Quantum mechanics is one of the biggest breakthroughs in the history of physics. At the beginning of the 20th century its founding fathers like Planck, Bohr, De Broglie, Schrodinger, Heisenberg and Dirac were not only confronted with unexpected conceptual difficulties but also had to introduce new mathematical techniques in the area of physics to get their head around this revolutionary new way of thinking. But successful applications of the theory in for example the hydrogen atom showed that quantum mechanics was the way to go.

Today, almost a century later, physicists are still struggling to fully understand the implications of quantum mechanics. Especially in systems with many degrees of freedom it is extremely challenging to calculate the predictions made by this theory. The reason for this is that the complexity of the problem escalates very quickly with number of degrees of freedom, even exponentially to be precise.

In recent years insights from quantum information theory were used to tackle the quantum many-body problem. It was realized that in many cases the locality of interactions has a profound influence on the entanglement structure of the ground and other low energy states. Out of this arose the so called ‘tensor network states’, which are an ansatz for the ground state of the many-body system that capture the relevant entanglement structure. Tensor network states allow for a more efficient use of time and resources since one no longer has to look for the correct answer in the entire space of possible states, restricting to the physical subspace of tensor network states is sufficient. The local structure of these states also provide ways to get a deeper understanding of generic emergent phenomena in quantum many-body systems and their connection to entanglement.

Tensor network states are finding their way in an increasing number of fields in physics, ranging from condensed matter theory to quantum gravity. In the Ghent university quantum group of Prof. Verstraete we are at the forefront of many of these applications, with an expertise in both the theoretical and numerical aspects.

Below we list the topics of active research in our quantum group and provide additional details.

#### Quantum phases of matter

One of the main goals of condensed matter physics is the determination of all possible phases of matter. It was believed that, with a few notorious exceptions, the Landau paradigm of phase transitions was sufficient to understand the different possible phases of matter that exist in nature. Apart from the usual classical phase transitions, which occur at a certain critical temperature, there exist phase transitions between different quantum phases that persist even at zero temperature. They are driven by quantum instead of thermal fluctuations. Several of these quantum phases and phase transitions can still be understood in the Landau framework.

However, recently it has become clear that there exists a plethora of quantum phases that cannot be distinguished by using just local order parameters. If we require certain symmetries to be respected, we see that the phase diagram of matter becomes more complicated. Indeed, it turns out that there are so called symmetry protected topologically (SPT) ordered phases that are in the same phase as the trivial system if we allow all kind of perturbations. That is, we can smoothly interpolate between an SPT state and a completely trivial state without crossing a phase transition. However, once we impose a certain symmetry on the allowed paths, we cannot interpolate without crossing a phase transition. We see that these systems are in a different phase than the trivial system. Possible probes to classify these systems are their defects or edge properties. Well known examples of SPT phases are the Haldane phase in bosonic 1D spin chains and free fermionic topological insulators. It turns out that with the help of tensor networks the bosonic SPT systems can be easily understood and classified. The edge or defect properties appear as some traits of just a single local tensor. Hence, given a tensor network description of a quantum state, the determination of its SPT phase is readily accomplished. At least for 1D spin systems, a complete classification of all possible phases is known and given by the different SPT phases. In more dimensions, a lot is already known about SPT, although no full classification is yet obtained. Remarkably, in higher dimensions, the story of phases of matter is still way more complicated than just the different SPT phases.

The most famous example of a class of phases that cannot be distinguished by Landau’s symmetry arguments is the illustrious fractional quantum hall effect. Different phases in such systems can only be distinguished by their topological properties, or by measuring extended operators. In the case of the fractional quantum hall effect this role is played by the quantised hall conductance. However, no local measurement in the bulk of such a system can discriminate between the different phases. Furthermore, these states even fall beyond the paradigm of SPT states, as no symmetry is required to differentiate them from a trivial system. Such systems are called (intrinsically) topologically ordered. Since the discovery of the fractional quantum hall effect, a wealth of other examples of topological phases phases, both fermionic as bosonic, have been proposed and in exceptional cases also realised in experiments. All these phases share a certain amount of properties, the combination of which (or a subset of them) qualifies a system to be topologically ordered. These topologically ordered systems are to be contrasted with the usual topologically trivial systems, to which (at least in the absence of symmetries) Landau’s classification principle applies. The properties that are most commonly associated with intrinsically topological ordered Hamiltonians are the following. The Hamiltonian is assumed to be gapped, this is important for the stability of the phase. Next, if we put the system on a manifold, for instance a torus, the Hamiltonian has a ground state degeneracy that only depends on the topological properties of the manifold. Locally, these states are indistinguishable, as there is no order parameter. Furthermore, the elementary excitations of the Hamiltonian are anyons, special quasiparticles with very exotic statistics. Anyons only exist in two spatial dimensions, at least if we want them to be particles and not extended objects. For this reason, the study of topologically ordered systems is most naturally pursued in two spatial dimensions.

Apart from the previously mentioned macroscopic properties of topological order, we now have a microscopic understanding of its origins. The contemporary framework used to describe topological order is heavily influenced by quantum information theory. We now understand that the crucial feature of topologically ordered states is the existence of long range quantum correlations, or entanglement. The connection between topological order and quantum information is twofold. On one hand, one of the most interesting aspects of topologically ordered states are exactly their possible applications in quantum information theory. The lack of an order parameter implies that such states are protected from local perturbations. When constructed on topologically non trivial manifolds, the ground state space can be used to store quantum information. Moreover, the exotic anyonic statistics of the quasiparticle excitations can be used to perform intrinsically fault-tolerant quantum computation. On the other hand, the appearance of entanglement makes it hard to understand such quantum states by conventional many body methods such as mean field theory. Moreover, it is not easy to give an efficient description of highly entangled states, as usually an exponential number of parameters is required.

Luckily, tensor networks provide an efficient, approximate or exact, description of several interesting classes of topologically ordered states. We now know that the bosonic topologically ordered states in 2D, which are hoped to be useful for quantum computation, can be very efficiently represented by a certain class of tensor networks, the Projected Entangled Pair States (PEPS). Such a representation allows us to describe these states not only on fixed points, but also away from these idealised situations. Furthermore, once the PEPS description of the ground states of a topologically ordered system are determined, we can use these tensors to find all the elementary topological sectors of the system, that is the anyons, and determine the fundamental anyonic statistics of the system. Importantly, the PEPS description also makes it possible to study these systems numerically, we can for instance use tensor network methods to determine phase transitions and dispersion relations. The crucial insight that is required for all these accomplishments is that all relevant information is contained in the symmetry properties of a local tensor. As always, the true power of the tensor network methods is that even topological properties, undetectable by local operators, can easily be determined from the traits of just one local tensor.

At this point in time, we can say that we understand the phase diagram of quantum matter in 1D, this is given by the SPT systems. In 2D, we already know a lot of both SPT and intrinsically topologically ordered systems and although a complete understanding is still beyond our reach, progress is made every day. There are, however, two main open questions. One is the interplay between SPT and intrinsic topological order, the so called symmetry enriched topological phases. This complicates the phase diagram of quantum matter even more, only preliminary steps have been taken in this direction. The same is true for the second big open question, the phase diagram in higher dimensions. Not a lot is known in 3D and beyond, but it is clear that tensor networks can play an important role in the understanding of these systems.

#### Effective particles in quantum spin systems

It is one of the central insights of condensed-matter physics that the low-energy properties of a (quantum) material are not so much determined by the ground state, but rather by the elementary excitations. Furthermore, in contrast to the ground state, which is typically of a highly non-trivial strongly-correlated nature, these excitations can often be described in a simple way. In most cases, they can be pictured as localized and weakly-interacting particles with a non-extensive energy, living on this strongly-correlated background state. In this regard, understanding the physical properties of large quantum systems – dynamical correlation functions, phase transitions, real-time evolution, low-temperature behaviour, etc. – amounts to finding the effective theory describing the dynamics of these particles.

Because strongly-correlated ground states remain inaccessible for an accurate description, finding this effective theory is typically achieved in a perturbative way. The paradigmatic example is that of the Fermi-liquid theory for describing interacting electrons, where the effective particles are first defined in the non-interacting limit. It appears that these particles survive the turning on of the interactions, although they get a finite lifetime. The effective properties of the “quasi-particles” – effective mass and charge, lifetime, scattering processes, etc. – can be determined through perturbative techniques that do not explicitly take into account the background state on which they live (i.e. the ground state of the interacting system).

In the setting of low-dimensional quantum systems, this situation has changed somewhat with the advent of tensor-network states as a variational description for strongly-correlated ground states. Indeed, having access to an accurate ground-state wavefunction opens up the possibility of constructing the effective particles directly in the interacting case, without relying on some trivial limit. It is not clear, however, that these particles should still have a simple, particle-like description. That such a local description remains possible has been proven rigorously for the case of gapped quantum spin systems, a result which seems to be the non-relativistic analog of similar results in axiomatic quantum field theory. Furthermore, the tensor-network structure of a ground state wavefunction provides a way to locally modify it and create a simple variational ansatz for constructing “particles” on a non-trivial vacuum state. Despite its simplicity – change one tensor in the tensor network state and make a momentum superposition – this ansatz has proven to capture the elementary excitations to an unprecedented precision. Indeed, with this variational approach, the exact eigenstates of the fully interacting Hamiltonian are directly targetted, such that these particles have an infinite lifetime.

In one dimension, where matrix product states (MPS) provide an extremely efficient framework for simulating low-energy physics, this particle picture of elementary excitations has been further developed towards the study of the particles’ interactions. More specifically, it was shown how to construct two-particle eigenstates variationally by solving the corresponding scattering problem and how to compute the two-particle S matrix. In addition, the variational expression of the two-particle wavefunction allows for the computation of the low-energy part of different spectral functions.

Moreover, having full access to the two-particle S matrix allows for the construction of an “approximate Bethe ansatz” to describe a finite density of particle excitations on top of an MPS background state. This approach towards an effective particle description assumes that, at small densities, a gas of effective particles can be taken to be integrable (i.e. no diffractive or three-particle scattering processes), such that the Bethe ansatz wavefunction provides a good description for it. It was shown that e.g. magnetization processes of SU(2) invariant spin chains can be captured accurately with this approach.

In two dimensions, the formalism of projected entangled-pair states is computationally more demanding. Recently, through the introduction of a new contraction scheme, it has proven possible to compute the necessary overlaps and to variationally determine excitation spectra of two-dimensional quantum spin systems.

**References**

J. Haegeman, B. Pirvu, D. J. Weir, J. I. Cirac, T. J. Osborne, H. Verschelde, and F. Verstraete, “Variational matrix product ansatz for dispersion relations,” PRB 85, 100408 (2012)

L. Vanderstraeten, J. Haegeman, T. J. Osborne, and F. Verstraete, “S-matrix from matrix product states,” PRL 112, 257202 (2014)

L. Vanderstraeten, J. Haegeman, and F. Verstraete, “Scattering particles in quantum spin chains,” ArXiv 1506.01008 (2015)

L. Vanderstraeten, F. Verstraete, and J. Haegeman, “Excitations and the tangent space of projected entangled-pair states,” ArXiv 1507.02151 (2015)

#### Quantum computing

#### Variational optimization on tensor network manifolds

When using tensor network states as a numerical tool, one of the first task is often to find the best ground state approximation of a given system (Hamiltonian) as a tensor network state. The variational principle dictates that a good approximation can be obtained by minizing the energy expectation value over the set of states within a given class (e.g. the set of tensor networks with a specified network structure and given virtual dimensions). This amounts to solving an optimization problem over a set of tensors which encode (typically) a smooth submanifold of the full Hilbert space.

In principle, the ground state only gives access to static information (though its encoding into a tensor network state seems to also contain qualitative information about the dynamics). To obtain a complete understanding of the physics of quantum many body systems, we also want to capture dynamical information, such as non-equilibrium evolution and quantum quenches, excited states and spectral functions at both zero and finite temperature. The framework of tensor networks is sufficiently versatile to model most of these situations and to provide an approximate description of the relevant physics. For all of these, the best approximation is typically obtained as solution of some kind of optimization problem.

The development of new and improved algorithms to solve these challenging optimization problems is an ongoing research topic within our group. While all of these methods start from a good physical intuition, they require to combine insights and ideas from various fields:

- optimization theory: least square methods, gradient methods, …
- linear algebra: matrix factorizations, iterative linear solvers and eigensolvers, low rank approximations, …
- numerical differential equations: non-linear ODEs and PDEs, implicit schemes, …
- differential geometry: tangent spaces, connections, curvature, …
- computer architecture: parallellism, memory layout, …

While the case of one-dimensional lattice systems is well established over the last 20+ years (starting with the development of the famous Density Matrix Renormalization Group of Steve White in 1992) , there are certainly many open questions and room for improvement for the case of higher-dimensional lattice systems (using tensor networks such as projected entangled-pair states or the multiscale entanglement renormalization ansatz) or for the continuous formulation of tensor network states in order to study quantum field theories.

#### Quantum field theory and lattice gauge theories

#### Entanglement, holography and quantum gravity

Since the revolutionary discovery of the AdS/CFT correspondence in late 1997, a substantial amount of work has been devoted to clarifying the ramifications and subtle intricacies of the holographic principle. The correspondence highlights a nontrival mapping between the physics of a gravitational theory in the bulk of a spacetime manifold and a boundary quantum field theory living on its boundary. Although initially conceived in the context of string theory, its concepts and insights have reverberated throughout all of physics, leading to novel approaches and applications both in condensed matter physics and high-energy physics. Holography attempts to provide a unified description of gravity and field theoretic degrees of freedom, and allows us to translate highly intractable questions on one side of the duality into fairly straightforward calculations on the other side, which in many cases provides fruitful intuition on the physics inherent to both sides of the duality. Today, almost twenty years later, there is no shortage of overwhelming evidence in support of the conjecture, yet we are nowhere near a profound understanding of the inner workings of holography.

It is important to stress the ubiquitous influence of black holes during both the initial discovery of the AdS/CFT correspondence, and in its subsequent developments continuing up to this day. Black holes are unique objects in nature in so much as they force us to reconcile our ideas on gravity, which is described geometrically by Einstein’s general relativity, and the microscopic realm, where quantum mechanics guides our thought. Aside from technical difficulties, attempts at a satisfactory description of quantum gravity are plagued by paradoxes and inconsistencies, which force us to sharply assess every silent assumption put forward in familiar physics. Taking black hole thermodynamics seriously, it has been shown in the last decade that entanglement entropy in holographic systems encodes geometric features of the bulk geometry, in turn inspiring a flourishing interest in the connections between spacetime en entanglement.

Because entanglement entropy measures how quantum information is organized in a quantum state, and is, as such, an extremely complicated quantity to calculate in general quantum field theories, the mere fact that its calculation simplifies so much in holographic conformal field theories, even to the point of becoming geometric, reveals a deep property of strongly-coupled systems. From this non-perturbative vantage point, a lot of information in a quantum state seems to be redundant, rendering it amenable to a description in terms of suitably designed tensor networks, which provide an intrinsically non-perturbative description of quantum many-body systems. Recent work already suggests a refined proposal for spacetime entanglement that can be markedly different from the usual perturbative viewpoint, and proposes enticing connections to quantum error-correcting codes.

Tensor networks could thus serve as a promising tool to embark on answering important open questions in holography. What quantum states have good geometric duals? To what extent is this geometry determined by information theoretic and entropic quantities of (conformal) field theories? What information about the state is truly necessary for its non-perturbative description? Does there exist a holographic interpretation of the renormalization group in terms of an information theoretic coarse-graining scheme? What is the meaning of area in spacetime and the relation to black hole entropy? As tensor networks always remain close to numerics, they provide an ideal playground for the analytical and numerical study of toy models of holographic theories, forging ever tighter bonds between quantum information and black hole physics.