Periodic Reporting for period 1 - ResQu (Resource Sensitive Quantum Computing) Reporting period: 2017-09-01 to 2019-08-31 Summary of the context and overall objectives of the project The ResQu project has aimed to develop a resource-theoretic perspective on quantum information and computing.We are still far from being able to construct universal quantum machines that are large-scale enough to implement the most impressive applications that we know from theory they should eventually be capable of performing. A typical perspective is that we are unable to produce the basic quantum systems (qubits) in sufficient quantities, upon which a universal set of operations (quantum gates) can be performed with high-fidelity.This project has developed an alternative perspective, whereby fidelities and numbers of qubits are less relevant resources than the extent to which a variety of non-classical behaviours can be achieved, such as non-locality and contextuality, among others.The objectives of this project have been: (1) to develop the tools and formalisms that with which non-classical behaviours can be identified; (2) to understand how these behaviours can give rise to computational advantages over classical systems in a quantifiable way; (3) to propose new examples of quantum-over-classical computational advantage that is realisable within the scope of existing technologies. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The project has delivered significant progress on each of its objectives, and contributed to a growing body of research into resource theories for non-classicality and the roots of quantum-over-classical computational advantage.The project has notably led to the identification of a novel form of non-classical behaviour, known as sequential-transformation contextuality or dynamic contextuality. It is a form of non-classical behaviour that can be exhibited in to the evolution of a system under transformations (e.g. by quantum gates).Dynamic contextuality was found to enable quantum-over-classical advantages relating to the computation of certain privileged classes of boolean functions. A new form of cooperative game known as the CHSH* game was also proposed, in which dynamic contextuality gives rise to quantum-over classical advantage. These relationships were found to be quantifiable: the degree of advantage can be precisely related to the degree of dynamic contextuality. A mathematically robust formalism for treating and quantifying contextuality in continuous variable quantum systems, which provide some of the most promising candidates for implementing quantum computations and informatic protocols, was also developed. The project has likewise contributed methods for resource-theoretic reasoning about contextuality, as a non-classical behaviour that is known to have computational utility. In particular these allow for reasoning about how contextuality evolves when one performs operations on systems, and also for reasoning about which (contextual) behaviours can simulate others.Overall the project has led to publications in leading venues in both Computer Science and Physics, including LICS, Physical Review Letters, and Physical Review A. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The project has contributed to a foundational approach to quantum computing in which one aims to understand the sources of quantum computational power in order to harness and realise its full potential. To this end the project succeeded in identifying a novel non-classical behaviour of quantum systems that can act as a source of quantum-over-classical computational advantage, and proposed tasks in which this advantage can be realised and demonstrated. The extent to which this behaviour can be exploited in other tasks and applications of quantum computing remains to be further explored.The project has also delivered new methods for reasoning about how sources of quantum computational power and advantage (in particular contextual behaviours) can be manipulated via computational operations, and how quantum resources can be used to simulate others. It has additionally developed tools that will eventually allow these methods to be applied in continuous variable systems. Visualising classical and quantum strategies in the CHSH* game
