Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains
Published 2017-10-12Version 1
Quantum error correction was invented to allow for fault-tolerant quantum computation. Topological ordered systems turned out to give a natural physical realization of quantum error correcting codes (QECC) in their grounspaces. More recently, in the context of the AdS/CFT correspondence, it has been argued that eigenstates of CFTs with a holographic dual should also form QECCs. These two examples lead to the question of how generally do eigenstates of many-body models form quantum codes. In this work we establish new connections between quantum chaos and translation-invariance in many-body spin systems, on one hand, and approximate quantum error correcting codes (AQECC), on the other hand. We first observe that quantum chaotic systems exhibiting the Eigenstate Thermalization Hypothesis (ETH) have eigenstates forming quantum error-correcting codes. Then we show that AQECC can be obtained probabilistically from the spectrum of every translation-invariant spin chains, even for integrable models, by taking translation-invariant energy eigenstates. Applying this result to 1D classical systems, we show that local symmetries can be used to construct parent Hamiltonians which embed these codes into the low-energy subspace of gapless 1D spin chains. As explicit examples we obtain local AQECC in the ground space of the 1D ferromagnetic Heisenberg model and the Motzkin spin chain model with periodic boundary conditions, thereby yielding non-stabilizer codes in the ground space and low energy subspace of physically plausible 1D gapless models.