Kolmogorov complexity growth of black holes (Brown–Susskind conjecture)

Establish that, in the Brown–Susskind model where a black hole is associated with the group SU(n), the space is partitioned into cells labeled by an index c with Kolmogorov complexity K(c), and dynamics proceed as a random walk on the graph G generated by simple gate applications (U2 = gU1), the Kolmogorov complexity of the visited cell increases linearly for an exponential amount of time before reaching a maximum and thereafter fluctuates continuously.

Background

Brown and Susskind introduce a model assigning black holes to elements of SU(n), partitioning that space into cells, each labeled by an index c whose Kolmogorov complexity K(c) measures the complexity of the corresponding unitary transformations. The cells form vertices of a graph G, with edges induced by simple gate actions, and a black hole’s evolution is modeled as a random walk on G.

They conjectured that black hole complexity grows linearly for an exponential time, saturates, and then fluctuates, aligning with the Complexity/Volume Correspondence relating complexity to the Einstein–Rosen bridge volume. While this has been proven for quantum circuit complexity, the Kolmogorov complexity version is explicitly stated to remain open. The paper also notes partial progress: if G is an expander, linear growth of Kolmogorov complexity follows.