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.

中文翻译：

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来自长虫洞的近似量子代码


我们讨论了近似量子纠错码族，它们是由非交换项组成的某些量子多体哈密顿量的近简并基态而出现的。对于精确码，纠错条件可以根据低温热场双态中双边互信息的消失来表述。我们考虑通过要求该互信息很小而获得的近似代码的距离概念，并且我们评估 SYK 模型和一系列低秩 SYK 模型的互信息。外推到接近零的温度后，我们发现当费米子的数量 $N$ 趋于无穷大时，两种模型都会以恒定的速率产生费米子码。对于SYK，距离缩放为$N^{1/2}$，而对于低秩SYK，距离可以任意接近线性缩放，例如$N^{.99}$，同时保持恒定速率。我们还考虑了无低能平凡态属性的类比，我们将其称为无低能绝热可及状态属性，并表明这些模型确实具有可以在不随时间缩放的时间内绝热准备的低能态。系统大小$N$。我们讨论这些代码的全息模型，其中大代码距离是简单量子引力模型中出现长虫洞几何形状的结果。