM from file.
+
+- `save_dem(dem, path)`
+  Save a DEM to file.
+
+### Parity Check Matrix
+
+- `build_parity_check_matrix(dem, split_by_separator=True, merge_hyperedges=True) -> (H, priors, obs_flip)`
+  Build parity check matrix from DEM.
+  - `H`: Binary matrix of shape `(n_detectors, n_errors)`
+  - `priors`: Error probabilities of shape `(n_errors,)`
+  - `obs_flip`: Observable flip indicators of shape `(n_errors,)`
+
+- `build_decoding_uai(H, priors, syndrome) -> str`
+  Build UAI model string for MAP decoding from parity check matrix and syndrome.
+
+---
+
+## Batch Decoders
+
+High-performance batch decoding for multiple syndromes.
+
+### BatchBPDecoder
+
+```python
+from bpdecoderplus.batch_bp import BatchBPDecoder
+
+decoder = BatchBPDecoder(H, priors, device='cpu')
+marginals = decoder.decode(syndromes, max_iter=30, damping=0.2)
+```
+
+- `BatchBPDecoder(H, priors, device='cpu')`
+  Initialize batch BP decoder with parity check matrix and priors.
+
+- `decode(syndromes, max_iter=100, damping=0.2) -> torch.Tensor`
+  Decode batch of syndromes, returns marginal probabilities.
+
+### BatchOSDDecoder
+
+```python
+from bpdecoderplus.batch_osd import BatchOSDDecoder
+
+osd_decoder = BatchOSDDecoder(H, device='cpu')
+solution = osd_decoder.solve(syndrome, marginals, osd_order=10)
+```
+
+- `BatchOSDDecoder(H, device='cpu')`
+  Initialize OSD decoder with parity check matrix.
+
+- `solve(syndrome, marginals, osd_order=10) -> np.ndarray`
+  Find error pattern using Ordered Statistics Decoding.
+
+---
+
+## PyTorch BP (Low-level API)
+
+Low-level belief propagation implementation for factor graphs.
 
 ### UAI Parsing
 
@@ -43,3 +113,30 @@ This reference documents the public API exported from `bpdecoderplus.pytorch_bp`
 
 - `apply_evidence(bp, evidence: Dict[int, int]) -> BeliefPropagation`
   Return a new BP object with evidence applied to factor tensors.
+
+---
+
+## Syndrome Database
+
+Functions for generating and loading syndrome datasets.
+
+- `sample_syndromes(circuit, num_shots, include_observables=True) -> (syndromes, observables)`
+  Sample syndromes from a circuit.
+
+- `save_syndrome_database(syndromes, observables, path, metadata=None)`
+  Save syndrome database to `.npz` file.
+
+- `load_syndrome_database(path) -> (syndromes, observables, metadata)`
+  Load syndrome database from `.npz` file.
+
+---
+
+## Circuit Generation
+
+Functions for generating surface code circuits.
+
+- `generate_circuit(distance, rounds, p, task='z') -> stim.Circuit`
+  Generate a rotated surface code memory circuit.
+
+- `generate_filename(distance, rounds, p, task) -> str`
+  Generate standardized filename for circuit parameters.