Phase transitions in time complexity of Brownian circuits

Abstract

Brownian circuits perform computations using stochastic transitions driven by thermal fluctuations. While the energetic costs of such fluctuation-driven computation have been extensively studied within stochastic thermodynamics, much less is known about its computational complexity, in particular, how computation time scales with circuit size. In this work, the computation time for explicitly designed Brownian circuits is numerically investigated via the first-passage time to a completed state. For arithmetic circuits such as adders, varying the forward transition rate induces a sharp change in the scaling behavior of the mean computation time with circuit size, from linear to exponential. This change can be interpreted as an easy-hard transition in computational time complexity. The transition suggests that, for meaningful computational tasks, achieving efficient polynomial-time computation generally requires a finite forward bias corresponding to a nonzero energy input. As a counterexample, we show that arbitrary logical operations can be reduced to an effective one-dimensional stochastic process in which the zero-bias limit lies within the computationally efficient (easy) regime. However, achieving such a one-dimensional normal form unavoidably leads to an exponential increase in circuit size. These results reveal a fundamental trade-off between computation time, circuit size, and energy input in Brownian circuits and demonstrate that phase transitions in time complexity provide a natural framework for characterizing the cost of fluctuation-driven computation.