### 2021 Winter Semester Seminar

#### Kade Head-Marsden (Harvard University):

###### 05.11.2021, 13:30 CET, Ernst Mach lecture hall, Strudlhofgasse 4, 2nd floor, room 3248

Quantum algorithms for non-unitary time evolution of quantum systems

Abstract: Open quantum system evolution in the presence of an environment is crucial to understanding and improving many processes including the communication of quantum information and the transfer of energy. Quantum computing platforms have emerged as a promising route to modelling and predicting the behaviour of such systems. However, mapping inherently non-unitary dynamics into the unitary framework of gate-based quantum algorithms is a challenging task. Here, I will discuss two different density matrix gate-based quantum algorithms to predict the dynamics of open quantum systems. The first algorithm is dilation based where the Hilbert space of interest is expanded to map non-unitary evolution into a unitary framework [1,2]. The second algorithm is based on the decomposition of a non-unitary matrix into Hermitian and anti-Hermitian components [3]. I will discuss the theory behind these algorithms, their extension from the Markovian to the non-Markovian regime, and applications relevant in chemistry and physics [4].

[1] Z. Hu, R. Xia, and S. Kais, Sci. Rep., 10, 3301 (2020)

[2] K. Head-Marsden, S. Krastanov, D. A. Mazziotti, and P. Narang, Phys. Rev. Res., 3 (1), 013182 (2021)

[3] A.W. Schlimgen, K. Head-Marsden, L. Sager, P. Narang, and D. A. Mazziotti (submitted 2021) arXiv:2106.12588

[4] Z. Hu, K. Head-Marsden, D. A. Mazziotti, P. Narang, and S. Kais (submitted 2021) arXiv:2101.05287v2

#### Alex Turzillo (Max Planck Institute of Quantum Optics, Munich):

###### 12.11.2021, 11:15 CET, lecture hall: Kurt Gödel lecture hall (Strudlhofgasse 4, ground floor, room 3E63)

Symmetry-protected topological order in open quantum systems

Abstract: This talk will discuss the robustness of symmetry protected topological (SPT) order to evolution by noisy channels. By studying the evolution of non-local string order parameters, we find that one-dimensional SPT order is destabilized by generic couplings to the environment as well as by couplings satisfying a weak symmetry condition that directly generalizes from closed systems. We introduce a stronger symmetry condition on channels that ensures SPT order is preserved.

#### Bram Vanhecke (University of Vienna):

###### 19.11.2021, 11:15 CET, lecture hall: Kurt Gödel lecture hall (Strudlhofgasse 4, ground floor, room 3E63)

Frustrated systems and tensor networks

Abstract: Classical frustrated systems are characterised by an extensive ground state degeneracy. As such, they could still have interesting physics at zero temperature. Generally these systems are hard to study numerically, and certainly characterising the ground state ensemble can be challenging as there are no thermal fluctuations. We show how tensor network methods may be applied to frustrated systems in two and three dimensions with roughly the same ease as they may be applied to unfrustrated classical systems. In particular we will show how to study the ground state ensemble directly, which requires circomventing a sort of sign-problem in tensor networks.

#### Ilya Kull (University of Vienna):

###### 26.11.2021, 11:15 CET, online

An area law for 2D frustration-free spin systems

Abstract: I will try to give an overview of a recent proof of an entanglement entropy area law in the ground states of 2D frustration-free systems (2103.02492). For the most part, I will be talking about previous results in the same vein (1301.1162 and 1905.11337). The proof techniques are very similar in all three papers (the 2D problem is reduced to 1D) and employ the same arsenal of tools and insights. Those tools will be the focus of the talk.

#### David Blanik (University of Vienna):

###### 03.12.2021, 9:30 CET, online

An informal introduction to (co)homology and applications in physics

Abstract: I will give a very basic introduction to the theory of homology and cohomology groups. In regards to their applications in physics, I will focus mainly on the topics of group cohomology and the BRST construction.

#### Wen-Tao Xu (University of Vienna):

###### 10.12.2021, 11:15 CET, online

Matrix product operator algebras, non-abelian anyons and their splitting

Abstract: Using Kitaev’s quantum double model as an example, I will introduce the matrix product operator (MPO) algebras, which describe the symmetry of topological tensor network states on the entanglement level. The anyonic excitations can be constructed from the irreps of the MPO algebras. From this construction, the internal degrees of freedom of non-Abelian anyons can be specified. Through boson condensation, the internal degrees of freedom of a non-Abelian anyon can split into Abelian anyons. I will also discuss the possible ways to detect splitting.

#### José Garre Rubio (University of Vienna):

###### 17.12.2021, 11:15 CET, online

Classifying non-onsite symmetries with tensor networks

Abstract: In this talk I will give an introduction to the classification of non-onsite symmetries of spin chains. The approach is based on tensor networks: the symmetry operators are represented as matrix product operators and the states are matrix product states. I will show the structure of the invariants classifying these phases and the connection to symmetry protected topological phases. Finally, I will comment on the gauging of these phases and possible generalizations where the symmetry is not restricted to the group case.

#### Mingru Yang (University of Vienna):

###### 14.01.2022, 11:15 CET, online

Weak ergodicity breaking, quantum many-body scars, and Hilbert space fragmentation

Abstract: It has been thought thermalization is the inevitable fate of many interacting quantum systems, whose dynamics allow them to fully explore the vast configuration space regardless of the initial state--the behavior known as quantum ergodicity, with the only two counterexamples being integrable systems and many-body localization. However, recent experiments in quantum simulators based on 51 Rydberg atoms found persistent revivals for certain initial states, revealing the existence of a set of special non-thermal eigenstates in the bulk of the many-body energy spectrum, dubbed "quantum many-body scars", which is a new mechanism of ergodicity breaking. In the presentation, I will give an introduction to this weak ergodicity breaking phenomenon and three formalisms to try to unify the towers of quantum many body scar states in different models. I will show how matrix product states and tangent space methods become useful tools in the analytical construction of these states and giving a semiclassical description. I will also point out some open questions that might be interesting for us to explore.

#### András Molnár (University of Vienna):

###### 21.01.2022, 11:15 CET, online

Matrix product operators, tensor fusion categories and weak Hopf algebras for topological order

Abstract: Topologically ordered phases first appeared in condensed matter physics (such as in the explanation of the fractional quantum Hall effect) and they provided novel kind of phase transitions not explained by Landau's local order parameter. These systems are intrinsically protected against perturbations, and thus they can also be used for quantum computing, where precise control of states is needed. Theoretical models explaining topological order include Kitaev's toric code as well as the string-net models. While the toric code can rather straightforwardly be generalized to a model built from Hopf algebras, the string-net models are another kind of generalization of the toric code using fusion categories. In this talk we try to reconcile these two seemingly different models using the well-known result that fusion categories arise exactly as representations of (weak) Hopf algebras. We use matrix product operators (MPO) to illustrate these results, and in fact, we provide an alternative interpretation of MPO-injective PEPS, tensor network states describing the aforementioned topologically ordered states.

#### Manuel Mekonnen (University of Vienna):

###### 28.01.2022, 11:15 CET, online

Quantum Control: Decomposing the dynamics of two spin-1/2 systems

Abstract: Starting from a certain state in a controlled quantum system (e.g. with an external field present) it is a priori not clear which final states can be reached, i.e. which unitaries are available to use in order to transform the initial state. In order to find the reachable states of the system, one analyses the given Hamiltonian dynamics to characterise the so-called dynamical Lie algebra which allows one to understand if the system is controllable or not. In the latter case, the theory also provides tools to decompose the dynamical Lie algebra in order to construct the possible transformations. These can then be implemented as quantum gates in quantum computers via the controls of the system. In this talk I will provide an overview of quantum control theory which includes revisiting and introducing important notions from Lie algebra and Lie group theory. To put the concepts into a physical setting we will consider an application of the theory by looking at the dynamics of two spin-1/2 systems.