QuantumCliffordJuMPExt: compute the minimum distance of QLDPC using Mixed Integer Programming (MIP) via GNU Linear Programming Kit

[Fe-r-oz]

This PR implements the Mixed Integer programming (MIP) approach to compute the minimum distance of QLDPC using GNU's Linear Programming Kit. This method is used by Panteleev and Kalachev and by Bravyi et. al.. The latter reference mentioned that this MIP approach was originally developed in 2011 and used in this reference

This PR allows the users to compute the minimum distance with ease and also provide the necessary documentation along with the relevant modern usecases. The tests have been conducted in #435

ILP/MILP methods: https://quantumcomputing.stackexchange.com/questions/37289/compute-the-exact-minimum-distance-of-a-qecc-with-integer-linear-programming-met

P.S. Mixed Integer Programming (MIP) or mixed integer linear program refer to same thing in the literature, in this context.

Edit:

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 83.13%. Comparing base (5fcf353) to head (988751a).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #439      +/-   ##
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+ Coverage   83.00%   83.13%   +0.13%     
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  Files          71       72       +1     
  Lines        4565     4636      +71     
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+ Hits         3789     3854      +65     
- Misses        776      782       +6     

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[Fe-r-oz]

Please help review this PR, thank you :)

Thanks to GNU for open-souring GLPK. I have tested all the instances of 2bga codes using distance and added other tests for correctness. Many more tests are in #435. Referring to #353 as it's related a issue. This PR aims to provide the first reasonable step. This MIP method is used by Panteleev and Kalachev and by Bravyi et. al.. This MILP method that Bravyi used was also asked on stack exchange. The end goal is to be able to compute minimum distance of QECC/QLDPC codes with n being $10^3$ or $10^4$: QDistRnd or MaxSAT solvers or upcoming algorithms appear to be helpful for that. There is a lot more work to be done in this direction. I will be happy if this PR is one of my PRs you are most excited about, along with in-place Ted's Pauli measurement algorithm PR. :)

Possible Future TODOs include:

• More performance optimizations if possible
• comparison with distance solvers from stim.
  • It uses Microsoft's z3 SAT solver. This method formulate task the problem into maxSAT problem. Julia does have the z3.jlbinding! stim also uses pysat for maxSAT problem.
  • stim also uses pysat to convert distance-finding problem into a maxSAT problem
• comparison with QDistRnd Package from GAP: we can use Oscar to directly use QDistRnd package. User would have to provide the matrix is MTX format.
• integrate latest solvers: Grassl, Eric and others are working towards it and have some papers in this new directions
• implement minimum distance solvers in pure Julia
• Benchmark Lifted product codes Lifted and lifted product codes via Hecke's GroupAlgebra #356
• Benchmark 2BGA codes with finitely presented groups and with Direct product of General Groups, e.g. direct product of two finite groups using Oscar add QuantumCliffordOscarExt to provide convenient API for finitely presented groups via specific group presentation #400
• Benchmark Haah's cubic code using abelian group ℤ₃ˣ³ via Hecke's Group Algebra #388,
• Benchmark Coprime Bivariate Bicycle code via Hecke's Group Algebra #378
• Benchmark Multivariate Bicycle code via Hecke's Group Algebra #381
• Benchmark Implementing Bivariate Bicycle Codes using 2BGA as the parent via Hecke's Group Algebra #399
• Benchmark add tests and examples of Cₘ × C₂ 2BGA codes #392
• Benchmark Honeycomb color code via Hecke's Group Algebra #436
• Benchmark more examples of Bivariate Bicycle codes #441
• Benchmark Haah's cubic code via LP construction method #442
• Benchmark color codes with varying distances according to formula