[[2ᵐ - 1, 1, 3]] quantum Reed-Muller code

[Fe-r-oz]

Initial Results

• [[15, 1 ,3]] color code, aka QuantumReedMuller code by QRM(4)
• Steane7() is actually the QRM(3) - smallest QuantumReedMuller code
• paper followed: code conversion between Quantum Reed Muller code and Steane Code
• ECC Zoo Entry: https://errorcorrectionzoo.org/c/quantum_reed_muller

goal: implement most things from the paper as part of testing. I am waiting for RecursiveReedMuller to be reviewed so that I can use it to built these codes.

julia> parity_checks(QuantumReedMuller(4))
+ X_X_X_X_X_X_X_X
+ _XX__XX__XX__XX
+ ___XXXX____XXXX
+ _______XXXXXXXX
+ Z___Z___Z___Z__
+ _Z___Z___Z___Z_
+ __Z___Z___Z___Z
+ ___ZZZZ____ZZZZ
+ ____Z_Z_____Z_Z
+ _____ZZ______ZZ
+ _______ZZZZZZZZ
+ ________Z_Z_Z_Z
+ _________ZZ__ZZ
+ ___________ZZZZ

julia> parity_checks(Steane7())
+ ___XXXX
+ _XX__XX
+ X_X_X_X
+ ___ZZZZ
+ _ZZ__ZZ
+ Z_Z_Z_Z

julia> parity_checks(QuantumReedMuller(3))
+ X_X_X_X
+ _XX__XX
+ ___XXXX
+ Z_Z_Z_Z
+ _ZZ__ZZ
+ ___ZZZZ

julia> canonicalize!(parity_checks(Steane7())) ==  parity_checks(QuantumReedMuller(3))
true

Construction Method Followed from the paper, It looks like

julia> RM13 = generator(RecursiveReedMuller(1, 3));

julia> RM23 = generator(RecursiveReedMuller(3 - 2, 3));

julia> QRM_steane = CSS(RM13[2:end, 2:end], RM23[2:end, 2:end])
CSS(Bool[1 0 … 0 1; 0 1 … 1 1; 0 0 … 1 1], Bool[1 0 … 0 1; 0 1 … 1 1; 0 0 … 1 1])

julia> Steane = Stabilizer(QRM_steane)
+ X_X_X_X
+ _XX__XX
+ ___XXXX
+ Z_Z_Z_Z
+ _ZZ__ZZ
+ ___ZZZZ

[codecov]

Codecov Report

Attention: Patch coverage is 76.92308% with 9 lines in your changes missing coverage. Please review.

Project coverage is 82.79%. Comparing base (dddaedb) to head (65f8a2c).
Report is 8 commits behind head on master.

Files	Patch %	Lines
src/ecc/codes/classical/recursivereedmuller.jl	65.21%	8 Missing ⚠️
src/ecc/codes/quantumreedmuller.jl	93.75%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #302      +/-   ##
==========================================
- Coverage   82.85%   82.79%   -0.06%     
==========================================
  Files          60       62       +2     
  Lines        3971     4010      +39     
==========================================
+ Hits         3290     3320      +30     
- Misses        681      690       +9     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[Fe-r-oz]

PyMatching Decoder should not be used for QRM codes.

Please check out the decoder graphs for these codes!

BeliefPropDecoder

Table Decoder

More Samples

[Fe-r-oz]

The PR is now ready for review, Thank you!

[Krastanov]

This looks like a great addition and pretty close to ready for merge (hopefully the test change I made did not break anything -- these were meant as stylistic changes mostly).

I think there is only one thing left to do. You showed that the belief propagation decoder seems to work well for this code. Let's add it to the list of codes that are actually tested with the belief propagation decoder.

[Krastanov]

ah, good catch

[Fe-r-oz]

BitFlipDecoder gives better results than BeliefPropDecoder. There were a lot of assertion errors when comparing max(e...) < noise/4 for BeliefPropDecoder so instead I added BitFlipDecoder that gives better results. Hope this is satisfactory, I this is ready for review. Thank you!

julia> mem_errors = 0.001:0.005:0.1;
julia> codes = [QuantumReedMuller(3), QuantumReedMuller(4), QuantumReedMuller(5)];
julia> results = zeros(length(codes), length(mem_errors), 2);
julia> for (ic, c) in pairs(codes)
           for (i,m) in pairs(mem_errors)
               setup = ShorSyndromeECCSetup(m, 0)
               decoder = BitFlipDecoder(c)
               r = evaluate_decoder(decoder, setup, 1000)
               results[ic,i,:] .= r
           end
       end
julia> make_decoder_figure(mem_errors, results, "QRM(3), QRM(4), QRM(5) code with a BitFlipDecoder decoder")
julia> for (ic, c) in pairs(codes)
           for (i,m) in pairs(mem_errors)
               setup = ShorSyndromeECCSetup(m, 0)
               decoder = BeliefPropDecoder(c)
               r = evaluate_decoder(decoder, setup, 1000)
               results[ic,i,:] .= r
           end
       end
julia> make_decoder_figure(mem_errors, results, "QRM(3), QRM(4), QRM(5) code with a BeliefPropDecoder decoder")

[Fe-r-oz]

Thanks for polishing the PR. That is very much appreciated!