### 2023 Summer Semester

During the 2023 summer semester, the seminar generally takes place in person according to the schedule and details listed below. In addition, seminar talks might occasionally be given online as announced here.

#### Seminar calendar for the 2023 Summer Semester

#### Paul Brehmer (RWTH Aachen University)

Reduced basis modeling of quantum spin systems based on DMRG

Abstract: Within the reduced basis modeling approach, an effective low-dimensional subspace of a quantum many-body Hilbert space is constructed in order to investigate, e.g., the ground-state phase diagram. The basis of this subspace is built from solutions of snapshots, i.e., ground states corresponding to particular and well-chosen parameter values. Here, we show how a greedy strategy to assemble the reduced basis and thus to select the parameter points can be implemented based on density-matrix-renormalization-group (DMRG) calculations. Once the reduced basis is computed, observables required for the computation of phase diagrams can be computed with a computational complexity independent of the underlying Hilbert space for any parameter value. We illustrate the efficiency and accuracy of this approach for different one-dimensional quantum spin-1 models, including anisotropic as well as biquadratic exchange interactions, leading to rich quantum phase diagrams.

#### Nick Jones (University of Oxford)

The MPS skeleton and exact results in topological chains

Abstract: Matrix-product states (MPS) have proven to be tremendously useful theoretical and practical tools. The MPS skeleton of a phase diagram is the family of Hamiltonians that have exact (finite-bond-dimension) MPS ground states in the thermodynamic limit. I will characterise this skeleton in a class of topological free-fermion chains, unearthing an interesting intersection between two rather different notions of exact solvability. As well as outlining the construction of the MPS ground state on the skeleton, I will explain how, on the MPS skeleton, we can find exact closed formulas for string-correlation functions in the ground state (order parameters for the different phases) and have an analytic method to compute the entanglement spectrum of a finite subsystem.

Based on arxiv:2105.12143 with Julian Bibo, Bernhard Jobst, Frank Pollmann, Adam Smith and Ruben Verresen and arxiv:2105.13359 with Ruben Verresen.

(date/time/location: 20.02.2023, 14:30, Boltzmanngasse 5, 5th floor, seminar room nr. 3510)

#### Zhao Zhang (SISSA Trieste)

Beyond area-law entanglement in higher dimensions and holographic tensor networks

Abstract: Height models and random tiling are well-studied objects in classical statistical mechanics and combinatorics that lead to many interesting phenomena, such as arctic curve, limit shape and Kadar-Parisi-Zhang scaling. We introduce quantum dynamics to the classical hexagonal dimer, six-, and nineteen-vertex models to construct frustration-free Hamiltonians with unique ground state being a superposition of discrete random surfaces subject to a Dirichlet boundary configuration. The local Hilbert space is further decorated by a color degree of freedom, matched in pairs between lattice sites of the same height, generating long range entanglement that makes area law violation of entanglement entropy possible. The scaling of entanglement entropy between half systems is analyzed with the gradient surface tension in the scaling limit and under a q-deformation that weighs random surfaces by the volume below, it undergoes a phase transition from area law to volume scaling. At the critical point, the scaling is L logL due to the so-called "entropic repulsion” of Gaussian free fields conditioned to stay nonnegative. An exact holographic tensor network description of the ground state is proposed with one extra dimension perpendicular to the lattice. I will also discuss an alternative realisation with vertex models, inhomogeneous deformation to obtain sub-volume intermediate scaling, and possible generalizations to higher dimension.

(date/time/location: 23.02.2023, 14:30, Erwin Schrödinger lecture room, Boltzmanngasse 5, 5th floor, room 3500)

#### Jared Jeyaretnam (University College London)

Symmetry-protected topological order and strong zero modes in non-ergodic systems

Abstract: At zero temperature, symmetry-protected topological (SPT) order can encode quantum information in an edge strong zero mode, robust to perturbations respecting some symmetry. On the other hand, phenomena like many-body localisation (MBL) and quantum scarring can arrest the approach to thermal equilibrium, contrary to the ergodic dynamics expected of generic quantum systems. This raises the possibility that by combining SPT order with such ergodicity breaking phenomena, one might be able to construct a quantum memory that is robust at finite temperature.

In the first part of this talk, I will look at an interacting spin-1/2 chain hosting SPT order, identifying long-lived bulk coherence in the dynamics and the quantum scars responsible, and show that these scars exhibit signatures of SPT order even at finite energy density.

In the second part of this talk, I will focus on a topological transition between two MBL phases. Through a Clifford-circuit based renormalisation group approach, we identify many-body resonances in the basis of localised eigenstates, showing that these proliferate in the vicinity of the transition and cause delocalisation. Additionally, we characterise the SPT strong zero mode. This has important implications for the stability of MBL and transitions between MBL phases with different topological orders.

(date/time/location: 27.02.2023, 11:30, Erwin Schrödinger lecture room, Boltzmanngasse 5, 5th floor, room 3500)

#### Oliver Holmegaard Schwarze (University of Copenhagen)

Exploiting symmetries in quantum state tomography

Abstract: Quantum State tomography is the task of reconstructing the state of a quantum system from measurements. Generally, this process is very resource intensive due to the number of state copies and measurement settings needed, as wells as the complexity of the post-processing. This motivates the development of efficient tomography algorithms which utilize knowledge of the target state as much as possible. In some scenarios, the target state to be determined possess some symmetry which is known beforehand. I will present a way of using prior knowledge of the target state's symmetry to perform efficient quantum state tomography, which improves over similar general algorithms in terms of sample, post-processing and measurement complexity. These speedups are demonstrated through simulation for the specific case of permutationally invariant quantum states.

(date/time: 21.03.2023, 13:30)

#### Tim Lüders (University of Greifswald)

Higher Geometry and Symmetries in Quantum Mechanics

Abstract: We discuss how general principles of symmetry in quantum mechanics lead naturally to twisted notions of a group extension. From a mathematical point of view, these kinds of twisted group extensions are best understood in terms of Jandl gerbes, which are certain objects from equivariant higher categorical differential geometry. As an application, we will sketch how twisted group extensions can be used to classify topological insulators in terms of their symmetries.

(date/time: 01.06.2023, 11:30)

#### Christopher Wächtler (UC Berkeley)

Topological synchronization of classical and quantum systems

Abstract: For many quantum mechanical applications dissipation is often regarded as an undesirable yet unavoidable consequence because it potentially degrades quantum coherences and renders the system classical. However, interactions with the environment can also be considered a fundamental resource for striking collective effects typically impossible in Hamiltonian systems. A hallmark of such collective behavior in nonequilibrium systems is the phenomenon of synchronization: in the complete absence of any time-dependent forcing from the outside, a group of oscillators adjusts their frequencies such that they spontaneously oscillate in unison. With the recent developments in quantum technology which allow one to exquisitely tailor both the system and environmental properties, synchronization has emerged in the quantum domain with various different examples ranging from nonlinear oscillators to spin-1 systems, superconducting qubits and optomechanics. However, to observe synchronization in large networks of classical or quantum systems demands both excellent control of the interactions between nodes and accurate preparation of the initial conditions due to the involved nonlinearities and dissipation. This limits its applicability for future devices. In this talk, I will present a potential route towards significantly enhancing the robustness of synchronized behavior in open nonlinear systems that utilizes the power of topological insulators, which exhibit an insulating bulk but conducting surface states, known as topological edge states. These edge states display a surprising immunity to a wide range of local deformations and even circumvent localization in the presence of disorder. By combining nontrivial topological lattices with nonlinear oscillators, we show that synchronized motion emerges at the lattice boundaries in the classical (mean field) as well as the quantum regime. Furthermore, the synchronized edge modes inherit the topological protection known from closed systems with remarkably robust dynamics against local disorder and even random initial conditions. Our work demonstrates a general advantage of topological lattices in the design of potential experiments and devices as fabrication errors and longterm degradation are circumvented in this way. This is especially important in networks where specific nodes need special protection.

(date/time/location: 12.06.2023, 11:30, seminar room nr. 3510, Strudlhofgasse 4, 5th floor)

#### Lorenz Mayer (University of Cologne)

The Hubbard Model on a Klein Bottle

Abstract: In the last two decades, a variety of topological phases have been described, predicted, classified, proposed, and measured. This talk will deal with fermions in one spatial dimension, in the presence of anti-unitary symmetries. Anti-unitary symmetries, compared to those represented unitarily, pose a few unique challenges, affect the structure of states differently, and are less studied. G-protected topological phases are classified by the group cohomology of G. Given a state, its class can be determined by computing the (projective) G-representation associated to any edge in the system. As this is often inconvenient to do, it is helpful to consider generalized order parameters. Topological quantum field theory offers some help in their construction, roughly, relating order parameters and partition functions on special manifolds. For anti-unitary symmetries, these have to be non-orientable surfaces. The tool of choice are fermionic versions of matrix product states. Any state can be approximated by sequences of such states. By imposing a bounded correlation length on such a sequence, the limit state is guaranteed to correspond to a gapped phase; moreover, the same condition allows to prove continuity of the generalized order parameters.

(date/time/location: 02.08.2023, 11:30, Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Boltzmanngasse 9, 2nd floor, Boltzmann lecture hall)

#### Lukas Kienesberger (LMU Munich)

Quantum Limits of Position and Polarizability Estimation in the Optical Near Field

Abstract: Optical near fields are at the heart of various applications in sensing and imaging. We investigate dipole scattering as a parameter estimation problem and show that optical near-fields carry more information about the location and the polarizability of the scatterer than the respective far fields. This increase in information originates from and occurs simultaneously with the scattering process itself. Our calculations also yield the far-field localization limit for dipoles in free space.

(date/time: 03.08.2023, 11:30)

#### Laura Herzog (LMU Munich)

Restricted Active Space Quantum Chemistry Density Matrix Renormalization Group for Fermions in a Harmonic Trap and Arbitrary Pair Interaction

Abstract: The Density Matrix Renormalization Group (DMRG), a method widely used in condensed matter physics, has found applications in Quantum Chemistry (QC) despite the absence of an area law. While it proves to be effective, applying DMRG to QC faces challenges due to long-range interactions and the lack of translation invariance. To overcome this, a transition from simplistic Full Configuration (FCI) to active space methods is useful. These methods focus on a Fermi sea-like ground state, accommodating fixed occupied sites deep within the Fermi Sea and empty sites distant from the Fermi level. Our approach aims to combine multiple methods, including a restricted active space approach, exploiting symmetries via so-called projected purification, and quantum information-inspired orbital rotations.

In this project, we investigate a simple QC-like system – spinless fermions in a one-dimensional harmonic trap with arbitrary pair interactions. Despite its simplicity compared to molecules, this system provides interesting physics and serves as an excellent testing ground for novel computational techniques. By studying the entanglement structure of this system, we hope to gain insights into why QC-DMRG works effectively even without the presence of an established area law.

(date/time: 05.09.2023, 11:30)

#### Refik Mansuroglu (Friedrich-Alexander Universität Erlangen-Nürnberg)

Classically assisted Quantum Simulation

Abstract: Nonequilibrium time evolution of large quantum systems is a strong candidate for quantum advantage. Although product formulas have been proven efficient for fault tolerant quantum simulation, their application on near-term devices is limited by gate errors. To solve this, variational quantum algorithms have been put forward, but their quantum optimization routines suffer from trainability and sampling problems. I discuss classical processing routines for quantum simulation that circumvent the need of quantum optimization. In the most general setting, rigorous error bounds can be expanded in a perturbative regime for suitable time steps. The resulting cost function is efficiently computable on a classical computer and it can be shown that there always exists potential for optimization with respect to a Trotter sequence of the same order and that the cost value has the same scaling as for Trotter in simulation time and system size. In the presence of translaional symmetry or if only local properties are of interest, we are further able to leave the perturbative regime by reducing the problem to classically feasible system sizes.

(date/time/location: 06.09.2023, 11:30, Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Boltzmanngasse 9, 2nd floor, Schrödinger lecture hall)

#### Miao Hu (Heidelberg University)

Local unitary invariant polynomials and randomized measurements of O-O invariant states for entanglement detection

Abstract: Entanglement is the key resource on which quantum technologies rely. One method to detect the entanglement of an unknown state is to consider the expected outcome for random local projections (a type of randomized measurements) performed on the state and analyze the resulting probabilistic distribution. This can be done by computing cumulants (connected correlations) of this distribution. We have explicitly derived these cumulants for the parametric families of states that interpolate among the maximally mixed state, isotropic states, and Werner states (called O-O invariant states, whose entanglement properties are known). We have also studied a spanning set for these cumulants, the set of polynomials of tensors invariant under local unitary transformations, called trace invariants. We have found that for finite-dimensional Hilbert spaces, the trace invariants of O-O invariant states are graph polynomials --- the Bollobás-Riordan polynomials, which contain information about how graphs drawn on surfaces are connected and their topological properties. We have also found the limits of these trace invariants for large dimension, which allowed us to compute the dominant values of the cumulants introduced above. The computed cumulants show that when the parameter for an unknown isotropic state is large enough, it can be recovered by performing random local projections.

(date/time/location: 07.09.2023, 11:30)