A lower bound on the overhead of addressable surgery on sparse stabiliser codes

Abstract

We show that, for stabilisers codes, there always exists a logical subspace requiring $\tilde{\Omega }(k^2)$ ancillary qubits to be measured with a sparse surgery ancilla. For CSS codes the hard subspace can be taken to be generated by all-X, or all-Z operators. As a result, for codes with large $k$, constant overhead ancillas can only realise a small fraction of possible measurements.