Quantum Complexity as Hydrodynamics

TL;DR

The paper defines a quantum complexity measure for $SU(N)$ gates by mapping Nielsen's geometric approach to a two-dimensional hydrodynamic problem on the torus, enabling a well-defined large-$N$ limit with polynomial penalty factors. By constructing a non-commutative plane-wave basis $J_{}$ and using the Euler-Arnold formalism, the authors show that the low-energy, low-momentum sector of $SU(N)$ is isomorphic to the volume-preserving diffeomorphisms $\mathrm{SDiff}(\mathds{T}^2)$, so complexity reduces to finding geodesics in a hydrodynamic metric. They derive the Euler-Arnold equation $\Delta\dot{\mathpzc{h}}=-\{\mathpzc{h},\Delta\mathpzc{h}\}$, discuss the role of polynomial penalty factors, and analyze curvature data to demonstrate ergodicity and the presence of conjugate points, aligning the construction with holographic complexity expectations. The study also develops a sub-Riemannian interpretation for finite $N$, identifies a hydrodynamic $\mathcal{O}(1)$ sector that survives the $N\to\infty$ limit, and outlines how a physical, all-to-all qudit Hamiltonian fits into this framework. Overall, the work provides a tractable, geometrically rich approach to operator complexity with a clear holographic interpretation and potential avenues for finite-$N$ corrections.

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.

Figures (4)

• Figure 1: Color density plot of the critical value $N_c$ at which the normalized Ricci curvature of a given direction $\vec{m}$ in $\mathfrak{su}(N)$ becomes negative over the $\mathds{Z}^2$ lattice, spanned by the vector components $m_1,m_2$. The interpolation between the integer points of the lattice is there to guide the eye. The color flare at the lower left corner is an artifact of this interpolation.
• Figure 2: Example for the final distribution of $C$'s (solid red arrows) and $S$'s (dashed blue arrows) inside the fundamental cell (dashed black) for $N=5$.
• Figure 3: Distribution of signs of sectional curvatures of $SU(N)$ in directions given by pairs of generators of the $\mathfrak{su}(N)$ algebra for $2\leq N\leq 39$. The computations exhibited changes of less than $0.1\%$ for large $N$, strongly suggesting that the percentages stabilize. Lines are a visual guide, not an interpolation. A slight majority of positively curved directions settles, but this is inconclusive with regards to the stability of geodesics on the manifold of unitaries.
• Figure 4: Internal qudit levels with exemplary transitions. The physical Hamiltonian $H$ is assumed to implement all-to-all transitions, each of which is determined, up to eight-fold degeneracy, by the wave vector of the associated excitation.