Complexity measures in QFT and constrained geometric actions

We study the conditions under which, given a generic quantum system, complexity metrics provide actual lower bounds to the circuit complexity associated to a set of quantum gates. Inhomogeneous cost functions — many examples of which have been recently proposed in the literature — are ruled out by our analysis. Such measures are shown to be unrelated to circuit complexity in general and to produce severe violations of Lloyd’s bound in simple situations. Among the metrics which do provide lower bounds, the idea is to select those which produce the tightest possible ones. This establishes a hierarchy of cost functions and considerably reduces the list of candidate complexity measures. In particular, the criterion suggests a canonical way of dealing with penalties, consisting in assigning infinite costs to directions not belonging to the gate set. We discuss how this can be implemented through the use of Lagrange multipliers. We argue that one of the surviving cost functions defines a particularly canonical notion in the sense that: i) it straightforwardly follows from the standard Hermitian metric in Hilbert space; ii) its associated complexity functional is closely related to Kirillov’s coadjoint orbit action, providing an explicit realization of the “complexity equals action” idea; iii) it arises from a Hamilton-Jacobi analysis of the “quantum action” describing quantum dynamics in the phase space canonically associated to every Hilbert space. Finally, we explain how these structures provide a natural framework for characterizing chaos in classical and quantum systems on an equal footing, find the minimal geodesic connecting two nearby trajectories, and describe how complexity measures are sensitive to Lyapunov exponents.

Attention-Based Temporal Encoding Network with Background-Independent Motion Mask for Action Recognition

Convolutional neural network (CNN) has been leaping forward in recent years. However, the high dimensionality, rich human dynamic characteristics, and various kinds of background interference increase difficulty for traditional CNNs in capturing complicated motion data in videos. A novel framework named the attention-based temporal encoding network (ATEN) with background-independent motion mask (BIMM) is proposed to achieve video action recognition here. Initially, we introduce one motion segmenting approach on the basis of boundary prior by associating with the minimal geodesic distance inside a weighted graph that is not directed. Then, we propose one dynamic contrast segmenting strategic procedure for segmenting the object that moves within complicated environments. Subsequently, we build the BIMM for enhancing the object that moves based on the suppression of the not relevant background inside the respective frame. Furthermore, we design one long-range attention system inside ATEN, capable of effectively remedying the dependency of sophisticated actions that are not periodic in a long term based on the more automatic focus on the semantical vital frames other than the equal process for overall sampled frames. For this reason, the attention mechanism is capable of suppressing the temporal redundancy and highlighting the discriminative frames. Lastly, the framework is assessed by using HMDB51 and UCF101 datasets. As revealed from the experimentally achieved results, our ATEN with BIMM gains 94.5% and 70.6% accuracy, respectively, which outperforms a number of existing methods on both datasets.

The tunneling effect for Schrödinger operators on a vector bundle

In the semiclassical limit $$\hbar \rightarrow 0$$ ħ → 0 , we analyze a class of self-adjoint Schrödinger operators $$H_\hbar = \hbar ^2 L + \hbar W + V\cdot {\mathrm {id}}_{\mathscr {E}}$$ H ħ = ħ 2 L + ħ W + V · id E acting on sections of a vector bundle $${\mathscr {E}}$$ E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points $$m^1,\ldots m^r \in M$$ m 1 , … m r ∈ M , called potential wells. Using quasimodes of WKB-type near $$m^j$$ m j for eigenfunctions associated with the low lying eigenvalues of $$H_\hbar $$ H ħ , we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension $$\ell + 1$$ ℓ + 1 . This dimension $$\ell $$ ℓ determines the polynomial prefactor for exponentially small eigenvalue splitting.

MINIMAL GEODESIC FOLIATION ON T2 IN CASE OF VANISHING TOPOLOGICAL ENTROPY

On a Riemannian 2-torus (T2, g) we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper [12] we already obtained that the asymptotic direction and therefore also the rotation number exists for all geodesics. In this paper we show that for all r ∈ ℝ ∪ {∞} the universal cover ℝ2 is foliated by minimal geodesics of rotation number r. For irrational r ∈ ℝ all geodesics are minimal, for rational r ∈ ℝ ∪ {∞} all geodesics stay in strips between neighboring minimal axes. In such a strip the minimal geodesics are asymptotic to the neighboring minimal axes and generate two foliations.

AN ENHANCED LIPSCHITZ EMBEDDING CLASSIFIER FOR MULTI-EMOTION SPEECH ANALYSIS

This paper proposes an Enhanced Lipschitz Embedding based Classifier (ELEC) for the classification of multi-emotions from speech signals. ELEC adopts geodesic distance to preserve the intrinsic geometry at all scales of speech corpus, instead of Euclidean distance. Based on the minimal geodesic distance to vectors of different emotions, ELEC maps the high dimensional feature vectors into a lower space. Through analyzing the class labels of the neighbor training vectors in the compressed low space, ELEC classifies the test data into six archetypal emotional states, i.e. neutral, anger, fear, happiness, sadness and surprise. Experimental results on clear and noisy data set demonstrate that compared with the traditional methods of dimensionality reduction and classification, ELEC achieves 15% improvement on average for speaker-independent emotion recognition and 11% for speaker-dependent.

On the differentiability of a distance function

Let M be a simply connected complete K?hler manifold and N a closed complete totally geodesic complex submanifold of M such that every minimal geodesic in N is minimal in M. Let U? be the unit normal bundle of N in M. We prove that if a distance function ? is differentiable at v ? U?, then ? is also differentiable at -v.

GRASSMANNIANS OF A FINITE ALGEBRA IN THE STRONG OPERATOR TOPOLOGY

If [Formula: see text] is a type II1 von Neumann algebra with a faithful trace τ, we consider the set [Formula: see text] of self-adjoint projections of [Formula: see text] as a subset of the Hilbert space [Formula: see text]. We prove that though it is not a differentiable submanifold, the geodesics of the natural Levi–Civita connection given by the trace have minimal length. More precisely: the curves of the form γ(t) = eitxpe-itx with x* = x, pxp = (1 - p)x(1 - p) = 0 have minimal length when measured in the Hilbert space norm of [Formula: see text], provided that the operator norm ‖x‖ is less or equal than π/2. Moreover, any two projections which are unitary equivalent are joined by at least one such minimal geodesic, and only unitary equivalent projections can be joined by a smooth curve. Finally, we prove that these geodesics have also minimal length if one measures them with the Schatten k-norms of τ, ‖x‖k = τ((x* x)k/2)1/k, for all k ∈ ℝ, k ≥ 0. We also characterize curves of unitaries which have minimal length with these k-norms.