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In this work we establish lower bounds on the size of Clifford circuits thatmeasure a family of commuting Pauli operators. Our bounds depend on theinterplay between a pair of graphs: the Tanner graph of the set of measuredPauli operators, and the connectivity graph which represents the qubitconnections required to implement the circuit. For local-expander quantumcodes, which are promising for low-overhead quantum error correction, we provethat any syndrome extraction circuit implemented with local Clifford gates in a2D square patch of $N$ qubits has depth at least $\Omega(n/\sqrt{N})$ where $n$is the code length. Then, we propose two families of quantum circuitssaturating this bound. First, we construct 2D local syndrome extractioncircuits for quantum LDPC codes with bounded depth using only $O(n^2)$ ancillaqubits. Second, we design a family of 2D local syndrome extraction circuits forhypergraph product codes using $O(n)$ ancilla qubits with depth $O(\sqrt{n})$.Finally, we use circuit noise simulations to compare the performance of afamily of hypergraph product codes using this last family of 2D syndromeextraction circuits with a syndrome extraction circuit implemented with fullyconnected qubits. While there is a threshold of about $10^{-3}$ for a fullyconnected implementation, we observe no threshold for the 2D localimplementation despite simulating error rates of as low as $10^{-6}$. Thissuggests that quantum LDPC codes are impractical with 2D local quantumhardware. We believe that our proof technique is of independent interest andcould find other applications. Our bounds on circuit sizes are derived from alower bound on the amount of correlations between two subsets of qubits of thecircuit and an upper bound on the amount of correlations introduced by eachcircuit gate, which together provide a lower bound on the circuit size.

Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes