Projects

The main objectives of QUEVADIS can roughly be divided into 5 categories.
WP1: Mathematical theory of fixed points and convergence rates
WP2: Dissipative quantum state engineering
WP3: Dissipative quantum computing
WP4: Quantum effects driven by dissipation
WP5: Experimental realization

Mathematical theory of fixed points and convergence rates
Development of a mathematical theory with which we can study convergence rates of dissipative Markovian quantum systems by generalizing results obtained for classical Markov systems.
A mathematical theory of fixed points and respective rates of convergence is vital for our project. Fixed points (steady states) will be used to encode the outcome of a quantum computation, they are to be engineered with pre-determined properties, and their dependence on external changes shall be investigated. In all these cases it will be crucial to know how fast the space of fixed points (or limit cycles) is reached and how robust this procedure is. Both issues depend, mathematically speaking, on the spectral gap of the evolution operator which has thus to be assessed in various scenarios.
We aim at deriving quantum counterparts of various results known for classical Markov chains with a particular emphasis on processes where the dissipation acts locally (on overlapping regions) and in a translational-invariant way – both restrictions are motivated by typical practical constraints.
Apart from the applications inside this project, the fundamental importance of the set of fixed points of a quantum evolution makes plausible that our study finds natural applications in a plethora of different topics, like complexity theory, quantum error correction, or even wavelet theory.

Dissipative quantum state engineering
Investigation of how many-body quantum states can efficiently be generated by exploiting dissipative evolution.
Instead of fixing a dynamic and asking for the steady states, this WP aims at ‘reverse engineering’. That is, given a desired quantum state or a predetermined property, find ways of engineering it via dissipation. This is particularly relevant in the context of quantum simulators, which will be one of the main uses of quantum processing devices. We believe that the use of dissipative processes can help in constructing novel algorithms for simulating strongly correlated quantum systems (such as the ubiquitous Hubbard model).
First of all, we will investigate in detail the requirements and convergence rates for the simulation of ground states of frustration free Hamiltonians such as in the case of matrix product states, projected entangled pair states, stabilizer states and quasi-free states. Next, we plan to tackle one of the central outstanding problems in the field of quantum computation, namely the question of how to simulate thermal (i.e. Gibbs) states of general strongly correlated quantum Hamiltonians on a quantum device (e.g., a quantum computer). To achieve this, the dissipation will have to be complemented by some coherent dynamics and/or coherent control. More broadly, we will investigate the uses and limitations of quantum Metropolis devices.

Dissipative quantum computing
Study of dissipative quantum computation and issues of fault tolerance.
One of the most intriguing problems in the context of quantum computation is the investigation of the origin of the power of a quantum computer, and, related to this, of the minimal resources needed for building a universal quantum computater. The concept of teleportation and cluster state quantum computation revealed that measurements alone on highly entangled states are sufficient to achieve universal quantum computation, but recently we have been able to show that in principle less resources are necessary: time-independent dissipative (Markovian) processes, without any additional coherent dynamics, are enough for universal quantum computation . This new model of quantum computation apparently violates most of the celebrated DiVincenzo criteria for building a quantum computer, and therefore opens up a wealth of new research problems: 1) Investigate in detail the minimal resources needed to allow for universal quantum computation, i.e. characterize the minimal set of dissipative gadgets needed. Is it e.g. possible to replace the space-dependence of the dissipative processes by time-dependance? 2) Is it possible to make robust, i.e. fault tolerant, quantum memories and/or a quantum computer by exploiting dissipation, eventually by combining standard ideas of error correction with dissipation? 3) Come up with natural algorithms to be implemented on dissipative quantum computers (like quantum Metropolis devices, cf. WP2).

Quantum effects driven by dissipation
Investigation of relaxation processes and introduce natural quantum analogues of classical non-equilibrium hopping processes and study the associated phase diagrams.
The central question to be addressed in this project is to identify novel effects or phenomena that arise due to dissipation, such as dissipatively driven quantum phase transition or relaxation effects. Non-equilibrium dissipative processes have been well studied in classical physics, and are encountered in a wide variety of situations. In the case of 1-dimensional hopping (traffic) processes, particularly intriguing phenomena and phase transitions can occur, and most phenomena can be understood within the framework of the universality class corresponding to directed percolation. In this project we plan to study a quantum version extension of such systems by supplementing the incoherent diagonal hopping terms by non-commuting hopping terms and / or coherent terms. The associated phase diagram will most likely be very nontrivial. We plan to develop renormalization group methods, both of analytical and numerical kind, for analyzing such models. This will allow understanding what kind of quantum perturbations are relevant, marginal or irrelevant. We will also focus on properties of steady states which are caused by dissipative dynamics, such as (broken) symmetries or asymptotic freeness. It is expected that many questions and specific effects only arise during the course of investigation of the other WPs.

Experimental realization
Investigation of how all of this can be implemented in experiments.
Most of the previous workpackages involve the development of an abstract theory which characterizes and quantifies the possibilities of exploiting dissipative processes in order to observe intriguing quantum phenomena and carry out certain quantum information tasks. The present workpackage will connect such a theory with particular physical systems in both directions: it will study the experimental implications of the rest of the project, as well as use current experimental set ups to motivate some of the research carried out in other workpackages. The objectives of the present workpackage are:
(i) To find practical systems where the ideas developed in the project could be implemented.
(ii) Make specific predictions for what should be observed in experiments.
(iii) Trigger experimental research in the subject of the project.
(ii) Raise important questions that are relevant in practical situations and should be theoretically developed.