I will discuss the integrability property of a stochastic and quantum deformation of the Rule 54 cellular automaton: the simplest microscopic (deterministic) reversible model in 1+1 discrete space and time dimensions with strong local interactions. First, I will introduce the Rule 54 model and its two deformations:    

• In the stochastic case, I couple the system to stochastic boundary reservoirs and show that the resulting non-equilibrium steady states can be constructed explicitly in matrix product form.   
• In the quantum case, I explain how the model can be embedded into the Yang-Baxter integrability framework and provide an expression of the Lax operator, the operator responsible for the existence of an extensive number of conserved quantities that constrain the dynamics. It turns out that Yang-Baxter integrability is more common than previously thought!

Based on 2603.25424 with T. Prosen.