On the Addressability Problem on CSS Codes

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Recent discoveries in asymptotically good quantum codes have intensified research on their application in quantum computation and fault-tolerant operations.

• This study focuses on the addressability problem within CSS codes: we ask what circuits might implement logical gates on strict subsets of logical qubits.
• With some notion of fault-tolerance, we prove several impossibility results: for CSS codes with non-zero rate, one cannot address a logical $H$, $HS$, $SH$, or $\mathsf{CNOT}$ to any non-empty strict subset of logical qubits using a circuit made only from 1-local Clifford gates.
• Furthermore, we show that one cannot permute the logical qubits in a code purely by permuting the physical qubits, if the rate of the code is (asymptotically) greater than 1/3 and the distance is at least 3.