Three Lectures on Complexity and Black Holes

Abstract: Given at PiTP 2018 summer program entitled "From Qubits to Spacetime." The first lecture describes the meaning of quantum complexity, the analogy between entropy and complexity, and the second law of complexity. Lecture two reviews the connection between the second law of complexity and the interior of black holes. I discuss how firewalls are related to periods of non-increasing complexity which typically only occur after an exponentially long time. The final lecture is about the thermodynamics of complexity, and "uncomplexity" as a resource for doing computational work. I explain the remarkable power of "one clean qubit," in both computational terms and in space-time terms. The lectures can also be found online at \url{https://static.ias.edu/pitp/2018/node/1796.html} .

• The paper presents a framework linking quantum complexity with black hole interiors through analogies to entropy and a 'second law of complexity'.
• The lectures detail the use of inner product and relative complexity metrics to quantitatively relate complexity growth with black hole geometry.
• The discussion highlights 'uncomplexity' as a resource, emphasizing its role in advancing both quantum computation and our understanding of quantum gravity.

Complexity and Black Holes

Leonard Susskind's lectures on complexity and black holes provide an in-depth exploration of the intricate relationship between quantum complexity and black hole physics. These lectures, elaborated at the PiTP 2018 summer program, explore the conceptual analogies and the theoretical frameworks that tie quantum computational complexity to the structure and evolution of black holes. The discourse is structured in three parts, each addressing a specific facet of this profound connection.

In the first lecture, Susskind introduces the notion of quantum complexity, drawing an analogy with entropy and establishing the concept of a "second law of complexity." Here, quantum complexity is linked with the navigational challenges within the enormous space of quantum states. Two metrics are introduced to help conceptualize this space: the inner product metric and the relative complexity metric. Each has distinct applications and implications for understanding quantum states, particularly in the context of black holes. The second law of complexity asserts that complexity predominantly increases, akin to the second law of thermodynamics, with significant implications for quantum systems.

The second lecture transitions into the field of black holes, specifically engaging with how the second law of complexity interrelates with black hole interiors. Susskind discusses how the increase in complexity parallels the growth of the interior volume of a black hole, challenging the classical intuition that associates this growth with entropy. This lecture underscores the notion that while black holes might reach thermal equilibrium, their internal complexity continues to evolve, particularly driven by quantum gravity aspects that elude classical thermodynamic descriptions.

Lastly, the third lecture introduces the concept of "uncomplexity," juxtaposing it with negative entropy. Uncomplexity is conceived as a valuable resource, much like negentropy, and is instrumental for executing computational tasks. The lecture demonstrates the intriguing power of "one clean qubit" in both computational and space-time contexts, highlighting how even a seemingly negligible component can significantly alter computational capabilities and affect the geometric properties linked to black holes.

Throughout these lectures, Susskind refrains from making sensationalized claims about the groundbreaking nature of this exploration. Instead, the lectures systematically build on existing frameworks, bridging complexity theory and the physics of black holes. Quantitative aspects are frequently addressed, such as the rate of complexity growth being proportional to $ST$, which mirrors the expected rate of computation from theoretical predictions. These explicit connections substantiate the proposed complexity-geometry dualities.

The implications of Susskind’s lectures are profound. On a theoretical ground, they suggest that understanding complexity may be vital to deciphering spacetime structures within black holes, potentially offering insights into quantum gravity. Practically, it hints at novel approaches for handling quantum information and computation, especially within the limits defined by black hole physics. Looking toward the future, these insights raise numerous questions, especially regarding the application of these principles beyond specific contexts like AdS/CFT and towards more general spacetime theories.

In conclusion, Susskind’s exploration deepens the conceptual ties between quantum computational complexity and black hole physics, proposing complexity as a central pillar in understanding black hole interiors and, by extension, the underlying fabric of quantum gravity. The lectures invite continued exploration and potential experimental inquiries into these complex interdependencies.