Published August 12, 2026
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Can Spacetime Be an Optimal Code? Gravity from Quantum Error Correction
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Description
If space behaves like an error-correcting code, it is tempting to go one step further and guess that the laws of gravity are just the statement that this code is the best one possible. That would be an appealing shortcut: the equations of general relativity would drop out of an optimization principle, the way a soap film settles into the shape of least area. This paper asks, carefully and in a setting where everything can be computed, whether that guess actually holds. The answer is a clean and instructive no. Optimizing a code does pick out a definite geometry, which is already a real result, but the geometry it picks is never the one general relativity describes.
The work is carried out in a controlled toy world, where a geometry is just a set of weighted connections on a network and a code's performance is a single number to be improved under a fixed budget. It first pins down why the target is so demanding: among all the local laws a geometry could obey, the one general relativity uses is singled out by a rigidity theorem, and that rigidity is forced by a deep symmetry, the freedom to relabel points without changing the physics. It then studies the two most natural ways of scoring how well a code protects information. One measures bottlenecks, how much can pass through the tightest cut; the other measures redundancy, how many independent paths hold things together. Both turn out to be well behaved enough that improving the score selects a single global geometry. Yet neither reproduces gravity. The bottleneck score only ever sees the same minimal-surface information that the holographic area law already captures, and yields no equation of motion at all. The redundancy score does yield a genuine field equation, but it is the equation of old-fashioned Newtonian gravity, the faint weak-field echo of general relativity rather than the full theory. The paper then documents a wide search for some other route to the full theory, sorting the attempts into easy, hard, and predictive, and explains with references why each one either lands in territory already worked over thoroughly or runs into the very same missing ingredient.
The central results are proved in full with standard tools from network theory, its finite algebraic steps are elementary and independently checkable, and its numerical experiments, with reproducible code, confirm the predicted behaviour.
The deeper lesson is about where the difficulty really lives. Making a code optimal is not enough to produce Einstein's gravity, because optimality on its own knows nothing about the relabeling symmetry that makes general relativity what it is, and no natural measure of a code's performance carries that symmetry inside it. What optimization does deliver is genuine but more modest: a uniquely selected geometry, and, in one case, Newtonian gravity. The paper is candid that this is a boundary result rather than a breakthrough, and it turns that candor into a method. It keeps an explicit ledger of which claims are proved, which are borrowed, and which rest only on the computer, and it treats the honest mapping of a crowded research frontier, including the routes that lead nowhere, as a result worth reporting in its own right.