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I will attempt to provide some insight regarding your first question. For starters, both quantum surface codes and quantum block codes are stabilizer codes, which means that although they are significantly different in terms of their construction and utility, they still share some common ground. With regard to which code family is more promising, I believe ...


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A general controlled unitary Let $CU$ denote the 'controlled' version of the $n$-qubit unitary $U$: \begin{equation} CU = |0\rangle\langle0|\otimes I_{t} + |1\rangle\langle1|\otimes U_{t}, \end{equation} where the operation acts on a Hilbert space $\mathcal{H}_{c}\otimes \mathcal{H}_{t}$, with $c$ denoting the control qubit and $t$ denoting the target ...


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A stabilizer code is also called an additive code, because it is closed under the sum of its elements. The namesake is described on page 33 of "Stabilizer Codes and Quantum Error Correction" (link). Additionally, additive quantum codes are the quantum version of additive codes found in coding theory.


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