Quantum Complexity as Hydrodynamics

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.

17 pages, 4 figures, v2 corrected results on sectional curvatures, further details about large N decoupling limit added

References in corpus (2)

• Quantum Computation as Geometry
• Quantum Complexity and Negative Curvature

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