Anticoncentration and state design of random tensor networks

• URL: http://arxiv.org/abs/2409.13023v2
• Date: Tue, 8 Oct 2024 11:14:55 GMT
• Title: Anticoncentration and state design of random tensor networks
• Authors: Guglielmo Lami, Jacopo De Nardis, Xhek Turkeshi,
• Abstract summary: We investigate quantum random tensor network states where the bond dimensions scalely with the system size, $N$. For bond dimensions $chi sim gamma N$, we determine the leading order of the associated overlaps probability distribution and demonstrate its convergence to the Porter-Thomas distribution. We extend this analysis to two-dimensional systems using randomed Project Entangled Pair States (PEPS)
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We investigate quantum random tensor network states where the bond dimensions scale polynomially with the system size, $N$. Specifically, we examine the delocalization properties of random Matrix Product States (RMPS) in the computational basis by deriving an exact analytical expression for the Inverse Participation Ratio (IPR) of any degree, applicable to both open and closed boundary conditions. For bond dimensions $\chi \sim \gamma N$, we determine the leading order of the associated overlaps probability distribution and demonstrate its convergence to the Porter-Thomas distribution, characteristic of Haar-random states, as $\gamma$ increases. Additionally, we provide numerical evidence for the frame potential, measuring the $2$-distance from the Haar ensemble, which confirms the convergence of random MPS to Haar-like behavior for $\chi \gg \sqrt{N}$. We extend this analysis to two-dimensional systems using random Projected Entangled Pair States (PEPS), where we similarly observe the convergence of IPRs to their Haar values for $\chi \gg \sqrt{N}$. These findings demonstrate that random tensor networks with bond dimensions scaling polynomially in the system size are fully Haar-anticoncentrated and approximate unitary designs, regardless of the spatial dimension.

Related papers

• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Quantum State Designs with Clifford Enhanced Matrix Product States [0.0]
  Nonstabilizerness, or magic', is a critical quantum resource that characterizes the non-trivial complexity of quantum states. We show that Clifford enhanced Matrix Product States ($mathcalC$MPS) can approximate $4$-spherical designs with arbitrary accuracy.
  arXiv  Detail & Related papers  (2024-04-29T14:50:06Z)
• Projected state ensemble of a generic model of many-body quantum chaos [0.0]
  The projected ensemble is based on the study of the quantum state of a subsystem $A$ conditioned on projective measurements in its complement. Recent studies have observed that a more refined measure of the thermalization of a chaotic quantum system can be defined on the basis of convergence of the projected ensemble to a quantum state design.
  arXiv  Detail & Related papers  (2024-02-26T19:00:00Z)
• Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory [50.24983453990065]
  An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers. We present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states. By providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory.
  arXiv  Detail & Related papers  (2024-01-31T15:50:50Z)
• Simplex Random Features [53.97976744884616]
  We present Simplex Random Features (SimRFs), a new random feature (RF) mechanism for unbiased approximation of the softmax and Gaussian kernels. We prove that SimRFs provide the smallest possible mean square error (MSE) on unbiased estimates of these kernels. We show consistent gains provided by SimRFs in settings including pointwise kernel estimation, nonparametric classification and scalable Transformers.
  arXiv  Detail & Related papers  (2023-01-31T18:53:39Z)
• Superposed Random Spin Tensor Networks and their Holographic Properties [0.0]
  We study boundary-to-boundary holography in a class of spin network states defined by analogy to projected entangled pair states (PEPS) We consider superpositions of states corresponding to well-defined, discrete geometries on a graph.
  arXiv  Detail & Related papers  (2022-05-19T12:24:57Z)
• Entanglement phase transitions in random stabilizer tensor networks [5.951407144817049]
  We explore a class of random tensor network models with "stabilizer" local tensors which we name Random Stabilizer Networks (RSTNs) For RSTNs defined on a two-dimensional square lattice, we perform extensive numerical studies of entanglement phase transitions between volume-law and area-law entangled phases of the one-dimensional boundary states. Our results at large $D$ approach known universal data of percolation conformal field theory, while showing clear discrepancies at smaller $D$.
  arXiv  Detail & Related papers  (2021-07-26T18:00:00Z)
• Entanglement distribution in the Quantum Symmetric Simple Exclusion Process [0.0]
  We study the probability distribution of entanglement in the Quantum Symmetric Simple Exclusion Process. By means of a Coulomb gas approach from Random Matrix Theory, we compute analytically the large-deviation function of the entropy in the thermodynamic limit.
  arXiv  Detail & Related papers  (2021-02-09T10:25:04Z)
• Stochastic Approximation for Online Tensorial Independent Component Analysis [98.34292831923335]
  Independent component analysis (ICA) has been a popular dimension reduction tool in statistical machine learning and signal processing. In this paper, we present a by-product online tensorial algorithm that estimates for each independent component.
  arXiv  Detail & Related papers  (2020-12-28T18:52:37Z)
• Quantitative Propagation of Chaos for SGD in Wide Neural Networks [39.35545193410871]
  In this paper, we investigate the limiting behavior of a continuous-time counterpart of the Gradient Descent (SGD) We show 'propagation of chaos' for the particle system defined by this continuous-time dynamics under different scenarios. We identify two under which different mean-field limits are obtained, one of them corresponding to an implicitly regularized version of the minimization problem at hand.
  arXiv  Detail & Related papers  (2020-07-13T12:55:21Z)
• Generalized Sliced Distances for Probability Distributions [47.543990188697734]
  We introduce a broad family of probability metrics, coined as Generalized Sliced Probability Metrics (GSPMs) GSPMs are rooted in the generalized Radon transform and come with a unique geometric interpretation. We consider GSPM-based gradient flows for generative modeling applications and show that under mild assumptions, the gradient flow converges to the global optimum.
  arXiv  Detail & Related papers  (2020-02-28T04:18:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.