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 [1]. 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 AdS ``eternal'' black hole [2]. Here we explore another link between complexity and thermodynamics for circuits of given functionality, 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 a new 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 with same size and functionality that cannot be connected via 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.

翻译：电路复杂度——实现特定布尔计算所需的最小电路规模——是计算机科学中的基础概念。确定电路复杂度被认为是一个困难的计算问题[1]。近期，在黑洞研究背景下，电路复杂度被提升为一种物理属性，其中复杂度的增长体现在连接AdS“永恒”黑洞两侧的爱因斯坦-罗森桥（“虫洞”）的时间演化中[2]。本文针对具有给定功能性的电路，探索复杂度与热力学之间的另一层关联，使这一受物理学启发的方法适用于以功能性为关键要素的实际计算问题。特别地，我们的热力学框架为任意长度程序的混淆问题——密码学中的一个重要问题——提供了新视角：通过递归混合电路相邻部分（可视为两个装有“门气体”的容器混合）实现热化。该递归过程在保持整体电路功能性的同时，使平均复杂度达到平衡并导致电路熵饱和。热力学论证基于电路空间中的遍历性，我们推测由于碎片化现象，该遍历性仅限于非连通的遍历扇区。碎片化概念对电路混淆问题具有重要启示，因为它表明存在规模相同、功能相同却无法通过局部操作连接的电路。此外，我们论证除非复杂度类NP与coNP重合（这意味着计算复杂度理论的多项式谱系坍缩至第一层），否则碎片化不可避免。