Quantum Complexity as Hydrodynamics

TL;DR

This paper maps quantum operator complexity to two-dimensional hydrodynamics, revealing a large N limit that simplifies the geometry of unitary spaces and captures key features of holographic complexity.

Contribution

It introduces a hydrodynamic framework for quantum complexity using non-commutative plane waves and identifies large N limits leading to effective incompressible hydrodynamics.

Findings

Large N limit yields regular geometries on the unitary manifold.

Cost function captures ergodicity and conjugate points in complexity.

Hydrodynamic description offers a new perspective on quantum complexity.

Abstract

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.