## Research

I am interested in problems at the interface of **quantum information science**,** condensed matter physics** and **computational physics**.

My main focus now is the development of new numerical simulation techniques for quantum many-body systems using tensor networks, and the investigation of the entanglement properties and the physics of strongly correlated systems. This embraces important topics such as topological quantum order, useful phases of matter for quantum computation, development of new computer algorithms, and new properties of many-body entanglement.

Below you can find a quick summary of the main idea behind these topics, together with a selection of some (recent or not) research highlights along these lines.

**Tensor Networks**

A number of results in the field of quantum information science have lead to new and fresh ideas to attack old, long-standing problems in condensed matter physics. The study of quantum correlations (or entanglement) has motivated the development of a host of new promising methods to numerically simulate the physics of quantum many-body systems: the so-called tensor network methods. These techniques are based on representations of quantum states for quantum lattice systems in terms of networks of interconnected tensors and, contrary to other approaches, they capture the amount and structure of entanglement in the system. In fact, tensor networks are also the natural language to describe new exotic quantum phenomena such as quantum spin liquids and topologically-ordered states of matter.

In my research I develop and use tensor networks to describe the properties of a variety of systems, both analytically and numerically. From the numerical perspective, one first develops new algorithms based on this approach which make use of tensor networks such as Matrix Product States (MPS) for 1d systems and Projected Entangled Pair States (PEPS) for 2d systems. Then, one uses the developed methods to study physical systems of interest, such as the ones mentioned above.

* A Projected Entangled Pair State (PEPS), for a 3 x 3 square lattice of p-level particles.*

**Topological Order**

Topological order is an example of new physics beyond Landau’s symmetry-breaking paradigm of phase transitions. Systems exhibiting this new kind of order are linked to concepts of the deepest physical interest, e.g. quasiparticle anyonic statistics, topological quantum computation, lattice gauge theories, and the emergence of light and fermions through string-net condensation. Importantly, topological order finds a realization in terms of topological quantum field theories, which are the low-energy limit of quantum lattice models such as the Toric Code and quantum double models, as well as string-net models. A remarkable property about topological order is that it influences the long-range pattern of entanglement in the wave function of the system.

* Two topologically-different objects of genus one (torus) and two. Ground spaces of topologically-ordered systems defined on their surfaces have different degeneracies.*

**Entanglement in Quantum Many-Body Systems**

The study of many-particle entanglement plays a key role in order to better understand quantum phases of matter. In particular, its scaling and universality properties (e.g. the so-called *area-law* scaling of the entanglement entropy) are of fundamental interest to describe the inner structure of low-energy states of many-body Hamiltonians with local interactions. This allows to identify the relevant corner of quantum states in the many-body Hilbert space, which in turn, is also the supporting theoretical framework needed for tensor network simulation methods.

As an example, the so-called *entanglement spectrum*, originally proposed by Li and Haldane, aims to study the distribution of eigenvalues of the reduced density matrices of the system. Importantly, it has been realized that such a spectrum is directly related to the energy spectrum of quantum theories defined at the boundaries of the system, thus providing an explicit realization of the holographic principle in the context of condensed matter systems. Moreover, the analysis of the entanglement spectrum of 1d systems has also allowed to characterize a variety of quantum phases of matter (e.g. the so-called Haldane phases).

In a similar way, the so-called *geometric entanglement *has been recently proposed as a powerful tool to understand global (as opposed to bipartite) quantum correlations in many-body systems, and thus characterize exotic states of matter. This quantity aims to quantify the distance in the Hilbert space between the quantum state of the system and its closest product state with some given structure. A plethora of results have shown that this quantity obeys important scaling laws both in 1d and 2d systems, and is also able to account for the existence of topological order.

* (a) Bipartite versus (b) multipartite entanglement scenarios for 2d blocks of length L. *

### Phases of Matter for Quantum Computation

What defines universal computationability of a quantum computer? Or in other words, when can a device be used to perform universal quantum computation under practical assumptions? Certainly, this question is of paramount importance if we aim to develop large-scale quantum algorithms some day. A way to address this problem is through the exciting approach of measurement-based quantum computation (MBQC) introduced by Raussendorf and Briegel. In this setting, a quantum computation is performed by implementing local measurements on a highly-entangled quantum state, known as cluster state. As such, MBQC is a universal model of quantum computation. Therefore, it is relevant to know how much can one perturb or alter the cluster state before it can no longer be used as a resource for quantum computation. If we see the cluster state as the ground state of a many-body system, then the relevant question is whether the ground state of the perturbed or altered system is still qualified for quantum computation or not. Or, to put it differently: is it possible to find the full computational power of the cluster state in other quantum states of matter? A different perspective to this problem is also offered by topological quantum computation. In this setting, the relevant question is: which quantum phases of matter support anyons that can be used for topological quantum computation, and how robust and generic these phases are?

### Some research highlights

** Kitaev honeycomb tensor networks: exact unitary circuits and applications*. P. Schmoll, R. Orus [e-print:arXiv:1605.04315]

The Kitaev honeycomb model is a paradigm of exactly-solvable models, showing non-trivial physical properties such as topological quantum order, abelian and non-abelian anyons, and chirality. Its solution is one of the most beautiful examples of the interplay of different mathematical techniques in condensed matter physics. In this project we show how to derive a tensor network (TN) description of the eigenstates of this spin-1/2 model in the thermodynamic limit, and in particular for its ground state. In our setting, eigenstates are naturally encoded by an exact 3d TN structure made of fermionic unitary operators, corresponding to the unitary quantum circuit building up the many-body quantum state. In our derivation we review how the different “solution ingredients” of the Kitaev honeycomb model can be accounted for in the TN language, namely: Jordan-Wigner transformation, braidings of Majorana modes, fermionic Fourier transformation, and Bogoliubov transformation. The TN built in this way allows for a clear understanding of several properties of the model. In particular, we show how the fidelity diagram is straightforward both at zero temperature and at finite temperature in the vortex-free sector. Finally, we also discuss the pros and cons of contracting of our 3d TN down to a 2d Projected Entangled Pair State (PEPS) with finite bond dimension. Our results can be extended to generalizations of the Kitaev model, e.g., to other lattices, spins, and dimensions.

** Spin-S Kagome quantum antiferromagnets in a field with tensor networks*. T. Picot, M. Ziegler, D. Poilblanc, R. Orus [e-print:arXiv:1508.07189]

Spin-S Heisenberg quantum antiferromagnets on the Kagome lattice offer, when placed in a magnetic field, a fantastic playground to observe exotic phases of matter with (magnetic analogs of) superfluid, charge, bond or nematic orders, or a coexistence of several of the latter. In this context, we have obtained the (zero temperature) phase diagrams up to S=2 directly in the thermodynamic limit thanks to infinite Projected Entangled Pair States (iPEPS). We find incompressible phases characterized by a magnetization plateau vs field and stabilized by spontaneous breaking of point group or lattice translation symmetry(ies). The nature of such phases may be semi-classical or fully quantum. We also discuss the performance of our algorithm for the S=1/2 case at zero field, for which we provide a variational upper bound to the ground state energy comparable to previous results.

** Symmetry-protected intermediate trivial phases in quantum spin chains*. A. Kshetrimayum, H.-Hao Tu, R. Orus [e-print:arXiv:1511.06338]

Symmetry-protected trivial (SPt) phases of matter are the product-state analogue of symmetry-protected topological (SPT) phases. This means, SPt phases can be adiabatically connected to a product state by some path that preserves the protecting symmetry. Moreover, SPt and SPT phases can be adiabatically connected to each other when interaction terms that break the symmetries protecting the SPT order are added in the Hamiltonian. It is also known that spin-1 SPT phases in quantum spin chains can emerge as effective intermediate phases of spin-2 Hamiltonians. Here we show that a similar scenario is also valid for SPt phases. More precisely, we show that for a given spin-2 quantum chain, effective intermediate spin-1 SPt phases emerge in some regions of the phase diagram, these also being adiabatically connected to non-trivial intermediate SPT phases. We characterize the phase diagram of our model by studying quantities such as the entanglement entropy, symmetry-related order parameters, and 1-site fidelities. Our numerical analysis uses Matrix Product States (MPS) and the infinite Time-Evolving Block Decimation (iTEBD) method to approximate ground states of the system in the thermodynamic limit. Moreover, we provide a field theory description of the quantum phase transitions between the SPt phases. Together with the numerical results, such a description shows that the transitions can be described by Conformal Field Theories (CFT) with central charge c=1.

** The iPEPS algorithm, improved: fast full update and gauge fixing. *H. N. Phien, J. A. Bengua, H. D. Tuan, P. Corboz, R. Orus [e-print:arXiv:1503.05345]

Here we discuss how to improve the efficiency, stability and accuracy of the iPEPS algorithm for finding ground states of infinite-size 2d quantum lattice systems. For this, we first introduce the fast full update scheme, where effective environment and iPEPS tensors are both simultaneously updated (or evolved) throughout time. We show that this implies two crucial advantages: (i) dramatic computational savings with essentially no loss of accuracy, and (ii) improved overall stability. Besides, we extend the application of the local gauge fixing, successfully implemented for finite-size PEPS [M. Lubasch, J. Ignacio Cirac, M.-C. Bañuls, PRB 90, 064425 (2014)], to the iPEPS algorithm. The overall improved algorithm is remarkably stable, as well as 1-2 orders of magnitude faster than the old version.

** **Intermediate Haldane phase in spin-2 quantum chains with uniaxial anisotropy.* H.-Hao Tu, R. Orus, Physical Review B **84**, 140407(R) (2011) [eprint:arXiv:1107.2911].

* *All spin-1 topological phases in a single spin-2 chain.* A. Kshetrimayum, H.-Hao Tu, R. Orus [e-print:arXiv:1412.3370]

In these two papers we tackle the problem of identifying effective spin-1 intermediate Haldane (IH) phases in spin-2 quantum chains. The presence of such phases was conjectured by Oshikawa in 1992, and only elucitated very recently. In the first paper we propose a spin-2 model where, contrary to other approaches, we find a very large IH phase, as well as SO(5)-Haldane and polarized phases. Our study relies on calculations of degeneracies in the entanglement spectrum, and string-order parameters.

In the second paper, we show that in the same model one can actually find all the possible spin-1 symmetry-protected topological (SPT) phases protected by (Z2 x Z2) + T symmetry, called T0, Tx, Ty and Tz phases. We also find evidence for a gapless Ty SPT phase, as well as quantum critical points with central charge c=2 betweeen the SPT phases. An effective field theory in terms of four massless majorana fermions is proposed to describe these transisions. In this last paper we also compute SPT order parameters directly from the MPS, signaling the exact irreducible representation of the symmetry group protecting the different phases.

** Geometric Entanglement in Topologically Ordered States**. *R. Orus, T.-Chieh Wei, O. Buerschaper, M. van den Nest, New Journal of Physics **16**, 013015 (2014) [e-print:arXiv:1304.1339]

** Topological transitions from multipartite entanglement with tensor networks: sharper and faster.* R. Orus, T.-Chieh Wei, O. Buerschaper, A. Garcia-Saez, Physical Review Letters 113, 257202 (2014) [e-print:arXiv:1406.0585]

** Topological Minimally Entangled States via Geometric Measure.* O. Buerschaper, A. Garcia-Saez, R. Orus, T.-C. Wei, JSTAT P110099 (2014) [e-print:arXiv:1410.0484]

These three papers are different steps towards solving the following question: given a 2d quantum lattice Hamiltonian, how can we fully characterize its emergent topological properties at low energies as accuartely and efficiently as possible? Our approach to solve this problem is to use PEPS to characterize topological ground states, and a topological measure of entanglement which we call “topological geometric entanglement”.

In the first paper we prove that the geometric entanglement of blocks, which measures the distance in the Hilbert space between a given quantum state and the closest product state of blocks, has a topological correction for topologically-ordered systems. This is similar to the topological correction of the entanglement entropy of a block. We provide evidence of this for a wide variety of exactly-solvable lattice models, such as the toric code, double semion, color code, and quantum double models.

In the second paper we prove that this topological correction is robust and universal. We do this by proposing a new numerical method to compute it based on tensor networks, using PEPS on a torus and MPS with periodic boundary conditions. For a system undergoing a quantum phase transition, our method is orders of magnitude more accurate and more efficient than similar calculations of topological Rényi entropies.

In the third paper, we show that the topological geometric entanglement can be used to pinpoint minimally entangled states (MES). We show this for the double Fibonacci and toric code models. For large lattices, we use the method proposed in the previous paper. In the end, the identification of MES allows us to obtain all topological properties of the system such as topological sectors, anyonic statistics, S and U matrices, quantum dimensions, fusion rules...

** A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States.* R. Orus, Annals of Physics **349** (2014) 117–158 [e-print:arXiv:1306.2164].* *

** Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography.* R. Orus, European Physical Journal B 87, 280 (2014), [e-print:arXiv:1407.6552].

These two papers are useful reviews on several aspects of tensor network methods. The first one focuses on several practical issues of MPS and PEPS, as well as on their related numerical algorithm