### 2021 Summer Semester Seminar

#### Ilya Kull (University of Vienna)

Uncertainty and Trade-offs in Quantum Multiparameter Estimation

Abstract: In the talk, I will give a brief overview of the various kinds of uncertainty relations that have been studied since Heisenberg's seminal work almost 100 years ago. I will introduce classical and quantum parameter estimation theory and discuss its usefulness for formulating certain kinds of uncertainty relations. Finally, I will argue that quantum multi-parameter estimation theory provides us with yet another way to understand and quantify quantum uncertainty. I will show that known attainable bounds in quantum multi-parameter estimation encode trade-off relations for the precision with which we can estimate the different parameters describing a quantum state. This stands in contrast to classical multi-parameter estimation, where in the asymptotic limit optimal precision is attainable for all the parameters simultaneously.

(date/time/location: 10.03.2021, 14:00, via Zoom)

#### Anna Francuz (Jagiellonian University, Kraków):

Determining topological order from infinite Projected Entangled Pair States

Abstract: Topological order is a key ingredient of topological quantum computation. It gained recognition after it was realized, thanks to Alexei Kitaev, that quantum computational models can be written in the language of condensed matter systems. In my talk I will introduce topological order and methods to characterize it. Then I will briefly describe the numerical methods to determine both Abelian and non-Abelian topological order starting from a lattice Hamiltonian. With the 2D tensor network – Projected Entangled Pair States, the method allows to analyze states which were not achievable by 2D DMRG due to long correlation length.

(date/time/location: 17.03.2021, 14:00, via Zoom)

#### Mingru Yang (University of Vienna)

Improving time-dependent variational principle for long-range interactions

Abstract: I will talk about an improved scheme to do the time dependent variational principle (TDVP) in finite matrix product states (MPS) for two-dimensional systems or one-dimensional systems with long range interactions. In this method, the time-evolving state is represented in a MPS with its basis enriched by state-averaging with global Krylov vectors. It is shown that the projection error is significantly reduced so that precise time evolution can still be obtained even if a larger time step is used. Combined with the one-site TDVP, this approach provides a way to dynamically increase the bond dimension while still preserving unitarity for real time evolution. Benchmark results show the method can be more accurate and exhibit slower bond dimension growth than the conventional two-site TDVP. I will also discuss its application in studying the spin squeezing dynamics for two-dimensional systems with power-law decaying long-range interactions, which unveils the potential for experimental realization of practical quantum metrology.

References:

[1] MY and Steven R. White, Physical Review B 102 (9), 094315.

[2] MY, Sean R. Muleady, Steven R. White, Ana Maria Rey, to be published.

[3] Michael A Perlin, Chunlei Qu, Ana Maria Rey, Physical Review Letters 125 (22), 223401

(date/time/location: 24.03.2021, 14:00, via Zoom)

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

Introduction to symmetry enriched topological phases

Abstract: In this talk, I will introduce symmetry enriched topological (SET) phases and some work of mine on this topic. SET phases are characterized by a non-trivial interplay between topological order and global symmetries. I will start explaining the basic background of both topological order and global symmetries that are relevant to SET phases. Then, I will explain the general phenomenology that we find in SET phases and also I will give a concrete example where all the properties can be studied analytically. Afterwards, I will show some results about this example. Finally, I will comment on some open question that could be addressed

(date/time/location: 14.04.2021, 14:00, via Zoom)

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

Condensation and confinement order parameters for the non-Abelian topological states

Abstract: I will talk about the "order parameters" in the topological states, especially the non-Abelian topological states. At first, I will introduce the background, including the conventional Landau’s theory, the novel topological phases, and the anyons as well as their condensation and confinement. I will also review the basic knowledge of the PEPS (projected entangled pair state), and show how to describe anyons and detect their condensation and confinement using PEPS. Then, via two concrete examples: deformed double Fibonacci and double Yang-Lee topological states, I will show the condensation and confinement of non-Abelian anyons can be treated as the "order parameters" describing both the topological phases and their phase transitions. By mapping the topological states to the Potts models, the related critical exponents are predicted from conformal field theories. The predictions are also verified numerically.

(date/time/location: 21.04.2021, 14:00, via Zoom)

#### Manuel Mekonnen (University of Vienna)

Classical simulation of quantum circuits

Abstract: In this talk, I will introduce a certain class of quantum circuits, consisting only of nearest neighbour gates, that can be simulated efficiently by classical means. This is shown by using a Clifford algebra formalism that relies on the Jordan-Wigner representation and leads to a more general class of simulatable circuits, so called Gaussian quantum circuits. It then turns out that if we slightly relax the n.n. condition in our original circuits, allowing the same gates to act also on next to n.n. qubit lines, the resulting circuits are able to perform universal quantum computation. This small modification can be reached by qubit swapping alone and bridges the gap between efficient classical and quantum computation in this scenario.

The talk is based on a paper by Richard Jozsa and Akimasa Miyake, titled "Matchgates and classical simulation of quantum circuits" (https://arxiv.org/abs/0804.4050).

(date/time/location: 28.04.2021, 14:00, via Zoom)

#### Felix Hubmann & Julian Innerhofer (University of Vienna)

A Compact Fermion to Qubit Mapping

Abstract: Mappings between fermions and qubits are valuable constructions in physics. To date only a handful exist. In addition to revealing dualities between fermionic and spin systems, such mappings are indispensable in any quantum simulation of fermionic physics on quantum computers. The number of qubits required per fermionic mode, and the locality of mapped fermionic operators strongly impact the cost of such simulations. In this talk we will present some important basics on fermion to qubit mappings, including the Jordan-Wigner transform and a correspondence from fermionic operators to edge and vertex operators. In the later part of the presentation we will present a novel fermion to qubit mapping which outperforms all previous local mappings in both the qubit to mode ratio, and the locality of mapped operators.

The talk is based on the work of Charles Derby and Joel Klassen, titled “A Compact Fermion to Qubit Mapping”. -- https://arxiv.org/abs/2003.06939

(date/time/location: 05.05.2021, 14:00, via Zoom)

#### Antoine Tilloy (Max Planck Institute of Quantum Optics, Munich)

Variational method in relativistic quantum field theory without cutoff

Abstract: The variational method is a powerful approach to solve many-body quantum problems non perturbatively. However, in the context of relativistic quantum field theory (QFT), it needs to meet 3 seemingly incompatible requirements outlined by Feynman: extensivity, computability, and lack of UV sensitivity. In practice, variational methods break one of the 3, which translates into the need to have an IR or UV cutoff. In this talk, I will introduce a relativistic modification of continuous matrix product states that satisfies the 3 requirements jointly in 1+1 dimensions. I will then show how to apply the method to the self-interacting scalar field, without UV cutoff and directly in the thermodynamic limit.

(date/time/location: 12.05.2021, 14:00, via Zoom)

#### David Stephen (Max Planck Institute of Quantum Optics, Munich)

Measurement-based quantum computation with symmetry-protected topological phases

Abstract: The study of quantum phases of matter is deeply linked with the study of quantum computation. A prime example is Kitaev's toric code, which demonstrates how topologically ordered systems can be used to perform fault-tolerant topological quantum computation. More recently, a similar connection has been established between symmetry-protected topological (SPT) phases and the paradigm of measurement-based quantum computation (MBQC). In this talk, I will give an overview of this connection, starting by showing how it arises very naturally within the language of matrix product states. I will then describe a large class of 1D SPT phases in which every ground state is a resource for MBQC, and I will show that 2D SPT phases with subsystem symmetries allow for universal MBQC. These 2D phases are best understood using the language of quantum cellular automata. Finally, I will comment on other miscellaneous results in this field and potential future directions of research.

(date/time/location: 19.05.2021, 14:00, via Zoom)

#### Thorsten Wahl (Cambridge)

Novel Insights into Many-Body Localized Phases in One and Two Dimensions

Abstract: I will give an overview over our results on the description of many-body localized (MBL) systems with quantum circuits – a specific type of tensor networks. I will present how they can be used numerically to simulate the MBL-to-thermal transition observed in two-dimensional optical lattice experiments. We obtained a transition point consistent with the latest charge-density wave experiments and were also able to extract the mobility edge. I will explain why this description captures experimentally relevant time scales on which two-dimensional MBL is stable. I will also indicate how quantum circuits can be used to rigorously classify symmetry-protected topological MBL phases in one and two dimensions. Finally, I will demonstrate that the conventional notion of local integrals of motion has to be revised for topologically ordered MBL systems. I will argue that all of their eigenstates must have the same topological order, which cannot change unless sufficiently strong perturbations destroy MBL.

(date/time/location: 26.05.2021, 14:00, via Zoom)

#### Caroline de Groot (Max Planck Institute of Quantum Optics, Munich)

Symmetry protected topological phases in open systems with string order parameters

Abstract: It's increasingly become of interest to examine exotic quantum phases in non-equilibrium, thermal or dissipative settings, and here we make a contribution towards understanding symmetry protected topological (SPT) phases in open systems. We show that a notion of SPT order can exist for mixed states by considering its action under certain noisy channels. In this talk we will first discuss how how non-local order parameters (namely string order parameters) can uniquely detect SPT phases in closed systems. We will then move to the extension of a consistent notion of SPT phases in open systems, by first considering numerical evidence of the string order parameter with dissipation. This suggests a suitable requirement for channel symmetry to preserve SPT order. We prove that certain symmetry classes of channels can map between SPT phases (strong symmetry), while certain others (weak symmetry) totally destroy SPT order, and finally we suggest a consistent notion of mixed SPT states.

* from an upcoming work together with Alex Turzillo and Norbert Schuch.

(date/time/location: 02.06.2021, 14:00, via Zoom)

#### Angelo Lucia (Universidad Complutense de Madrid)

Thermalization in Kitaev’s quantum double models via Tensor Network techniques

Is it possible to have a 2D self-correcting memory? The general belief that this is not the case usually lies upon the absence of a strong enough energy barrier preventing errors and noise to quickly destroy the information encoded in the memory. But this is not sufficient, as there are entropic factors at play that could take the role of the energy barrier, by delaying the spread of errors. In order to really settle the question, one should look directly at the time required for the memory to thermalize.

In this talk, I will consider Kitaev's Quantum Double models with arbitrary group (including the non-abelian case), and by representing their Gibbs states as a PEPS I will show how to prove that the themalization dynamic given by Davies' generators has a spectral gap, therefore excluding the possibility of self-correction for this class of models.

Based on joint work with David Pérez-García and Antonio Pérez-Hernández.

(date/time/location: 02.06.2021, 14:00, via Zoom)

#### Bram Vanhecke (University of Ghent)

Entanglement scaling for lambda*phi^4

We study the λϕ^4 model in 0+2 dimensions at criticality, and effectuate a simultaneous scaling of UV and IR physics. We demonstrate that the order parameter ϕ, the correlation length ξ and quantities like ϕ^3 and the entanglement entropy exhibit useful double scaling properties. The calculations are performed with boundary matrix product state methods on tensor network representations of the partition function, though the technique is equally applicable outside the realm of tensor networks. We find the value alpha_c=11.09698(31) for the critical point, improving on previous results.

(date/time/location: 16.06.2021, 14:00, via Zoom)