Multiscale Entanglement Renormalisation Ansatz

This thesis reviews the multiscale entanglement renormalisation ansatz or MERA, a numerical tool for the study of quantum many-body systems and a discrete realisation of the AdS/CFT duality. The thesis covers an introduction to the necessary background concepts of entanglement, entanglement entropy and tensor network states, the structure and main features of MERA and its applications in condensed matter theory and holography. Also covered are details on the algorithmic implementation of MERA and some of its generalisations and extensions. MERA belongs to a class of variational ansatze for quantum many-body states known as tensor network states. It is especially well-suited for the study of scale invariant critical points. MERA is based on a real-space renormalisation group procedure called entanglement renormalisation, designed to systematically handle entanglement at different length scales along the coarse-graining flow. Entanglement renormalisation has be used for example to efficiently describe Kitaev states of the toric code, the prime example of topological order, and numerically study the ground state of the highly frustrated spin-1/2 Heisenberg model on a kagome lattice and various other one- and two-dimensional lattice models. The geometric and causal structure of MERA, which underlies its effectiveness as a numerical tool, also makes it a discrete version of the AdS/CFT duality. This duality describes a conformal field theory by a gravity theory in a higher dimensional space, and vice versa. The duality is manifest in the scaling of entanglement entropy in MERA, which is governed by a law highly analogous to the Ryu-Takayanagi formula for holographic entanglement entropy, in the connection between thermal states and a black-hole-like MERA and in the connection between correlation functions and holographic geodesics in a scale invariant MERA. The aim of this thesis is to lead the reader to an understanding of what MERA is, how it works and how it can be used. MERA's core features and uses are presented in a comprehensive and explicit way, and a broad view of possible applications and further directions is given. Plenty of references are also offered to direct the reader to further research on how MERA may relate to his/her interests.MERA, eli multiscale entanglement renormalisation ansatz, on numeerinen menetelmä monen kappaleen kvanttimekaniikan tutkimiseen sekä diskreetti todentuma AdS/CFT-dualiteetista. Tämä tutkielma on yleiskatsaus MERAan. Se käy läpi tarvittavat pohjatiedot lomittumisesta, lomittumisentropiasta ja tensoriverkkotiloista, kuvailee yksityiskohtaisesti MERAn rakenteen ja tärkeimmät ominaisuudet ja esittelee sen käyttömahdollisuuksia tiiviin aineen teorian ja holografian tutkimuksessa. Lisäksi tutkielma käsittelee MERAn toteutuksen numeerisena algoritmina sekä joitain sen yleistyksiä ja laajennuksia. MERA kuuluu niin sanottuihin tensoriverkkotiloihin, jotka ovat yritteitä monen kappaleen kvanttitiloille. Se on suunniteltu soveltumaan erityisen hyvin skaalainvarianttien kriittisten pisteiden kuvaamiseen. MERA pohjautuu lomittumisrenormalisaatioon (entanglement renormalisation), renormalisaatioryhmäprosessiin, joka on suunniteltu ottamaan huomioon lomittuminen karkeistamisprosessin eri pituusskaaloilla. Sitä on käytetty muun muassa toruskoodin (toric code) Kitaev-tilojen - topologisen järjestyksen malliesimerkin - kuvaamiseen sekä vahvasti turhautuneen kagomehilan kehre-1/2-Heisenbergin mallin ja monien muiden yksi- ja kaksiulotteisten hilamallien perustilojen etsimiseen. MERAn geometrinen ja kausaalinen rakenne, jonka varaan sen tehokkuus numeerisena yritteenä perustuu, tekee siitä myös diskreetin todentuman AdS/CFT-dualiteetista. AdS/CFT-dualiteetti kuvaa konformin kenttäteorian gravitaatioteoriaksi korkeampiulotteisessa avaruudessa ja päinvastoin. Tämä dualiteetti ilmenee skaalainvariantissa MERAssa useilla tavoilla: MERAssa lomittumisentropia skaalautuu holografisesti niin kutsutun Ryu-Takayanagi-yhtälön mukaan, termiset tilat kuvautuvat mustaa aukkoa muistuttavaksi MERAksi ja korrelaatiofunktiot MERAssa riippuvat holografisista geodeeseista tensoriverkon halki. Tutkielman tarkoituksena on johdattaa lukija ymmärtämään, mikä MERA on ja kuinka se toimii, esittää kattavasti ja yksityiskohtaisesti sen keskeisimmät ominaisuudet ja käyttötarkoitukset sekä antaa laaja yleiskuva sen sovelluskohteista. Matkan varrella tutkielma pyrkii jakamaan runsaasti viitteitä, joita seuraamalla lukija voi löytää lisätietoa siitä, miten MERA liittyy hänen kiinnostuksensa kohteisiin

Tensor Networks and the Renormalization Group

Tensor networks are a class of methods for studying many-body systems. They give a geometrical description of the internal structure of many-body states, operators, and partition functions, that can be used to implement efficient algorithms to simulate them numerically. In this thesis, after providing a very brief overview of the field, we present new tensor network methods for three different use cases. First, we present a new real-space renormalization group algorithm for tensor networks. Compared to existing methods, its advantages are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it on the 2D classical Ising model and find accuracy comparable with the best existing tensor network methods. Due to its simplicity and generalizability, we consider our algorithm to be an excellent candidate for implementation of real-space renormalization in higher dimensions, and discuss some of the details and the remaining challenges in 3D. Second, we show how certain topological conformal defects of critical lattice theories can be represented as tensor networks, using the 2D classical Ising model as an example. Furthermore, we show how such tensor network descriptions, combined with a renormalization group algorithm, can be used to obtain accurate estimates of the universal properties of these defects. We also show how coarse-graining of defects can be applied to any conformal defect (i.e. not just topological ones), and yields a set of associated scaling dimensions. Finally, we leave behind the focus on renormalization group methods, and present a method for spatially resolving the overlap between two tensor network states, to obtain localized information about the similarities and differences between them. For a given region, the similarity of two states in this region can be quantified by the Uhlmann fidelity of their reduced density matrices, and we show how such fidelities can be efficiently computed in many cases when the two states are represented as tensor networks. We demonstrate the usefulness of evaluating such subsystem fidelities with three example applications: studying local quenches, comparing critical and non-critical states, and quantifying convergence in tensor network simulations

Riemannian optimization of isometric tensor networks

Several tensor networks are built of isometric tensors, i.e. tensors satisfying $W^\dagger W = \mathrm{I}$. Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers. We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g. ground states of 1D quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context. We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes. We also provide open-source implementations of our algorithms.Comment: 18 pages + appendices, 3 figures; v3 submission to SciPost; v4 expand preconditioning discussion and add polish, resubmit to SciPos

Entanglement compression in scale space: from the multiscale entanglement renormalization ansatz to matrix product operators

The multiscale entanglement renormalization ansatz (MERA) provides a constructive algorithm for realizing wavefunctions that are inherently scale invariant. Unlike conformally invariant partition functions however, the finite bond dimension $\chi$ of the MERA provides a cut-off in the fields that can be realized. In this letter, we demonstrate that this cut-off is equivalent to the one obtained when approximating a thermal state of a critical Hamiltonian with a matrix product operator (MPO) of finite bond dimension $\chi$. This is achieved by constructing an explicit mapping between the isometries of a MERA and the local tensors of the MPO.Comment: 5 page

Entanglement compression in scale space : from the multiscale entanglement renormalization ansatz to matrix product operators

The multiscale entanglement renormalization ansatz (MERA) provides a constructive algorithm for realizing wave functions that are inherently scale invariant. Unlike conformally invariant partition functions, however, the finite bond dimension chi of the MERA provides a cutoff in the fields that can be realized. In this paper, we demonstrate that this cutoff is equivalent to the one obtained when approximating a thermal state of a critical Hamiltonian with a matrix product operator (MPO) of finite bond dimension chi. This is achieved by constructing an explicit mapping between the isometrics of the MERA and the local tensors of the MPO. In terms of energy scales, our results show that a finite bond dimension MERA is equivalent to introducing both an infrared and an ultraviolet scale, characterizing relevant and irrelevant perturbations on the underlying conformal field theory

Dynamics of Transmon Ionization

International audienceQubit measurement and control in circuit quantum electrodynamics (QED) rely on microwave drives, with higher drive amplitudes ideally leading to faster processes. However, degradation in qubit coherence time and readout fidelity has been observed even under moderate drive amplitudes corresponding to a few photons populating the measurement resonator. Here, we numerically explore the dynamics of a driven transmon-resonator system under strong and nearly resonant measurement drives and find clear signatures of transmon ionization where the qubit escapes out of its cosine potential. Using a semiclassical model, we interpret this ionization as resulting from resonances occurring at specific resonator-photon populations. We find that the photon populations at which these spurious transitions occur are strongly parameter dependent and that they can occur at low resonator-photon population, something that may explain the experimentally observed degradation in measurement fidelity