# Circuit complexity and functionality: A statistical thermodynamics perspective

## TL;DR

This paper explores how thermodynamics can be used to understand and analyze the complexity of circuits, especially in the context of cryptography.

## Contribution

The paper introduces a thermodynamic framework for circuit complexity and connects it to ergodicity and fragmentation in circuit spaces.

## Key findings

- Circuit obfuscation can be viewed as a thermalization process preserving functionality.
- Fragmentation in circuit spaces implies limitations on connecting circuits via local moves.
- Fragmentation is unavoidable unless NP and coNP coincide, impacting computational complexity theory.

## Abstract

The current paper a) defines a thermodynamic approach to the complexity of circuits of specific functionality; and b) examines the validity of the approach by introducing the notion of “ergodicity” in the space of circuits, which hinges on functionality-preserving “local mixing” models of circuit dynamics that we connect with word problems in combinatorial and geometric group theory. Our “circuit thermodynamic” framework provides an alternative paradigm for studies of circuit obfuscation, a concept in cryptography that underpins a broad range of powerful cryptographic tools.

Circuit complexity, defined as the minimum circuit size required for implementing a particular Boolean computation, is a foundational concept in computer science. Determining circuit complexity is believed to be a hard computational problem. Recently, in the context of black holes, circuit complexity has been promoted to a physical property, wherein the growth of complexity is reflected in the time evolution of the Einstein-Rosen bridge (“wormhole”) connecting the two sides of an anti-de Sitter “eternal” black hole. Here, we are motivated by an independent set of considerations and explore links between complexity and thermodynamics for functionally equivalent circuits, making the physics-inspired approach relevant to real computational problems, for which functionality is the key element of interest. In particular, our thermodynamic framework provides an alternative perspective on the obfuscation of programs of arbitrary length—an important problem in cryptography—as thermalization through recursive mixing of neighboring sections of a circuit, which can be viewed as the mixing of two containers with “gases of gates.” This recursive process equilibrates the average complexity and leads to the saturation of the circuit entropy, while preserving functionality of the overall circuit. The thermodynamic arguments hinge on ergodicity in the space of circuits which we conjecture is limited to disconnected ergodic sectors due to fragmentation. The notion of fragmentation has important implications for the problem of circuit obfuscation as it implies that there are circuits of same size and functionality that cannot be connected via a polynomial number of local moves. Furthermore, we argue that fragmentation is unavoidable unless the complexity classes NP and coNP coincide, a statement that implies the collapse of the polynomial hierarchy of computational complexity theory to its first level.