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The trivial stabilizer code is defined by $$T=\{|0\rangle^{\otimes(n-k)}\otimes|\Psi\rangle:|\Psi\rangle\in(\mathbb{C}^{2})^{k}\}\tag{1}$$ which is stabilized by the Pauli operators $Z_1, ...., Z_{n-k}$.

How can we prove Claim 2 on page 2 of this note?

It seems we need to apply the discussion on page 4, but how? In other words, if $g_1,...,g_{n-k}$ are the stabilizer generators of code $S$, we need to find a unitary operator $u$ such that $ug_iu^{\dagger}=Z_i$ for all $i$.

FDGod
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Star21
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  • $u$ is just the "encoder". To go from a set of stabilizers to encoder is not trivial but is well understood...it should be described in many places; look for "encoding circuit" or variations of that. – unknown Nov 13 '23 at 23:50
  • To uniquely define such a Clifford unitary $u$, you have to pick a logical basis for your code. Then, you can read off the binary/symplectic representation of $u$ and use that in standard compilation methods. See also https://quantumcomputing.stackexchange.com/questions/9534/stabilizer-circuit-synthesis-via-clifford-gates/15665#15665 – Markus Heinrich Nov 14 '23 at 07:50
  • Thank you very much! – Star21 Nov 19 '23 at 21:32

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