Abstract: Quantum circuit complexity is a measure of the implementation difficulty of unitary operators acting on a Hilbert space. While originally defined for finite dimensional Hilbert spaces, complexity has recently been argued to play an important role in understanding? (holographic) QFTs. Despite this, we lack a clear and well-defined connection between the complexity of systems with finite and infinite degrees of freedom. In this talk, I will show that there exists such a connection for the group of SU(N) unitaries as N goes to infinity. In particular I will show that generating SU(N) unitaries via a set of non-commutative plane waves and penalizing harsher the operators with larger wave-momentum, leads to a well-defined and tractable large-N limit. In this limit, we can use the Euler-Arnold approach to show that 2d inviscid incompressible hydrodynamics on a 2-torus emerges as an effective theory of complexity.? Examining complexity from the point of view of this effective theory, indicates that for large N our complexity measure captures two essential properties of holographic complexity measures: ergodicity and conjugate points.