Welcome to our Quantum Computing group website!

Our research focuses on quantum computing algorithms and software, numerical methods for simulating strongly correlated quantum systems, and transport properties of physical systems. We are part of Informatics 5 - Scientific Computing at TUM.

# Current openings

We are always looking for talented students - if you are interested in a student project, Bachelor, Master or PhD thesis, drop by or contact us.

# Research

## Quantum simulation and quantum algorithms

We explore and investigate algorithms for “quantum simulation”, i.e., using a quantum computer to simulate a target quantum system, like a (strongly correlated) system in condensed matter physics or chemistry. In particular, we study approaches based on qubitization and the quantum singular value transform, as well as embedding frameworks like DMET. We also investigate quantum algorithms for optimization and machine learning.

\[U_{\vec{\varphi}} = e^{i \varphi_0 Z} \prod_{k=1}^d W(a) e^{i \varphi_k Z}\]Selected publications and preprints:

- Simon Wiedemann, Daniel Hein, Steffen Udluft, Christian B. Mendl

*Quantum policy iteration via amplitude estimation and Grover search - Towards quantum advantage for reinforcement learning*

[arXiv:2206.04741]

## Numerical methods

Our expertise is in *tensor network* and *quantum Monte Carlo* methods, and we are also exploring neural network quantum states. For example, we have implemented matrix-product-operator techniques to study out-of-time ordered correlation (OTOC) functions and elucidate scrambling of quantum information, and have applied the determinant quantum Monte Carlo (DQMC) algorithm to a three-band Hubbard model of high-Tc cuprate superconductors.

Selected publications and preprints:

- Richard Milbradt, Lisa Scheller, Christopher Aßmus, Christian B. Mendl

*Ternary unitary quantum lattice models and circuits in 2 + 1 dimensions*

[arXiv:2206.01499] - Irene López Gutiérrez, Christian B. Mendl

*Real time evolution with neural-network quantum states*

Quantum 6, 627 (2022) [arXiv:1912.08831] - Edwin W. Huang, Christian B. Mendl, Shenxiu Liu, Steven Johnston, Hong-Chen Jiang, Brian Moritz, Thomas P. Devereaux

*Numerical evidence of fluctuating stripes in the normal state of high-Tc cuprate superconductors*

Science 358, 1161-1164 (2017) [arXiv:1612.05211]

Code on GitHub: hubbard-dqmc - Annabelle Bohrdt, Christian B. Mendl, Manuel Endres, Michael Knap

*Scrambling and thermalization in a diffusive quantum many-body system*

New J. Phys. 19, 063001 (2017) [arXiv:1612.02434]

Code on GitHub: tensor_networks

## Quantum computing software stack

Currently we are working on a quantum circuit simulation toolbox *Qaintum* written in Julia, and a Python package *qib* (still early stage) for translating high-level quantum algorithms to circuits and submitting these to hardware backends. We also work on hybrid quantum-classical programming languages and runtime environments.

Selected software projects:

*Qaintum*

Julia-based simulator, supporting gradient computation and density matrices*qib*- quantum library

Python package for quantum circuits and algorithms

## Statistical physics and generalized hydrodynamics

One-dimensional physical systems can exhibit interesting anomalous transport properties. A characteristic signature is the time-dependent response of the local energy to a small, localized perturbation of the equilibrated system, expressed by the energy-energy time-correlation. One approach to understand such behavior is a nonlinear extension of fluctuating hydrodynamics, which has attracted particular interest in recent years due to surprising connections to the KPZ universality class. Specifically, nonlinear fluctuating hydrodynamics predicts that the energy correlation function contains a left- und right-moving “sound peak” with asymptotic scaling form

\[(\lambda_\mathrm{s} t)^{-2/3} f_{\mathrm{KPZ}} \big((\lambda_{\mathrm{s}} t)^{-2/3}(x-\sigma ct)\big),\]where \(\lambda_{\mathrm{s}}\) is a model-dependent coefficient, \(\sigma = \pm 1\) denotes the left or right peak and \(f_{\mathrm{KPZ}}\) is the KPZ function. Together with Herbert Spohn, we have worked out the detailed connection between statistical physics models and KPZ theory and performed numerical molecular dynamics simulations.

Selected publications and preprints:

- Christian B. Mendl, Herbert Spohn

*High-low pressure domain wall for the classical Toda lattice*

SciPost Phys. Core 5, 002 (2022) [arXiv:2011.11008]

Accompanying Mathematica code: Toda-domainwall - Christian B. Mendl, Herbert Spohn

*Shocks, rarefaction waves, and current fluctuations for anharmonic chains*

J. Stat. Phys. 166, 841-875 (2017) [arXiv:1607.05205] - Christian B. Mendl, Herbert Spohn

*Low temperature dynamics of the one-dimensional discrete nonlinear Schrödinger equation*

J. Stat. Mech. (2015) P08028 [arXiv:1505.04218] - Christian B. Mendl, Herbert Spohn

*Dynamic correlators of Fermi-Pasta-Ulam chains and nonlinear fluctuating hydrodynamics*

Phys. Rev. Lett. 111, 230601 (2013) [arXiv:1305.1209]

Accompanying Mathematica code: fluct-hydro-chains