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تسجيل مستخدم جديدQuantum Complexity as Hydrodynamics
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As a new step towards defining complexity for quantum field theories, we consider Nielsens geometric approach to operator complexity for the $SU(N)$ group. We develop a tractable large $N$ limit which leads to regular geometries on the manifold of unitaries. To achieve this, we introduce a particular basis for the $mathfrak{su}(N)$ algebra and define a maximally anisotropic metric with polynomial penalty factors. We implement the Euler-Arnold approach to identify incompressible inviscid hydrodynamics on the two-torus as a novel effective theory for the evaluation of operator complexity of large qudits. Moreover, our cost function captures two essential properties of holographic complexity measures: ergodicity and conjugate points. We quantify these by numerically computing the sectional curvatures of $SU(N)$ for finite large $N$. We find a predominance of negatively curved directions, implying classically chaotic trajectories. Moreover, the non-vanishing proportion of positively curved directions implies the existence of conjugate points, as required to bound the growth of holographic complexity with time.
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