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Program - week 3Mari Carmen Bañuls - Tensor Networks and spectral properties Tensor networks are well known as efficient numerical tools for ground states of quantum many-body systems in low spatial dimensions. The ansatz is however not suitable to describe highly excited eigenstates, or out-of-equilibrium dynamics. Nevertheless, recently proposed TNS algorithms allow us to access spectral properties of the Hamiltonian over extensive energy ranges. They allow us to access spectral densities of many-body operators (e.g. densities of states) but also to probe the structure of physical operators in the energy eigenbasis, in particular the predictions of the eigenstate thermalization hypothesis, which particularizes the random matrix behaviour for the physical many-body setting. The slides are available here.
Dario Benedetti - From O(N)3 to SO(3) in tensor models In this talk we consider the mean field equations of a bosonic tensor model with quartic interactions and O(N)3 symmetry. For N different from 3, their nontrivial solutions necessarily represent possible patterns of spontaneous breaking of the O(N)3 symmetry down to a proper subgroup. Besides less interesting low-rank solutions, we find one explicit solution with SO(3) invariance, for which the tensor field is expressed in terms of the Wigner 3jm symbol, highlighting an intriguing link between tensor models and SO(3) recoupling theory. Moreover, such solution provides a tantalizing relation between different models sharing a similar large-N limit: models in which random tensors appear as fundamental variables (tensor models), models in which they appear as random couplings (p-spin and SYK models), and the Amit-Roginsky model, in which a cubic interaction is mediated by a Wigner 3jm symbol. Main reference: D. Benedetti and I. Costa, Phys.Rev.D 101 (2020) 086021 [arXiv:1912.07311]
Sylvain Carrozza - Averaging over local unitary groups in Random Tensor Networks TBA The slides are available here.
Leticia Cugliandolo - Hamiltonian dynamics of classical disordered models I will describe the dynamics of classical disordered macroscopic models (of p-spin kind) completely isolated from any environment reproducing, in a classical setting, the "quantum quench" protocol. We show that the asymptotic dynamics can reach a steady state of Gibbs-Boltzmann kind for non-integrable (p > 2, multi-body) interactions while, instead, they approach a stationary state of Generalized Gibbs Boltzmann type in integrable (p = 2, two-body) cases. Interesting phase transitions will be discussed in both cases. This talk summarizes work done in collaboration with D. Barbier, G. Lozano, N. Nessi, M. Picco, and A. Tartaglia.
Razvan Gurau - Free Cumulants For Random Tensors The slides are available here.
Aram Harrow - Randomized truncation Given a vector v, what is the closest k-sparse vector? The answer to this question is usually that we should take the largest k entries of v. It turns out that we can do better with randomized approximations. When approximating pure bipartite entangled states with states of low Schmidt rank, this means that mixed approximations outperform pure approximations. This fact has application to classical algorithms for matrix product states by improving the truncation step, and to quantum algorithms for Hamiltonian simulation. The slides are available here.
Tomohiro Hayase - Analysis of Deep Neural Networks With Random Tensors Combining random matrices and multilayer perceptrons (MLPs) forms the foundation theory of deep neural networks (DNN). So, what role do random tensors play in deep learning? In this talk, we introduce how random tensors appear in the analysis of the MLP-Mixer. The MLP-Mixer is a type of DNN used in image processing and is a simplified model of the Vision Transformer (ViT). In these models, input images are divided into tokens, arranged sequentially, and input as second-order tensors. The MLP-Mixer processes both within-token and between-token operations using MLP blocks. Despite its simple structure that replaces the attention mechanism of ViT with MLPs, the MLP-Mixer achieves performance close to ViT's, highlighting the importance of data volume and tokenization. Specifically, this talk presents experimental results showing that high sparsity and large hidden layer dimensions positively impact performance. To this end, we intentionally disrupt the model's structure using tensor products and random permutation matrices, verifying that these beneficial properties are not dependent on the model's specific structure. The slides are available here.
Maria Jivulescu - Entanglement detection based on projective tensor norms In my talk I will present the class of entanglement criteria based on applying local contractions to an input state and compute the projective tensor norm of the output. The idea is based on introducing the notion of entanglement testers, which captures in a unique framework entanglement criteria as realignment and SIC POVM . This is a joint work with Cecilia Lancien and Ion Nechita https://arxiv.org/abs/2010.06365 Connections to the theory of thresholds detection for different entanglement criteria can be also taken into consideration. The slides are available here.
Pax Kivimae - Instability and the Number of Real Eigenvalues of A Random Tensor We study the number of (real) eigenvalues, and stable eigenvalues, of a non-symmetric n-dimensional p-tensor (p>2) for large n. Particularly, when the tensor is taken to have i.i.d. standard Gaussian entries, we show that, typically, while the total number of eigenvalues grows exponentially large, not a single one is stable. Beyond this though, we show that if one considers a partially symmetric p-tensor, there is a critical value for the symmetry coefficient, such that if the symmetry is below this coefficient, the tensor typically has no stable eigenvalues, and if it is above this coefficient, the number of stable eigenvalues grows exponentially large, though still exponentially small relative to the total number of eigenvalues. This gives the first rigorous example of a transition from "absolute to relative instability", as coined by [1], and extends the picture produced in the symmetric case by [2]. [1] Ben Arous, Fyodorov and Khoruzhenko 2021
Joseph Landsberg - How to find hay in a haystack - tensor version The "hay in a haystack" problem is to find an explicit object that behaves like a random one in terms of its complexity. A tensor version of this problem is to find an explicit tensor in some tensor space that has maximal border rank (i.e., maximal complexity), in other words the border rank of a random tensor. A stronger version is to find an explicit sequence, of say (m,m,m) tensors as m goes to infinity, with maximal border rank. The state of the art for this problem is embarrassing. Arora and Barak refer to lower bounds as "Complexity theory's Waterloo". I'll report on the little that is known and try to explain why the problem is so difficult. The slides are available here.
Benjamin McKenna - Injective norm of real and complex random tensors The injective norm is a natural generalization to tensors of the operator norm of a matrix. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, where it is known as the geometric entanglement. We give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, corresponding to a lower bound on the geometric entanglement of random quantum states. The proof is based on spin-glass methods, the Kac—Rice formula, and recent progress coming from random matrices. Joint work with Stéphane Dartois. The notes are available here.
James Mingo - Real infinitesimal freeness Thanks to Voiculescu's celebrated asymptotic freeness theorem, free independence can be used to model the large $N$ interactions between random matrices, provided some independence and invariance assumptions are made. When one or more of the matrices are of bounded rank, then the interactions are at the infinitesimal scale. Recent papers of Shlyakhtenko and Collins, Hasebe, Sakuma showed that for unitarily invariant ensembles, the infinitesimal freeness of Belinschi and Shlyakhtenko can model the interactions. For orthogonally invariant ensembles a new model was needed. I will present this new model, which is joint work with Guillaume Cébron.
Alexander Müller Hermes - Entanglement annihilation and regularizing operator norms A quantum channel is called entanglement annihilating if all its k-fold tensor powers map arbitrary quantum states to fully separable quantum states. It is an open problem to determine whether the entanglement breaking channels are the only examples of entanglement annihilating channels. We explain how a variant of this question can be cast in the setting of general proper cones and their tensor products, and how we can solve it for certain types of cones. In particular, we will relate the question to certain operator ideals studied in the theory of Banach spaces and how they arise by procedures akin to regularizations in quantum information theory. (Based on joint work with Guillaume Aubrun).
Yoshiko Ogata - Boundary states of a bulk gapped ground state in 2-d quantum spin systems
Carlos Palazuelos - Quantum XOR games via tensor norms Quantum XOR games extend the model of classical XOR games by allowing the referee's questions to the players to be quantum states. Then, it is known that quantum XOR games exhibit a variety of behaviors that do not appear in the classical case. In this talk, we will explain how to describe these games using tensor norms and how this description can lead to new results on the relationship between quantum entanglement and classical communication. If time permits, some open problems concerning certain multipartite quantum states will also be discussed. The slides are available here.
Valentina Ros - Energy landscapes from random tensors: correlations between triplets of stationary points When random tensors are contracted with vectors, they generate random functions in high dimension (N >> 1). These functions are typically highly non-convex, with numerous stationary points (minima, maxima, and saddles). Such models have long been used as simplified representations of the energy landscapes of complex systems like glasses, serving as toy models for studying optimization dynamics with local algorithms such as Langevin dynamics (gradient descent with noise). While the dynamics is rather well understood in the mean-field limit (N → ∞), extending this understanding beyond the mean-field limit remains an open challenge. In this regime, when the noise is weak, the dynamics is expected to be dominated by rare "activated" processes—where the system relies on rare noise fluctuations to cross high energy barriers and escape metastable local minima. In this talk, I will present results on the local geometry of the energy landscape, specifically focusing on the joint distribution of triplets of stationary points conditioned to be at fixed distances from one another. These results are obtained with a combination of the Kac-Rice formalism with the replica method. After presenting the results, I will discussing to what extent this statistical distribution can give hints on the system's dynamics in the activated regime. The slides are available here.
Norbert Schuch - The role of positivity in tensor network contraction It is well known that quantum systems with negative entries in their Hamiltonian are generally much harder to simulate than those with positive entries only. I will show that a very similar transition in hardness also shows up in tensor network simulations: Random tensor networks with negative entries are generally much harder to contract than those with positive entries only. While such a transition is expected from a sampling argument -- akin to the Monte Carlo negative sign problem -- I will show that the actual hardness transition occurs much earlier, namely already for a vanishingly small positive bias. I will discuss two independent and entirely different explanations which remarkably both yield the same transition point -- first, by relating the problem to a transition in the scaling of correlations (entanglement) in the boundary, and second, through a generalization of Barvinok's algorithm for approximating permanents.
Stanislaw Szarek - Noncommutative Lambdap-sets and the additivity problems Constructions of counterexamples to additivity problems in quantum information theory along the lines pioneered by Hayden-Winter and Hastings requires solving various sharp instances of Dvoretzky's theorem for Schatten classes. The most efficient technology to obtain such solutions is to consider "random" subspaces, which is not fully satisfactory. In the commutative setting, similar questions led to the concept of Lambdap-sets, i.e., collections of characters on an Abelian group such that the p-norm and the 2-norm are approximately equivalent on the linear span of those characters. The ultimate results in this area were obtained by Rudin, Bourgain and Talagrand (1960s through 1990s). Some - but not all - of the constructions of Lambda_p-sets sets are explicit. In the present talk we will outline some of the above-mentioned results and report on the attempts to adapt the techniques to the non-commutative context, which will in particular lead to random (and pseudo-random) tensors. The slides are available here.
Sarah Timhadjelt - Non-hermitian expander obtained with Haar distributed unitaries A system in quantum mechanics is modeled by a state, i.e. a N dimensional with trace 1 positive semidefinite matrix where N is the number of possible values for an observable (e.g. momentum, level of energy). A transformation of such a system, after measurements for instance, is modeled by specific operators on matrices called quantum channels, preserving the set of states. These operators can be seen as the sum of tensor products of unit matrices. As for Markov operators, we are interested in the spectral gap of the quantum channel which can be seen as a quantifier of the distance of the operator to a rank one projector, and one way to optimize the gap is to consider Haar distributed unitaries. A proof of the optimality of the second largest eigenvalue or singular value in the non-Hermitian case is to use Schwinger-Dyson equations previously used by Hastings in the Hermitian case.
Peter Vrana - Tensor parameters in entanglement theory
Michael Walter - Random stabilizer tensors - duality and applications
Freek Witteveen - The resource theory of tensor networks Tensor networks provide succinct representations of quantum many-body states and are an important computational tool for strongly correlated quantum systems. Their expressive and computational power is characterized by an underlying entanglement structure, on a lattice or more generally a (hyper)graph, with virtual entangled pairs or multipartite entangled states associated to (hyper)edges. Changing this underlying entanglement structure into another can lead to both theoretical and computational benefits. In this talk I will explain results from arXiv:2307.07394, where we study a resource theory which generalizes the notion of bond dimension to entanglement structures using multipartite entanglement. It is a direct extension of resource theories of tensors studied in the context of multipartite entanglement and algebraic complexity theory, allowing for the application of the sophisticated methods developed in these fields to tensor networks. The slides are available here.
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