2
$\begingroup$

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.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

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

Improve this answer
$\endgroup$
3
  • $\begingroup$ 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
    Feb 16 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 $. $\endgroup$
    – Ian Gershon Teixeira
    Feb 16 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 $\endgroup$
    – Ian Gershon Teixeira
    Jul 26 at 21:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.