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All stabilizer codes and also all non stabilizer codes that I am aware of, for example the ones here,

Example non-stabilizer code?

have a basis of codewords which are all uniform modulus superpositions of computational basis kets, in the sense that every nonzero coefficient has modulus $$ 1/\sqrt{|S|} $$ where $ |S| $ is the size of the support of the codeword.

In the particular case of stabilizer codes every nonzero coefficient has modulus $$ 1/\sqrt{2^r} $$ for some fixed $ r \leq n-k $. For a reference see https://quantumcomputing.stackexchange.com/a/27573/19675

What is an example of a (necessarily non-stabilizer) code for which the code space is not spanned by codewords which are all uniform modulus superpositions?

So to reiterate I'm looking for a $ d=2 $ code for which the code space is not spanned by codewords which are all uniform modulus superpositions.

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Since we’re leaving stabilizer codes behind, let’s go even farther away: spin codes are one example.

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  • $\begingroup$ Do non-uniform superpositions of (stabilizer) codewords count? Because then we can go back to stabilizer codes. To be a bit cheeky, a single unencoded qubit is a stabilizer code (with stabilizer group $\{I\}$) but clearly has code states that aren't "uniform". $\endgroup$
    – squiggles
    Commented Feb 16, 2023 at 2:15
  • $\begingroup$ Sure but the code space is still spanned by uniform codewords because you can pick $ |0> $ and $ |1> $. Also that trivial code does not have $ d \geq 2 $. $\endgroup$ Commented Feb 16, 2023 at 3:19
  • $\begingroup$ related papers explicitly demonstrating spin codes corresponding to multiqubit codes with nontrivial distance: arxiv.org/abs/2304.08611 and arxiv.org/abs/2305.07023 $\endgroup$ Commented Jul 26, 2023 at 21:31

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