Final Report Summary - QOC4QIP (Quantum Control Applied to Quantum Information) The ultimate goal of the project QOC4QIP was to provide a general theoretical framework for the application of quantum optimal control to open quantum systems, especially those of current interest for the implementation of quantum information processing (QIP). The project could be divided into two main themes, each with its own relevance and identity. The first, in chronological order, was the development of a general approach for treating decoherence without invoking any assumption on the system-environment interaction and allowing for a non-trivial bath structure. The second theme was the development of optimal control strategies for open systems aimed at the implementation of quantum information tasks and at the application to a specific physical system.The first 12 months of the project have been devoted to the first theme. The challenge at this stage was to devise a theoretical, and consequently numerical, dynamical renormalisation approach to treat decoherence arising either from a bath of spins or of harmonic oscillators and allowing for arbitrary system-bath coupling and bath structure (e.g. interacting modes, arbitrary physical lattices). The starting point of the investigation was the surrogate Hamiltonian approach according to which at finite times the open system dynamics can be simulated using an effective generator including only a finite number of relevant environmental modes. As opposed to the Born-Markov approximation, this method leads to a controllable approximation. The first year of the project has been dedicated to the generalisation of the surrogate Hamiltonian method, in particular to the development of a general prescription for building up the effective generator. This part of the project required the combination of concepts from diverse areas of physics such as complex quantum system dynamics, graph theory, renormalisation group approaches and decoherence theory. The results of this work have been finalised into a paper (arXiv: 1111.4059v2) which is currently under review for Physical Review Letters. In the paper, we show that time naturally induces a renormalisation of the system-environment coupling, provided a renormalisation condition is obeyed. The renormalisation condition, of interactions being either local or defined on a finite continuum support, is in general fulfilled for both discrete and continuous environments. As a consequence, we could derive, surprisingly enough, a generalisation of the Lieb-Robinson bound to non-local interactions. The unified approach of our paper allows for a controllable approximation of arbitrary non-Markovian dynamics with an a priori estimate of the worst case computational cost for truncation based algorithms. These results have been presented by the young researcher in several talks, both invited and contributed, and posters, at conferences and in university seminars.The second year of the project started out with a focus on numerical simulations, instrumental to the goal of applying optimal control to open systems. As a first step, the young researcher carried out the numerical implementation of the spin bath dynamics. To this end, she has contributed the complete spin part to the Fortran90 group library qdyn. Subsequently, the young researcher has actively been involved in the code development generalising optimal control algorithms to open quantum systems. This extensive implementation period allowed the young researcher not only to greatly enhance her numerical skills but also to become more thoroughly acquainted with the application of optimal control to realistically modelled physical systems and to benefit from the group's leading expertise in this field. Based on the expertise gained at this stage, the young researcher could be assigned the lead role in two large scale numerical projects dedicated to applying optimal control to QIP with superconducting circuits. The goal of this work, which is being continued in form of a collaboration between the young researcher and the host, is to determine the best possible quantum state tomography for superconducting qudits and to develop and apply an optimisation algorithm targeting the complete set of perfect entanglers. Both projects are embedded in larger collaborations with other theoretical and experimental groups in Germany, Ireland, Israel and the United States (US).In parallel to the numerical work, it became clear that the theoretical framework for applying optimal control theory to QIPC tasks was yet incomplete. Specifically, it was not clear how to relate the fidelity measures for open quantum systems to an optimisation functional in an efficient, i.e. numerically feasible way. Therefore, the young researcher has worked out with one the group's PhD students (D. Reich) the definition of a proper optimal control functional for QIP tasks in open systems and clarified its relation to standard fidelity measures for quantum operations. It quickly turned out that our findings are relevant well beyond the initial optimal control focus, addressing the problem of efficient quantum device characterisation. This represents a highly important and timely issue for QIP, due to the dramatic experimental progress in realising the building blocks of quantum computers and the ensuing necessity to certify what has been implemented in the lab. The quest for an efficient scheme to perform this characterisation is extremely topical since methods based on quantum process tomography require exponential resources and are thus unfeasible. As a result of our investigation, we propose a maximally targeted approach, exploiting the facts that: (i) one wants to evaluate a fidelity measure, and not characterise the whole process; and (ii) the target is a unitary operation as opposed to a general dynamical map. The algebraic theory developed in the course of this work has allowed us to classify all currently discussed approaches to quantum process tomography and device characterisation within a common framework, making seemingly distinct approaches comparable. Combining the minimum requirements for estimating the gate fidelity identified by us with Monte Carlo sampling furthermore enables a classification in terms of experimental resources. These results have been finalised into one manuscript which has been submitted to Physical Review Letters and a second one which is currently in preparation.
