Which Laws of Physics Can't Be Simulated on Any Kind of Computer?

The question whether there are physical laws that cannot be simulated on any kind of computer is both intriguing and complex. While modern high-performance computers can simulate various phenomena, such as atomic explosions, there are levels of complexity that currently elude our computational capabilities. This article explores the boundaries of computational physics and highlights key theories and principles that shed light on this question.

Understanding the Noncomputability of Physics

The concept of noncomputable physics owes a significant debt to Roger Penrose's work. In his books, Penrose discusses the limits of formal systems and their applicability to physical reality. He argues that certain physical phenomena may transcend the computational capacity of any digital computer or even of the human brain.

Examples and Current Limitations

Currently, scientists can simulate atomic explosions using ultra-high-level computers, but simulating stars proves challenging due to the sheer number of particles involved. While advancements in technology might enable more accurate simulations in the future, the fundamental limits of computational physics remain an open question.

The Holographic Principle and Simulability

The holographic principle is a fascinating concept in theoretical physics that suggests the information content of a volume can be perfectly represented by a boundary of that volume. This principle could imply that storing and simulating a vast universe might be possible with less computational resources than previously thought. However, this idea might represent the minimum requirement, necessitating an immensely powerful computational system to achieve it.

Computational Complexity and the Limits of Simulability

The computational systems can be categorized based on their complexity, with the highest category capable of simulating any other system, albeit at different speeds. This category, known as the top level of computational hierarchy, represents the pinnacle of simulability. However, even this class of systems may face limitations due to energy constraints, as seen in the brain's finite energy.

John Watrous' Quantum Computational Complexity delves into the realm of quantum computers and their unique capabilities. Quantum computers can handle certain problems more efficiently than classical computers, as illustrated by the discovery of Shor's algorithm for integer factoring and discrete logarithm problems. These problems fall within the class BQP (bounded error quantum polynomial time), which represents a set of problems that quantum computers can solve but classical computers cannot.

David Deutsch's Constructor Theory of Information

Constructor Theory of Information by David Deutsch and Chiara Marletto explores the idea that information is not just information but a universal constructor. This theory posits that the laws of physics should be described in terms of what is and what is not constructible. This framework offers a new perspective on the limits of simulation and computation.

While the exponential growth of computational power continues, also known as Moore's Law, it is unclear how sustainable this trend will be. The energy constraints on biological and computational systems may pose significant challenges to achieving higher levels of computational complexity.

Exploring the realms of noncomputable physics, the holographic principle, and quantum computational complexity, we uncover the intricate boundaries that separate what can be simulated and what cannot. These concepts challenge our understanding of both computation and the physical world, inviting further exploration and research.