[[2ʲ, 2ʲ - 2j - 2, 4]] Gottesman code#282
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## master #282 +/- ##
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Thanks, @Krastanov, for providing clarification in #287! Gottesman applies H to the state to get Family of Distance 4 codes. Previously, I was using which can now be changed |
[[2ʲ, 2ʲ - 2j - 2, 4]] Gottesman code
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| """The family of `[[2ʲ, 2ʲ - 2j - 2, 4]]` Gottesman codes, also known as a 'Class of Distance Four Codes', as described in [Gottesman's 1997 PhD thesis](@cite gottesman1997stabilizer) and in [gottesman1996class](@cite). | |||
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| The stabilizer generators of the original `[[2ʲ, 2ʲ - j - 2, 3]]` Gottesman codes are incorporated to create a set of generators for this distance-four code. The resulting stabilizer set, denoted by S, incorporates the following elements: The first two generators are the Pauli-`X` and Pauli-`Z` operators acting on all qubits, represented by `Mₓ` and `Mz`, respectively. The next `j` generators correspond to `M₁` through `Mⱼ`, which are directly inherited from the `[[2ʲ, 2ʲ - j - 2, 3]]` Gottesman code's stabilizers. This inclusion ensures that `S` retains the inherent distance-three property of the original Gottesman code. The final `j` generators are defined as `Nᵢ = RMᵢR`, where `i` ranges from `1` to `j`. Here, `R` signifies a Hadamard Rotation operation applied to all `2ʲ` qubits, and `Mᵢ` refers to one of the existing generators from the second set `(M₁ to Mⱼ)`. By incorporating the stabilizers of a distance-three code, the constructed set `S` inherently guarantees a minimum distance of three for the resulting distance-four Gottesman code. | |||
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A see also to Gottesman and an explanation about the different would be useful.


I found a small quantum code, aka a Distance 4 code from Gottesman 1997 thesis. This class of [[2ʲ, 2ʲ - j - 2, 3]]` Gottesman codes is derived from [[2ʲ, 2ʲ - 2j - 2, 4]] Gottesman codes . The method is taken from the his thesis:
Xand Pauli-Zoperators acting on all qubits, represented byMₓandMz, respectively.jgenerators correspond toM₁throughMⱼ, which are directly inherited from the[[2ʲ, 2ʲ - j - 2, 3]]Gottesman code's stabilizers. This inclusion ensures thatSretains the inherent distance-three property of the original Gottesman code.So, this is Gottesman(4) that we have in the library

jgenerators are defined asNᵢ = RMᵢR, whereiranges from1toj. Here,Rsignifies a Hadamard Rotation operation applied to all2ʲqubits, andMᵢrefers to one of the existing generators from the second set(M₁ to Mⱼ).Finally, Hadamard transform turns X operator to Z and Z operator to X, so for 1 to j Rows of (M₁ to Mⱼ)
