[[k + 4, k, 2] quantum code (H Code)

[Fe-r-oz]

Implemented [[k + 4, k, 2]] CSS quantum codes, also known as HCode introduced by Cody Jones in his 2013 Paper: Multilevel distillation of magic states for quantum computing

[Fe-r-oz]

Please consider reviewing, @Krastanov. Tested instances via ecc_base. ECC Zoo has an entry about HCodes! Thanks!

[Fe-r-oz]

Decoder gives max(e...) < noise / 4 even for small instances, k =2 and k= 4. Turning it into Draft. When I tested locally, the assertion was not happening.

julia> parity_checks(HCode(2))
+ XXXX__
+ XX__XX
+ ZZZZ__
+ ZZ__ZZ

julia> parity_checks(HCode(4))
+ XXXX____
+ XX__XXXX
+ ZZZZ____
+ ZZ__ZZZZ

[Fe-r-oz]

julia> parity_checks(HCode(2))
+ XXXX__
+ XX__XX
+ ZZZZ__
+ ZZ__ZZ

X1X2X3X4II
X1X2IIX5X6
Z1Z2Z3Z4II
Z1Z2IIZ5Z6

Authors: Our distillation circuit is based on a family of concatenated quantum codes (H code) that possess a transversal Hadamard operation, enabling each of these codes to distill the eigenstate of the Hadamard operator.

Checking graphs for the smallest HCode whose Stabilizer is shown above. What do these graphs tells us about their decoding performance? There is no error in the code_properties though. But it raises then question, why is their decoding performance like this?