#quantum complexity

Computational complexity has become an intriguing concept in black hole physics, offering new insights into the nature of black holes and their interiors. In computational theory, complexity refers to the resources required to solve a problem, such as time or computational steps. When applied to quantum systems, it specifically looks at how many quantum operations (or gates) are needed to transform one quantum state into another. This concept becomes particularly interesting when examining black holes.

Black holes have long been a subject of fascination due to their mysterious nature and the information paradox. The paradox arises from the apparent loss of information when matter falls into a black hole, seemingly violating the principle of information conservation in quantum mechanics. Recent theoretical developments suggest that the complexity of a quantum state can be associated with the geometric properties of a black hole's interior. The idea is that as a black hole evolves, its interior volume grows, which can be thought of as an increase in the complexity of the quantum state representing the black hole.

This perspective provides a new way to think about what happens inside a black hole. Instead of being a place where nothing escapes, it becomes a region where processes are highly complex. This complexity might hold the key to understanding how information is stored and potentially retrieved from black holes. Traditionally, the event horizon of a black hole is seen as a boundary beyond which nothing can return. However, by considering computational complexity, this view is nuanced. The horizon acts more like a filter or censor that prevents an outside observer from easily accessing the information within. It's not that information cannot escape; rather, decoding or accessing it requires immense computational resources.

The introduction of complexity also touches on fundamental principles like the quantum extended Church-Turing Thesis, which posits that any physical process can be efficiently simulated by a quantum computer. In the context of black holes, this principle suggests that while information might not be lost, simulating or retrieving it is computationally prohibitive for an observer outside the horizon.

These insights have profound implications for our understanding of both black holes and fundamental physics. They suggest new ways to reconcile general relativity with quantum mechanics and provide potential pathways for resolving longstanding paradoxes. Research in this area is ongoing, with physicists exploring how these concepts can be tested or further developed through theoretical models and potentially even experimental evidence in quantum computing and gravitational physics.

Leonard Susskind: Quantum Complexity - Quantum PCP, Area Laws, and Quantum Gravity (Simons Institute, March 2024)