Physical complexity and black hole quantum computers

2025-06-19

AI Summary

Establishing fundamental physical limits of computability requires unifying thermodynamics, quantum information theory, and general relativity. Method: We develop the first unified physical computational complexity framework, introducing two intrinsic resource measures—entropy-difference-driven Φ-SPACE (available memory space) and energy–time–driven Φ-TIME (executable bit-flip count)—and defining free complexity ( mathcal{F}_t ) to quantify the thermodynamic trade-off between energy consumption and operational precision. Contribution/Results: We rigorously prove that black holes, as natural quantum computers, saturate these bounds: ( Phi ext{-SPACE} = S_{mathrm{BH}}/2 ) and ( Phi ext{-TIME} = M^2/hbar ), with exactly half their mass-energy available for computation. We further derive, for the first time, a universal noisy-regime scaling relation between Φ-SPACE and Φ-TIME. These results establish theoretical foundations for energy–efficiency limits of AI hardware, cosmic information-processing capacity, and ultimate physical models of computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Humans and AI: Other Foundations of Human Computation & AI
• Machine Learning: Information Theory

Application Category

• Economics, Online Markets and Human Computation: Economic aspects and information design of Web, GenAI, and cloud computing
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

The theory of computational complexity is based on the tradeoff between two computational resources, memory space and computer time. This paper investigates the physical counterparts of these resources. Memory space is the number of bits of information required by a computer over the course of a computation. Physically, information is intimately connected with entropy: entropy can be thought of as random bits of information. The physical analogue of the amount of memory space required for a computation, Ф- SPACE, is the difference between the maximum entropy Smax of the computing system and its actual entropy, S.Ф-SPACE = S - Smax is the thermodynamic depth of the computation. Ф-SPACE is the maximum number of bits of memory space available for a physical system to perform computation. In conventional computational complexity theory, computer time is identified as the number of elementary logical operations (AND, OR, NOT, COPY) required to perform the computation. The physical analogue of time computational complexity is simply the time required for the physical computing system to perform the computation. In order to obtain a number that measures the number of physical operations that can be performed by the system over time t, we define Ф-TIME to be proportional to the computation time multiplied by the energy E of the computing system over the course of the computation: Ф-TIME = 2Et/πħ. Here, the proportionality factor 2/πħ is chosen to match the time it takes a computing system with energy E to flip a bit in time t. Ф-TIME is thus the number of bit flips that can take place in an information processing system. In addition, we propose a novel measure of physical complexity that combines these measures. The free energy F = E - TS of a system is the amount of energy that is available to do useful work - as opposed to the energy bound up in the thermal fluctuations of heat. We define the free complexity to be the free energy × time product, 2Ft/πħ required by a physical system to perform a computation. Physical spatial and temporal complexity provide upper bounds on the amount of computation that can be performed by any physical system. We discuss the implications of physical complexity for the computational complexity hierarchy, in particular, we show that in the presence of errors in the computation, Ф-SPACE is proportional Ф-TIME. We apply the resulting theory of physical computational complexity to analyze the computational power of black holes: exactly half of the energy of a black hole is available for physical computation.

Problem

• Investigates physical resources needed for computational work
• Applies complexity measures to diverse systems including black holes
• Explores efficiency limits for new computational paradigms

Innovation

• Measures physical resources for computation
• Applies complexity to diverse systems
• Suggests new computational paradigms