  As a new step towards defining complexity for quantum field theories, we map
Nielsen operator complexity for $SU(N)$ gates to two-dimensional hydrodynamics.
We develop a tractable large $N$ limit that leads to regular geometries on the
manifold of unitaries as $N$ is taken to infinity. To achieve this, we
introduce a basis of non-commutative plane waves for the $\mathfrak{su}(N)$
algebra and define a metric with polynomial penalty factors. Through the
Euler-Arnold approach we identify incompressible inviscid hydrodynamics on the
two-torus as a novel effective theory of large-qudit operator complexity. For
large $N$, our cost function captures two essential properties of holographic
complexity measures: ergodicity and conjugate points.
