Optimising, Renormalising, Evolving and Quantising Tensor Networks

Using the time-dependent variational principle as a guiding principle, I will discuss the various differential geometrical structures that are present in variational tensor network manifolds (mostly matrix product states). I will also discuss the relevance of these structures for numerical simulations and point towards current and possible future applications thereof.

I will discuss our recent proposal for the new definition of partial transpose for fermionic systems. As applications, i will discuss the construction of topological invariants and entanglement measure using the fermionic partial transpose. As target systems, I plant to discuss Majorana chains, and, if time allows, (2+1)d fermionic topological phases such as time-reversal symmetric topological insulators.

Implementing a global symmetry into a tensor network, i.e. enforcing that it is conserved at every step, often provides for a significant computational speedup which has been the main aim of such implementations during the last approximately 15 years. In this presentation, I will show how such a symmetry implementation cannot just be used to increase computational efficiency but also to detect spontaneous symmetry-breaking in challenging situations where data from calculations without those broken symmetries alone may be inconclusive. Examples will include DMRG and iPEPS calculations.

I will describe a simple model of locally interacting quantum spin 1/2 chain, namely Floquet-Ising chain with transverse and longitudinal fields, where the spectral form factor can be computed exactly (in the thermodynamic limit) due to a particular space-time duality symmetry of the model, and shown to match the prediction of Random Matrix Theory. Our result implies ergodicity for any finite amount of disorder in the longitudinal field, rigorously excluding the possibility of many-body localization in the model.

The wavefunction is the central quantity in quantum mechanics, but is forbidding in terms of its size and complexity. Many numerical techniques actually avoid it completely in favor of other approaches. But tensor networks are an interesting technique to work directly with many-body wavefunctions in a principled and efficient way. After motivating the many-body problem, I will introduce tensor networks, using the matrix product state as the main example. Then I will discuss numerical applications of tensor networks, emphasizing areas where they hold significant promise for future developments. These areas include applications to physics, chemistry, and machine learning of real-world data.

The saddle points of a conventional Feynman path integral are not entangled, since they comprise a sequence of classical field configurations. We combine insights from field theory and tensor networks by constructing a Feynman path integral over a sequence of matrix product states. The paths that dominate this path integral include some degree of entanglement. This new feature allows several insights and applications: i. A Ginzburg-Landau description of deconfined phase transitions. ii. The emergence of new classical collective variables in states that are not adiabatically continuous with product states. iii. Features that are captured in product-state field theories by proliferation of instantons are encoded in perturbative fluctuations about entangled saddles. We develop a general formalism for such path integrals and a couple of simple examples to illustrate their utility. Ref: 1607.01778

Due to the area-law entanglement of all eigenstates of many-body localized (MBL) systems, tensor networks have emerged as a natural tool for their simulation. While in higher dimensions, theoretical arguments seem to rule out MBL, experiments are suggestive of its existence. I will give an overview over our research results on quantum circuits with long gates (a specific type of tensor networks), which allow to describe all eigenstates of MBL systems at once. This facilitates the simulation of MBL system sizes which are far beyond the reach of exact diagonalization. Moreover, quantum circuits can also be used as an analytical tool to prove the stability of symmetry protected MBL phases - a high-temperature generalization of symmetry protected topological phases. In two dimensions, quantum circuits constitute an approach which is well-suited to describe the MBL-like behavior observed on experimental time scales. I will present the first large-scale numerical examination of a disordered Bose-Hubbard model in two dimensions realized in cold atoms, which shows entanglement based signatures of MBL. A careful analysis of the eigenstate entanglement structure provides an indication of the putative phase transition in a parameter range potentially consistent with experiments.

It is generally expected that an out-of-equilibrium quantum system will `thermalise' after some length of time, unless it is integrable (i.e. possesses infinitely many conserved charges). However it is known that rare states can exist in a model's spectrum that defy this tendency, yielding nonthermal behaviour and anomalous dynamics. I will discuss how MPS techniques can complement analytical methods, to provide a clear picture of rare states and nonthermal behaviour driven by confinement in the quantum Ising model in 1D and 2D. In the latter case I will describe time evolution of a 2D system using MPS algorithms combined with exact results for quantum chains.

We consider the general problem of the fate of conserved quantities in integrable models that are perturbed by terms that nominally breaking their integrability. We will begin by considering quantum quenches in one dimensional Bose gases where we prepare the gas in the ground state of a parabolic trap and then release it into a small cosine potential (PRX 5, 041043 (2015)). This cosine potential breaks the integrability of the 1D gas which absent the potential is described by the Lieb-Liniger model. We explore the consequences of this cosine potential on the thermalization of the gas. We argue that the integrability breaking of the cosine does not immediately lead to ergodicity inasmuch as we demonstrate that there are residual quasi-conserved quantities post-quench. We demonstrate that the quality of this quasi-conservation can be made arbitrarily good. We will also consider the lessons drawn from this work in the context of the recent finding of rare states in the one and two dimensional quantum Ising model (arXiv:1804.09990).

What is the energy cost of extracting entanglement from complex quantum systems? In other words, given a state of a quantum system, how much energy does it cost to extract m EPR pairs? This is an important question, particularly for quantum field theories where the vacuum is generally highly entangled. Here we discuss a framework for understanding the energy cost of entanglement extraction. First, we look at a toy model, and then we tackle the antiferromagnetic Heisenberg and transverse field Ising models numerically to calculate the energy cost of entanglement extraction using matrix product states. Finally, we give a physical argument to find the energy cost of entanglement extraction in some condensed matter and quantum field systems. The energy cost for those quantum field theories depends on the spatial dimension, and in one dimension, for example, it grows exponentially with the number of EPR pairs extracted.

Matrix Product Operators were introduced by Fannes, Nachtergaele and Werner in their seminal 1992 paper. In this talk, I will show how their study can give new insights in different subjects within quantum information, such as channels capacities, quantum cellular automata or topological order in quantum spin systems.

The past decade has seen a great interest in the question about whether and how quantum many-body system locally thermalize. It has been driven by theoretical findings involving the long sought demonstration that many-body localization (MBL) exists as well as the derivation of exact bounds on chaos. In my talk, I will introduce matrix-product state (MPS) based methods that allow for an efficient numerical simulation of the quantum thermalization dynamics. Firstly, I will show that, contrary to the common belief that the rapid growth of entanglement restricts simulations to short times, the long time limit of local observables can be well captured using the MPS based time-dependent variational principle. Secondly, I will discuss how mixed states can be represented using dynamically disentangled purified states. These novel methods allow to extract transport coefficients efficiently.

A fat-tree tensor network method is combined with a variational Monte Carlo method with a pair-product function, which show better accuracy and efficiency than each method. We show state-of-the-art accuracy for the challenging issue of the competing order between superconductivity and other orders in strongly correlated electron systems.

In this talk I will use the path integral associated to a cMERA tensor network to provide an operational definition for the complexity of a cMERA circuit/state which is relevant to investigate the complexity of states in quantum field theory. In this framework, it is possible to explicitly establish the correspondence (Minimal) Complexity = (Least) Action. I will show that the cMERA complexity action functional can be seen as the action of a Liouville field theory, thus showing connections with two dimensional quantum gravity. Concretely, the Liouville mode is identified with the variational parameter defining the cMERA circuit. The rate of complexity growth along the cMERA renormalization group flow is obtained and shown to saturate limits which are in close resemblance to the fundamental bounds to the speed of evolution in unitary quantum dynamics, known as quantum speed limits. Finally, if time allows, I will show that the complexity of a cMERA circuit measured through these complexity functionals, can be casted in terms of the variationally-optimized amount of left-right entanglement created along the cMERA renormalization flow. These results suggest that the patterns of entanglement in states of a QFT could determine their dual gravitational descriptions through a principle of least complexity.

Thermal properties of systems are believed to emerge differently in classical and quantum systems. Classical systems become thermal due to their chaotic nature leading to ergodic exploration of their phase space. In quantum systems the eigenstate thermalization hypothesis suggests that dephasing between eigenstates gradually reveals their underlying thermal properties. A more unified picture can be obtained by studying quantum dynamics projected onto classical Hamiltonian mechanics using the time-dependent variational principle. I will explore how the chaotic behaviour of the projected dynamics - characterised by its Lyapunov spectrum - provides an alternative perspective on thermalizing quantum systems.

Physical systems differing in their microscopic details often display strikingly similar behaviour when probed at macroscopic scales. Those universal properties, largely determining their physical characteristics, are revealed by the Renormalization Group (RG) procedure, which systematically retains “slow” degrees of freedom and integrates out the rest. However, given a new microscopic model these slow degrees of freedom may be difficult to identify. In this talk I'll demonstrate a machine learning (ML) algorithm capable of identifying the relevant degrees of freedom and executing RG steps iteratively without any prior knowledge about the model. The algorithm is based on a model-independent, information-theoretic characterisation of a real-space RG procedure. We apply the algorithm to several classical statistical physics problems in particular the Ising model in 1D and 2D, and dimers on a square lattice and show that it indeed extract the correct slow degrees of freedom. For the 2D Ising model we further demonstrate an RG flow and extract the critical exponent. Time permits I'll point out that standard training of deep neural networks does not lead to a neural network which performs RG. This in contrast to prior claims in the literature.

The technological success of machine learning techniques has motivated a research area in the condensed matter physics and quantum information communities, where new tools and conceptual connections between machine learning and many-body physics are rapidly developing. In this talk, I will discuss the use of generative models for learning quantum states. In particular, I will discuss a strategy for learning mixed states through a combination of informationally complete positive-operator valued measures and generative models. In this setting, generative models enable accurate learning of prototypical quantum states of large size directly from measurements mimicking experimental data.

In very short time artificial neural networks have achieved impressive results when tasked with calculating phase diagrams. These algorithms are mainly considered as black box algorithms. Hence, we cannot trust the results of artificial neural networks blindly. Here, we discuss how to interpret neural networks when they are tasked with classifying phases in a supervised manner. Further, we introduce an unsupervised neural network, the so called (variational) autoencoder, which can be interpreted naturally.