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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

A unique feature of the hybrid quantum Monte Carlo (HQMC) method is the potential to simulate negative sign free lattice fermion models with subcubic scaling in system size. Here we will revisit the algorithm for various models. We will show that for the Hubbard model the HQMC suffers from ergodicity issues and unbounded forces in the effective action. Solutions to these issues can be found in terms of a complexification of the auxiliary fields. This implementation of the HQMC that does not attempt to regularize the fermionic matrix so as to circumvent the aforementioned singularities does not outperform single spin flip determinantal methods with cubic scaling. On the other hand we will argue that there is a set of models for which the HQMC is very efficient. This class is characterized by effective actions free of singularities. Using the Majorana representation, we show that models such as the Su-Schrieffer-Heeger Hamiltonian at half filling and on a bipartite lattice belong to this class. For this specific model sub-cubic scaling is achieved.

The realization of the spin-Hall effect in quantum wells has led to a plethora of studies regarding the properties of the edge states of a two-dimensional topological insulator. These edge states constitute a class of one-dimensional liquids, called the helical liquid, where an electron's spin quantization axis is tied to its momentum. In contrast to one-dimensional conductors, magnetic impurities---below the Kondo temperature---cannot block transport and one expects the current to circumvent the impurity. To study this phenomenon, we consider the single-impurity Anderson model embedded into an edge of a Kane-Mele ribbon with up to $512\ifmmode\times\else\texttimes\fi{}80$ sites and use the numerically exact continuous time quantum Monte Carlo method to study the Kondo effect. We present results on the temperature dependence of the spectral properties of the impurity and the bulk system that show the behavior of the system in the various regimes of the Anderson model. A view complementary to the single-particle spectral functions can be obtained using the spatial behavior of the spin-spin correlation functions. Here we show the characteristic, algebraic decay in the edge channel near the impurity.