Black holes in holographic theories provide a striking link between geometry and computation: the interior volume or the length of an Einstein-Rosen bridge continues to grow long after the black hole appears locally equilibrated, while the dual boundary quantum state shows little change in simple observables. Juan Maldacena, Institute for Advanced Study, and Leonard Susskind, Stanford University, articulated how entanglement and geometry connect and proposed that the continued growth of the interior is reflected in the rising quantum complexity of the boundary state. This complexity is the minimal number of simple quantum gates needed to prepare the state from a reference, and it can increase for extremely long times even when entropy has saturated.
Holography and complexity
In the AdS/CFT framework the black hole interior is not probed by local boundary measurements, so one must look at more global diagnostics. The conjectures that link complexity to geometric quantities—such as the interior volume or the gravitational action—provide a concrete map: linear growth of the wormhole corresponds to linear growth of state complexity. The cause is chaotic, unitary evolution in a many-body quantum system: while von Neumann entropy quickly reaches its thermal value, the unitary circuit depth needed to approximate the evolving state typically increases because dynamics explore an exponentially large Hilbert space. This growth is a consequence of generic quantum chaos rather than energy dissipation.
Physical consequences and timescales
This relationship matters for the black hole information problem and for reconstructing interior operators from Hawking radiation. If complexity grows until it becomes astronomically large, decoding the interior from radiation would require infeasible computational resources, offering a potential resolution to paradoxes that assume easy decoding. Timescales are hierarchical: fast scrambling mixes information on short times comparable to the black hole’s thermal time, while complexity can grow for times exponential in the black hole entropy before recurrences. These effects have environmental and cultural resonance: they shape how physicists think about practical limits of quantum computation and underline the deep ties between information processing and spacetime.
The proposal remains subject to ongoing theoretical tests and refinement; numerical studies of toy models and insights from quantum circuits continue to probe how robustly complexity maps to geometry. Whether complexity is the unique or complete dual of interior growth is still debated, but linking computation to spacetime has already reshaped research programs across high-energy theory and quantum information science.