Approximate Quantum Codes From Long Wormholes

Abstract

We discuss families of approximate quantum error correcting codes which arise as the nearly-degenerate ground states of certain quantum many-body Hamiltonians composed of non-commuting terms. For exact codes, the conditions for error correction can be formulated in terms of the vanishing of a two-sided mutual information in a low-temperature thermofield double state. We consider a notion of distance for approximate codes obtained by demanding that this mutual information instead be small, and we evaluate this mutual information for the SYK model and for a family of low-rank SYK models. After an extrapolation to nearly zero temperature, we find that both kinds of models produce fermionic codes with constant rate as the number, $N$, of fermions goes to infinity. For SYK, the distance scales as $N^{1/2}$, and for low-rank SYK, the distance can be arbitrarily close to linear scaling, e.g. $N^{.99}$, while maintaining a constant rate. We also consider an analog of the no low-energy trivial states property which we dub the no low-energy adiabatically accessible states property and show that these models do have low-energy states that can be prepared adiabatically in a time that does not scale with system size $N$. We discuss a holographic model of these codes in which the large code distance is a consequence of the emergence of a long wormhole geometry in a simple model of quantum gravity.

81 pages, 22 figures, revised and substantially expanded version to be published in Quantum, main results are unchanged and new results added

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• Non-Stabilizerness of Sachdev-Ye-Kitaev Model
• Error Threshold of SYK Codes from Strong-to-Weak Parity Symmetry Breaking