A lower bound on the overhead of quantum error correction in low dimensions

Abstract

We show that a quantum architecture with an error correction procedure limited to geometrically local operations incurs an overhead that grows with the system size, even if arbitrary error-free classical computation is allowed. In particular, we prove that in order to operate a quantum error correcting code in 2D at a logical error rate of δ, a space overhead of Ω( log(1/δ)) is needed for any constant depolarizing noise p > 0.

Hierarchical memories: Simulating quantum LDPC codes with local gates

“…Finally, Baspin et al [BFS23] have recently generalized the result of Delfosse et al in another direction. In contrast to the constructive approach in this paper, they approach this problem top-down -given access to arbitrary local operations and classical communication (not merely Clifford operations), they study syndrome-extraction circuits for LDPC codes and their ability to suppress stochastic errors.…”

“…In contrast to the constructive approach in this paper, they approach this problem top-down -given access to arbitrary local operations and classical communication (not merely Clifford operations), they study syndrome-extraction circuits for LDPC codes and their ability to suppress stochastic errors. They prove the existence of a tradeoff between the parameters of the syndrome-extraction circuit and the subthreshold error scaling (See Theorem 28 of [BFS23]). For fixed gate error rate p, suppose we use an N, K, D code H N and desire a sub-threshold scaling of the logical failure rate p H (N ) = exp(−f (N )) for some function f (N ).…”

“…Let C N be the corresponding family of syndrome-extraction circuits. Assuming f (N ) = O(N ), we express Theorem 28 of [BFS23] in our notation…”