Quantum Complexity as Hydrodynamics

Understand

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.