There is a deep connection between quantum error correction and phases of matter for spatially local codes in finite dimensions. I will show how this analogy extends to more general settings: quantum codes with check soundness are absolutely stable phases of matter. These codes include constant-rate quantum low-density parity-check codes, which shows that the third law of thermodynamics is false: there exist absolutely stable phases of matter with constant entropy density at zero temperature. Our proof technique also establishes the robustness of the toric code phase to spatially nonlocal perturbations, and provides extremely strong bounds on the unitaries that rotate between two states near solvable points in the code phase. These latter bounds have some intriguing applications to condensed matter physics.
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