Quantum complexity as hydrodynamics

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