Related papers: On the geometry of tensor network states
We investigate a recent conjecture connecting the AdS/CFT correspondence and entanglement renormalization tensor network states (MERA). The proposal interprets the tensor connectivity of the MERA states associated to quantum many body…
The geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor,…
Tensor networks offer a variational formalism to efficiently represent wave-functions of extended quantum many-body systems on a lattice. In a tensor network N, the dimension \chi of the bond indices that connect its tensors controls the…
The current shift in the quantum optics community towards large-size experiments -- with many modes and photons -- necessitates new classical simulation techniques that go beyond the usual phase space formulation of quantum mechanics. To…
Tensor network methods have been a key ingredient of advances in condensed matter physics and have recently sparked interest in the machine learning community for their ability to compactly represent very high-dimensional objects. Tensor…
We show that the Hilbert space spanned by a continuously parametrized wavefunction family---i.e., a quantum state manifold---is dominated by a subspace, onto which all member states have close to unity projection weight. Its characteristic…
In this thesis we extend the formalism of tensor network algorithms to incorporate global internal symmetries. We describe how to both numerically protect the symmetry and exploit it for computational gain in tensor network simulations. Our…
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also…
Tensor networks prepare states that share many features of states in quantum gravity. However, standard constructions are not diffeomorphism invariant and do not support an algebra of non-commuting area operators. Recently, analogues of…
We consider the representation of operators in terms of tensor networks and their application to ground-state approximation and time evolution of systems with long-range interactions. We provide an explicit construction to represent an…
In the last decade it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks: separable elements, with connections (or interactions) between certain pairs of them.…
Tensor networks developed in the context of condensed matter physics try to approximate order-$N$ tensors with a reduced number of degrees of freedom that is only polynomial in $N$ and arranged as a network of partially contracted smaller…
Tensor network states are expected to be good representations of a large class of interesting quantum many-body wave functions. In higher dimensions, their utility is however severely limited by the difficulty of contracting the tensor…
The 1-form symmetry, manifesting as loop-like symmetries, has gained prominence in the study of quantum phases, deepening our understanding of symmetry. However, the role of 1-form symmetries in Projected Entangled-Pair States (PEPS),…
We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as $O(n^{\omega+\epsilon})$ time matrix multiplication, and in…
The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of…
Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant…
We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the…
We use a recently proposed class of tensor-network states to study phase transitions in string-net models. These states encode the genuine features of the string-net condensate such as, e.g., a nontrivial perimeter law for Wilson loops…
We show that, in discrete models of quantum gravity, emergent geometric space can be viewed as the entanglement pattern in a mixed quantum state of the "universe", characterized by a universal topological network entanglement. As a concrete…