Add docs for realtime decoding

[kvmto]

Add comprehensive real-time decoding documentation

Overview

Real-time decoding is a critical feature for quantum error correction on actual quantum hardware, enabling decoders to process syndromes and compute corrections within qubit coherence times. Based on the content of 075193cff0a0a087d577acecfc3cc575230409a2.

Motivation

Users need:

1. Clear API references to understand the three core decoding functions and configuration API
2. End-to-end guides showing the complete workflow from DEM generation to hardware deployment
3. Decoder selection guidance to choose between single_error_lut, multi_error_lut, nv-qldpc, and sliding_window
4. Backend-specific setup instructions, especially for Quantinuum hardware where decoders run on remote servers
5. Troubleshooting help for common issues like high error rates, latency, and configuration upload failures

Without this documentation, users cannot effectively deploy real-time decoding on quantum hardware or understand the critical differences between simulation and hardware backends.

Changes

Documentation Structure:

• Split real-time decoding API into language-specific reference files for clarity
• Consolidated three component docs into single comprehensive guide at examples_rst/qec/realtime_decoding.rst
• Added real-time decoding section to QEC introduction with quick examples

API Documentation (api/qec/):

• Created cpp_realtime_decoding_api.rst and python_realtime_decoding_api.rst
• Documented core decoding functions (enqueue_syndromes, get_corrections, reset_decoder)
• Documented configuration API (configure_decoders, decoder_config structures)
• Added helper functions section with real-time decoding usage examples
• Used manual .. function:: directives for consistent formatting

Comprehensive Guide (examples_rst/qec/realtime_decoding.rst - 859 lines):

• Workflow Overview: 4-stage process with narrative explaining separation of characterization vs. runtime
• Configuration: Step-by-step DEM generation, decoder setup, YAML serialization, and kernel usage
• Decoder Types: Detailed descriptions with performance characteristics and parameter tuning guidance
• Backend Selection:
  • Simulation backend for local development
  • Quantinuum backend: Configuration upload to REST API, server-side execution, emulation vs. hardware
• Best Practices: Decoder selection table, configuration validation patterns
• Troubleshooting: High logical error rates, latency issues, memory errors, Quantinuum upload failures

Bug Fixes:

• Fixed literalinclude directives using [Begin Documentation] markers
• Fixed Python API title underline length
• Fixed indentation errors in nv_qldpc_decoder_api.rst
  and some more

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[kvmto]

/ok to test 1cf7616

[bmhowe23]

Just a few quick comments...

[melody-ren]

Thanks Kevin! I think this is a very thorough guide on what the realtime decoding API offers and how to use it. I do recommend to match the existing documentation's tone and reduce the excessive use of "you/your" throughout the draft.

[kvmto]

/ok to test 6648209

[bmhowe23]

/ok to test f9535fd

[bmhowe23]

@justinlietz - double check me, but I think some of the matrix dimensions might be swapped.

[melody-ren]

Approving now since the majority of my comments have been addressed. Thanks!

[bmhowe23]

/ok to test 1c040d1