2D Local implementation of Bivariate Bicycle qLDPC codes: Toric Layouts

[Fe-r-oz]

This PR implements the 2D implementation of Bivariate Bicycle Codes via Toric Layout and Embeddings in ℤ² from the paper: https://arxiv.org/pdf/2404.17676. All these Toric layouts has been tested and cross checked with the original implementation. Some details are presented here: #352 (comment). The original python implementation is quite slow and causes many allocations. I have used sparse arrays and optimized using sparse matrices which causes almost 0 allocations and speed-up in run-time as well.

I think these low-hanging fruits from the paper will be useful in circuit simulation of these codes, but I have not familiar with those part of the paper yet.

Toric Layouts: (Lemma 4, Page 16)
The paper has not tried plotting these Toric Layouts. They only plotted the Embedding in ℤ² (the second graph). But I think, we can also plot the Toric Layouts as following:
Code instance: BivariateBicycle(15,3 ,[9 ,1 ,2] , [0 ,2 ,7])

Code instance: BivariateBicycle(12,12, [3 ,2 ,7] ,[3 ,1 ,2 ])

Embeddings in ℤ²
Towards the end, we will be able to Plot the Embeddings of these codes in 2D. Probably, this will be divided into smaller PR so each PR is well documented and tested separately.

• The code is properly formatted and commented.
• Substantial new functionality is documented within the docs.
• All new functionality is tested.
• All of the automated tests on github pass.

[codecov]

Codecov Report

Attention: Patch coverage is 98.43750% with 1 line in your changes missing coverage. Please review.

Project coverage is 83.25%. Comparing base (2b8e81f) to head (abb66ae).

Files with missing lines	Patch %	Lines
src/ecc/bivariate_toric_layout.jl	98.43%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #387      +/-   ##
==========================================
+ Coverage   83.03%   83.25%   +0.22%     
==========================================
  Files          70       71       +1     
  Lines        4404     4468      +64     
==========================================
+ Hits         3657     3720      +63     
- Misses        747      748       +1     

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[Fe-r-oz]

The PR is ready for review, @Krastanov. Thank you.