RESEARCH

The research pursued in our QI&QMBP group is located at the intersection of different fields: On the one hand, it lies at the interface of the field of Quantum Information and Computation with the study of Complex Quantum Many-Body Systems. On the other hand, the work pursued in our group closely combines methods from both Mathematics and Physics, complemented by concepts from Theoretical Computer Science.

Context

Quantum Computers, and Quantum Information Processing devices in general, use systems governed by the laws of quantum mechanics to carry out computations and other information processing tasks. Their computational power stems from the complex and rich nature of the quantum correlations - termed entanglement - present in those systems. They allow quantum computers to carry out certain tasks, such as factoring large numbers, exponentially faster than any classical computer, and hold the promise to revolutionize computational tasks such as the development of pharmaceuticals or novel materials, as well as complex optimization problems.

Quantum Many-Body Systems, on the other hand, have seen a similarly rapid development, driven by the observation of a range of exotic phenomena in quantum materials as well as by the steadily growing ability to engineer such models in experiments. The unconventional phases which these systems exhibit are not only interesting from a fundamental perspective, as they fall outside the established framework of phases of matter and exhibit exotic properties such as excitations with non-trivial statistics, but they also hold the promise to be useful for novel applications such as high-precision measurement devices or as a substrate for quantum information storage and processing.

What we do

In our research group, we explore the rich area at the interface between Quantum Information and Computing on the one side and the physics of Complex Quantum Many-Body Systems on the other side. The common link is given by the intricate quantum correlations formed between the many constituents in those systems (quantum bits in quantum computers, quantum particles in many-body systems). In particular, we apply the powerful concepts established in Quantum Information Theory, specifically Entanglement Theory, for classifying, quantifying, and manipulating entanglement, in order to assess key questions about the nature of quantum correlations in complex interacting many-body systems, and the way in which they manifest themselves in physical phenomena. This is complemented by the study of the connection between Many-Body Systems and Quantum Computing and Quantum Algorithms, considering Many-Body Systems both as a problem to be solved by Quantum Computers, and as a way to realize Quantum Computations in a particularly stable way.

A key tool in our endeavor are Tensor Networks, such as Matrix Product States (MPS) and Projected Entangled Pair States (PEPS): They explain how one can build up the complex entangled quantum many-body correlations between the individual quantum particles from local building blocks - tensors - and thus allow to assess the interplay between locality and entanglement, which is key to a comprehensive understanding of the phenomena displayed by those complex quantum systems.

The work pursued in our group is highly interdisciplinary, driven by the cross-disciplinary nature of our research interests at the interface of Quantum Information and Many-Body Physics. On the one hand, we use both mathematical and physical approaches to tackle the problems at hand, complemented by methods from theoretical computer science - this way, we are able to cover the diverse facets of the topic and to obtain both rigorous classification results and novel insights into physical applications. On the other hand, in our research we strive to closely combine analytical and numerical approaches - this way, we can go significantly beyond what can be obtained with a purely analytical or purely numerical approach alone, and obtain a maximum of results while maintaining the highest possible level of rigor.

Research questions

We investigate a wide range of questions covering mathematical, physical, and computational aspects of Quantum Information and Quantum Many-Body Systems.  A selection of topics we cover is given below.

Mathematical Aspects of Quantum Many-Body Systems

We characterize the mathematical structure of entanglement in quantum many-body systems:

  • We study the representation theory of entangled quantum many-body states using tensor networks and derive structure theorems characterizing the ambiguity in these representations ("fundamental theorems").
  • We use these theorems to characterize the interplay of physical symmetries and entanglement symmetry actions; this is relevant both for the classification of phases and for the implementation of unbiased numerical methods.
  • We characterize the possible strongly correlated phases of entangled quantum matter, by combining the structure theorems above with methods to demonstrate spectral gaps or other signatures of phase transitions.
  • We explore the suitability of tensor networks for describing quantum many-body systems through rigorous approximability results for different types of systems (zero and finite temperature, time evolution, critical systems, fermions); and we explore the fundamental limits to such descriptions.


Physical Aspects of Quantum Many-Body Systems

We investigate the physical ramifications of complex entanglement patters in quantum many-body systems:

  • We develop tools to detect and characterize exotic phases in quantum many-body systems using entanglement-based probes which can witness exotic quantum order directly at the entanglement level.
  • We work on simulation methods for quantum many-body systems which take into account their entanglement structure and the way in which it interplays with physical symmetries to obtain efficient and unbiased simulation methods.
  • We come up with novel models which exhibit exotic properties due to their intricate entanglement structure, and ways to detect and use those properties in practical scenarios.
  • We study how entangled quantum many-body systems can be used in Quantum Technologies, such as for the storage of quantum information, for quantum computing tasks, or in high-precision measurement devices.


Computational Aspects of Quantum Many-Body Systems

We examine the multifaceted connections between quantum many-body systems and the theory and practice of computation:

  • We study how to use Quantum Many-Body Systems for building Quantum Computers, for instance by using topologically ordered systems and their non-trivial excitations.
  • We investigate when certain phases of matter can serve as substrates for measurement-based quantum computation schemes, and which underlying properties stabilize these phases.
  • We explore how Quantum Many-Body problems can be simulated on quantum computers, using insights about their entanglement structure, with a particular focus on problems suitable for near-term noisy intermediate-scale quantum (NISQ) devices.
  • We complement this by studying the fundamental limitations imposed on our ability to simulate quantum many-body systems through Quantum Complexity Theory, and in particular Hamiltonian Complexity.