Quantum Error Detection at Low Energies

Motivated by the close relationship between error-correcting codes, topological order, holographic AdS/CFT duality and tensor networks, we initiate the study of approximate quantum error correcting codes in matrix product states (MPS). We first see that using open-boundary MPS to define boundary to bulk encoding maps only yields constant distance codes. These are degenerate ground spaces of gapped local Hamiltonians. Later, to get around this no-go result, we consider excited states of local gapped Hamiltonians, and low energy states of gapless Heisenberg-XXX model obtained by algebraic Bethe ansatz. All these codes protect against almost linear distance arbitrary (not necessarily geometrically local) errors even though they are not permutation-invariant. This suggests that a wide range of naturally occurring many-body systems possess intrinsic error-correcting features. This is a joint work with Martina Gschwedtner, Robert König and Eugene Tang, which can be found on arXiv:1902.02115v1 [quant-ph].